subroutine gamma_values ( n_data, x, fx ) !*****************************************************************************80 ! !! GAMMA_VALUES returns some values of the Gamma function. ! ! Discussion: ! ! The Gamma function is defined as: ! ! Gamma(Z) = integral ( 0 <= T < +oo) T^(Z-1) exp(-T) dT ! ! It satisfies the recursion: ! ! Gamma(X+1) = X * Gamma(X) ! ! Gamma is undefined for nonpositive integral X. ! Gamma(0.5) = sqrt(PI) ! For N a positive integer, Gamma(N+1) = N!, the standard factorial. ! ! In Mathematica, the function can be evaluated by: ! ! Gamma[x] ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 May 2007 ! ! Author: ! ! John Burkardt ! ! Source: ! ! https://people.sc.fsu.edu/~jburkardt/f_src/r8lib/r8lib.html ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! National Bureau of Standards, 1964, ! ISBN: 0-486-61272-4, ! LC: QA47.A34. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Cambridge University Press, 1999, ! ISBN: 0-521-64314-7, ! LC: QA76.95.W65. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 ! before the first call. On each call, the routine increments N_DATA by 1, ! and returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 25 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & -0.3544907701811032D+01, & -0.1005871979644108D+03, & 0.9943258511915060D+02, & 0.9513507698668732D+01, & 0.4590843711998803D+01, & 0.2218159543757688D+01, & 0.1772453850905516D+01, & 0.1489192248812817D+01, & 0.1164229713725303D+01, & 0.1000000000000000D+01, & 0.9513507698668732D+00, & 0.9181687423997606D+00, & 0.8974706963062772D+00, & 0.8872638175030753D+00, & 0.8862269254527580D+00, & 0.8935153492876903D+00, & 0.9086387328532904D+00, & 0.9313837709802427D+00, & 0.9617658319073874D+00, & 0.1000000000000000D+01, & 0.2000000000000000D+01, & 0.6000000000000000D+01, & 0.3628800000000000D+06, & 0.1216451004088320D+18, & 0.8841761993739702D+31 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & -0.50D+00, & -0.01D+00, & 0.01D+00, & 0.10D+00, & 0.20D+00, & 0.40D+00, & 0.50D+00, & 0.60D+00, & 0.80D+00, & 1.00D+00, & 1.10D+00, & 1.20D+00, & 1.30D+00, & 1.40D+00, & 1.50D+00, & 1.60D+00, & 1.70D+00, & 1.80D+00, & 1.90D+00, & 2.00D+00, & 3.00D+00, & 4.00D+00, & 10.00D+00, & 20.00D+00, & 30.00D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end subroutine gamma_log_values ( n_data, x, fx ) !*****************************************************************************80 ! !! GAMMA_LOG_VALUES returns some values of the Log Gamma function. ! ! Discussion: ! ! In Mathematica, the function can be evaluated by: ! ! Log[Gamma[x]] ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! National Bureau of Standards, 1964, ! ISBN: 0-486-61272-4, ! LC: QA47.A34. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Cambridge University Press, 1999, ! ISBN: 0-521-64314-7, ! LC: QA76.95.W65. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 ! before the first call. On each call, the routine increments N_DATA by 1, ! and returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) FX, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 20 real ( kind = 8 ) fx real ( kind = 8 ), save, dimension ( n_max ) :: fx_vec = (/ & 0.1524063822430784D+01, & 0.7966778177017837D+00, & 0.3982338580692348D+00, & 0.1520596783998375D+00, & 0.0000000000000000D+00, & -0.4987244125983972D-01, & -0.8537409000331584D-01, & -0.1081748095078604D+00, & -0.1196129141723712D+00, & -0.1207822376352452D+00, & -0.1125917656967557D+00, & -0.9580769740706586D-01, & -0.7108387291437216D-01, & -0.3898427592308333D-01, & 0.00000000000000000D+00, & 0.69314718055994530D+00, & 0.17917594692280550D+01, & 0.12801827480081469D+02, & 0.39339884187199494D+02, & 0.71257038967168009D+02 /) integer ( kind = 4 ) n_data real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 0.20D+00, & 0.40D+00, & 0.60D+00, & 0.80D+00, & 1.00D+00, & 1.10D+00, & 1.20D+00, & 1.30D+00, & 1.40D+00, & 1.50D+00, & 1.60D+00, & 1.70D+00, & 1.80D+00, & 1.90D+00, & 2.00D+00, & 3.00D+00, & 4.00D+00, & 10.00D+00, & 20.00D+00, & 30.00D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 fx = 0.0D+00 else x = x_vec(n_data) fx = fx_vec(n_data) end if return end subroutine get_unit ( iunit ) !*****************************************************************************80 ! !! GET_UNIT returns a free FORTRAN unit number. ! ! Discussion: ! ! A "free" FORTRAN unit number is a value between 1 and 99 which ! is not currently associated with an I/O device. A free FORTRAN unit ! number is needed in order to open a file with the OPEN command. ! ! If IUNIT = 0, then no free FORTRAN unit could be found, although ! all 99 units were checked (except for units 5, 6 and 9, which ! are commonly reserved for console I/O). ! ! Otherwise, IUNIT is a value between 1 and 99, representing a ! free FORTRAN unit. Note that GET_UNIT assumes that units 5 and 6 ! are special, and will never return those values. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 October 2008 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Output, integer ( kind = 4 ) IUNIT, the free unit number. ! implicit none integer ( kind = 4 ) i integer ( kind = 4 ) ios integer ( kind = 4 ) iunit logical ( kind = 4 ) lopen iunit = 0 do i = 1, 99 if ( i /= 5 .and. i /= 6 .and. i /= 9 ) then inquire ( unit = i, opened = lopen, iostat = ios ) if ( ios == 0 ) then if ( .not. lopen ) then iunit = i return end if end if end if end do return end function i4_log_10 ( i ) !*****************************************************************************80 ! !! I4_LOG_10 returns the integer part of the logarithm base 10 of an I4. ! ! Discussion: ! ! I4_LOG_10 ( I ) + 1 is the number of decimal digits in I. ! ! An I4 is an integer ( kind = 4 ) value. ! ! Example: ! ! I I4_LOG_10 ! ----- -------- ! 0 0 ! 1 0 ! 2 0 ! 9 0 ! 10 1 ! 11 1 ! 99 1 ! 100 2 ! 101 2 ! 999 2 ! 1000 3 ! 1001 3 ! 9999 3 ! 10000 4 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 June 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) I, the number whose logarithm base 10 ! is desired. ! ! Output, integer ( kind = 4 ) I4_LOG_10, the integer part of the ! logarithm base 10 of the absolute value of X. ! implicit none integer ( kind = 4 ) i integer ( kind = 4 ) i_abs integer ( kind = 4 ) i4_log_10 integer ( kind = 4 ) ten_pow if ( i == 0 ) then i4_log_10 = 0 else i4_log_10 = 0 ten_pow = 10 i_abs = abs ( i ) do while ( ten_pow <= i_abs ) i4_log_10 = i4_log_10 + 1 ten_pow = ten_pow * 10 end do end if return end function i4_modp ( i, j ) !*****************************************************************************80 ! !! I4_MODP returns the nonnegative remainder of I4 division. ! ! Discussion: ! ! If ! NREM = I4_MODP ( I, J ) ! NMULT = ( I - NREM ) / J ! then ! I = J * NMULT + NREM ! where NREM is always nonnegative. ! ! The MOD function computes a result with the same sign as the ! quantity being divided. Thus, suppose you had an angle A, ! and you wanted to ensure that it was between 0 and 360. ! Then mod(A,360) would do, if A was positive, but if A ! was negative, your result would be between -360 and 0. ! ! On the other hand, I4_MODP(A,360) is between 0 and 360, always. ! ! An I4 is an integer ( kind = 4 ) value. ! ! Example: ! ! I J MOD I4_MODP Factorization ! ! 107 50 7 7 107 = 2 * 50 + 7 ! 107 -50 7 7 107 = -2 * -50 + 7 ! -107 50 -7 43 -107 = -3 * 50 + 43 ! -107 -50 -7 43 -107 = 3 * -50 + 43 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 March 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) I, the number to be divided. ! ! Input, integer ( kind = 4 ) J, the number that divides I. ! ! Output, integer ( kind = 4 ) I4_MODP, the nonnegative remainder when I is ! divided by J. ! implicit none integer ( kind = 4 ) i integer ( kind = 4 ) i4_modp integer ( kind = 4 ) j integer ( kind = 4 ) value if ( j == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'I4_MODP - Fatal error!' write ( *, '(a,i8)' ) ' Illegal divisor J = ', j stop 1 end if value = mod ( i, j ) if ( value < 0 ) then value = value + abs ( j ) end if i4_modp = value return end function i4_uniform_ab ( a, b, seed ) !*****************************************************************************80 ! !! I4_UNIFORM_AB returns a scaled pseudorandom I4. ! ! Discussion: ! ! An I4 is an integer ( kind = 4 ) value. ! ! The pseudorandom number will be scaled to be uniformly distributed ! between A and B. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 November 2006 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Paul Bratley, Bennett Fox, Linus Schrage, ! A Guide to Simulation, ! Springer Verlag, pages 201-202, 1983. ! ! Pierre L'Ecuyer, ! Random Number Generation, ! in Handbook of Simulation, ! edited by Jerry Banks, ! Wiley Interscience, page 95, 1998. ! ! Bennett Fox, ! Algorithm 647: ! Implementation and Relative Efficiency of Quasirandom ! Sequence Generators, ! ACM Transactions on Mathematical Software, ! Volume 12, Number 4, pages 362-376, 1986. ! ! Peter Lewis, Allen Goodman, James Miller ! A Pseudo-Random Number Generator for the System/360, ! IBM Systems Journal, ! Volume 8, pages 136-143, 1969. ! ! Parameters: ! ! Input, integer ( kind = 4 ) A, B, the limits of the interval. ! ! Input/output, integer ( kind = 4 ) SEED, the "seed" value, which ! should NOT be 0. On output, SEED has been updated. ! ! Output, integer ( kind = 4 ) I4_UNIFORM_AB, a number between A and B. ! implicit none integer ( kind = 4 ) a integer ( kind = 4 ) b integer ( kind = 4 ), parameter :: i4_huge = 2147483647 integer ( kind = 4 ) i4_uniform_ab integer ( kind = 4 ) k real ( kind = 4 ) r integer ( kind = 4 ) seed integer ( kind = 4 ) value if ( seed == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'I4_UNIFORM_AB - Fatal error!' write ( *, '(a)' ) ' Input value of SEED = 0.' stop 1 end if k = seed / 127773 seed = 16807 * ( seed - k * 127773 ) - k * 2836 if ( seed < 0 ) then seed = seed + i4_huge end if r = real ( seed, kind = 4 ) * 4.656612875E-10 ! ! Scale R to lie between A-0.5 and B+0.5. ! r = ( 1.0E+00 - r ) * ( real ( min ( a, b ), kind = 4 ) - 0.5E+00 ) & + r * ( real ( max ( a, b ), kind = 4 ) + 0.5E+00 ) ! ! Use rounding to convert R to an integer between A and B. ! value = nint ( r, kind = 4 ) value = max ( value, min ( a, b ) ) value = min ( value, max ( a, b ) ) i4_uniform_ab = value return end function i4_wrap ( ival, ilo, ihi ) !*****************************************************************************80 ! !! I4_WRAP forces an I4 to lie between given limits by wrapping. ! ! Discussion: ! ! An I4 is an integer ( kind = 4 ) value. ! ! There appears to be a bug in the GFORTRAN compiler which can lead to ! erroneous results when the first argument of I4_WRAP is an expression. ! In particular: ! ! do i = 1, 3 ! if ( test ) then ! i4 = i4_wrap ( i + 1, 1, 3 ) ! end if ! end do ! ! was, when I = 3, returning I4 = 3. So I had to replace this with ! ! do i = 1, 3 ! if ( test ) then ! i4 = i + 1 ! i4 = i4_wrap ( i4, 1, 3 ) ! end if ! end do ! ! Example: ! ! ILO = 4, IHI = 8 ! ! I Value ! ! -2 8 ! -1 4 ! 0 5 ! 1 6 ! 2 7 ! 3 8 ! 4 4 ! 5 5 ! 6 6 ! 7 7 ! 8 8 ! 9 4 ! 10 5 ! 11 6 ! 12 7 ! 13 8 ! 14 4 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 September 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) IVAL, a value. ! ! Input, integer ( kind = 4 ) ILO, IHI, the desired bounds. ! ! Output, integer ( kind = 4 ) I4_WRAP, a "wrapped" version of the value. ! implicit none integer ( kind = 4 ) i4_modp integer ( kind = 4 ) i4_wrap integer ( kind = 4 ) ihi integer ( kind = 4 ) ilo integer ( kind = 4 ) ival integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo integer ( kind = 4 ) value integer ( kind = 4 ) wide jlo = min ( ilo, ihi ) jhi = max ( ilo, ihi ) wide = jhi - jlo + 1 if ( wide == 1 ) then value = jlo else value = jlo + i4_modp ( ival - jlo, wide ) end if i4_wrap = value return end subroutine i4int_to_r8int ( imin, imax, i, rmin, rmax, r ) !*****************************************************************************80 ! !! I4INT_TO_R8INT maps an I4INT to an R8INT. ! ! Discussion: ! ! The formula used is: ! ! R := RMIN + ( RMAX - RMIN ) * ( I - IMIN ) / ( IMAX - IMIN ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 January 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) IMIN, IMAX, the range. ! ! Input, integer ( kind = 4 ) I, the integer to be converted. ! ! Input, real ( kind = 8 ) RMIN, RMAX, the range. ! ! Output, real ( kind = 8 ) R, the corresponding value in [RMIN,RMAX]. ! implicit none integer ( kind = 4 ) i integer ( kind = 4 ) imax integer ( kind = 4 ) imin real ( kind = 8 ) r real ( kind = 8 ) rmax real ( kind = 8 ) rmin if ( imax == imin ) then r = 0.5D+00 * ( rmin + rmax ) else r = ( real ( imax - i, kind = 8 ) * rmin & + real ( i - imin, kind = 8 ) * rmax ) & / real ( imax - imin, kind = 8 ) end if return end subroutine i4mat_print ( m, n, a, title ) !*****************************************************************************80 ! !! I4MAT_PRINT prints an I4MAT. ! ! Discussion: ! ! An I4MAT is a rectangular array of I4's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 June 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, integer ( kind = 4 ) A(M,N), the matrix to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) a(m,n) integer ( kind = 4 ) ihi integer ( kind = 4 ) ilo integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo character ( len = * ) title ilo = 1 ihi = m jlo = 1 jhi = n call i4mat_print_some ( m, n, a, ilo, jlo, ihi, jhi, title ) return end subroutine i4mat_print_some ( m, n, a, ilo, jlo, ihi, jhi, title ) !*****************************************************************************80 ! !! I4MAT_PRINT_SOME prints some of an I4MAT. ! ! Discussion: ! ! An I4MAT is a rectangular array of I4's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 September 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, integer ( kind = 4 ) A(M,N), an M by N matrix to be printed. ! ! Input, integer ( kind = 4 ) ILO, JLO, the first row and column to print. ! ! Input, integer ( kind = 4 ) IHI, JHI, the last row and column to print. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ), parameter :: incx = 10 integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) a(m,n) character ( len = 8 ) ctemp(incx) integer ( kind = 4 ) i integer ( kind = 4 ) i2hi integer ( kind = 4 ) i2lo integer ( kind = 4 ) ihi integer ( kind = 4 ) ilo integer ( kind = 4 ) inc integer ( kind = 4 ) j integer ( kind = 4 ) j2 integer ( kind = 4 ) j2hi integer ( kind = 4 ) j2lo integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo character ( len = * ) title if ( 0 < len_trim ( title ) ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) end if if ( m <= 0 .or. n <= 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) ' (None)' return end if do j2lo = max ( jlo, 1 ), min ( jhi, n ), incx j2hi = j2lo + incx - 1 j2hi = min ( j2hi, n ) j2hi = min ( j2hi, jhi ) inc = j2hi + 1 - j2lo write ( *, '(a)' ) ' ' do j = j2lo, j2hi j2 = j + 1 - j2lo write ( ctemp(j2), '(i8)' ) j end do write ( *, '('' Col '',10a8)' ) ctemp(1:inc) write ( *, '(a)' ) ' Row' write ( *, '(a)' ) ' ' i2lo = max ( ilo, 1 ) i2hi = min ( ihi, m ) do i = i2lo, i2hi do j2 = 1, inc j = j2lo - 1 + j2 write ( ctemp(j2), '(i8)' ) a(i,j) end do write ( *, '(i5,a,10a8)' ) i, ':', ( ctemp(j), j = 1, inc ) end do end do return end subroutine i4vec_indicator0 ( n, a ) !*****************************************************************************80 ! !! I4VEC_INDICATOR0 sets an I4VEC to the indicator vector (0,1,2,...) ! ! Discussion: ! ! An I4VEC is a vector of I4's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 September 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Output, integer ( kind = 4 ) A(N), the array. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) a(n) integer ( kind = 4 ) i do i = 1, n a(i) = i - 1 end do return end subroutine i4vec_indicator1 ( n, a ) !*****************************************************************************80 ! !! I4VEC_INDICATOR1 sets an I4VEC to the indicator vector (1,2,3,...) ! ! Discussion: ! ! An I4VEC is a vector of I4's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 September 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Output, integer ( kind = 4 ) A(N), the array. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) a(n) integer ( kind = 4 ) i do i = 1, n a(i) = i end do return end subroutine i4vec_permute ( n, p, a ) !*****************************************************************************80 ! !! I4VEC_PERMUTE permutes an I4VEC in place. ! ! Discussion: ! ! An I4VEC is a vector of I4's. ! ! This routine permutes an array of integer "objects", but the same ! logic can be used to permute an array of objects of any arithmetic ! type, or an array of objects of any complexity. The only temporary ! storage required is enough to store a single object. The number ! of data movements made is N + the number of cycles of order 2 or more, ! which is never more than N + N/2. ! ! Example: ! ! Input: ! ! N = 5 ! P = ( 2, 4, 5, 1, 3 ) ! A = ( 1, 2, 3, 4, 5 ) ! ! Output: ! ! A = ( 2, 4, 5, 1, 3 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 July 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of objects. ! ! Input, integer ( kind = 4 ) P(N), the permutation. P(I) = J means ! that the I-th element of the output array should be the J-th ! element of the input array. ! ! Input/output, integer ( kind = 4 ) A(N), the array to be permuted. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) a(n) integer ( kind = 4 ) a_temp integer ( kind = 4 ) ierror integer ( kind = 4 ) iget integer ( kind = 4 ) iput integer ( kind = 4 ) istart integer ( kind = 4 ) p(n) integer ( kind = 4 ) perm1_check ierror = perm1_check ( n, p ) if ( ierror .ne. 0 ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'I4VEC_PERMUTE - Fatal error!' write ( *, '(a)' ) ' PERM1_CHECK returned error.' stop 1 end if ! ! Search for the next element of the permutation that has not been used. ! do istart = 1, n if ( p(istart) < 0 ) then cycle else if ( p(istart) == istart ) then p(istart) = - p(istart) cycle else a_temp = a(istart) iget = istart ! ! Copy the new value into the vacated entry. ! do iput = iget iget = p(iget) p(iput) = - p(iput) if ( iget < 1 .or. n < iget ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'I4VEC_PERMUTE - Fatal error!' write ( *, '(a)' ) ' A permutation index is out of range.' write ( *, '(a,i8,a,i8)' ) ' P(', iput, ') = ', iget stop 1 end if if ( iget == istart ) then a(iput) = a_temp exit end if a(iput) = a(iget) end do end if end do ! ! Restore the signs of the entries. ! p(1:n) = - p(1:n) return end subroutine i4vec_print ( n, a, title ) !*****************************************************************************80 ! !! I4VEC_PRINT prints an I4VEC. ! ! Discussion: ! ! An I4VEC is a vector of I4's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 May 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of components of the vector. ! ! Input, integer ( kind = 4 ) A(N), the vector to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) a(n) integer ( kind = 4 ) i character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' do i = 1, n write ( *, '(2x,i8,a,2x,i12)' ) i, ':', a(i) end do return end subroutine i4vec_transpose_print ( n, a, title ) !*****************************************************************************80 ! !! I4VEC_TRANSPOSE_PRINT prints an I4VEC "transposed". ! ! Discussion: ! ! An I4VEC is a vector of I4's. ! ! Example: ! ! A = (/ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 /) ! TITLE = 'My vector: ' ! ! My vector: ! 1 2 3 4 5 ! 6 7 8 9 10 ! 11 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 April 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of components of the vector. ! ! Input, integer ( kind = 4 ) A(N), the vector to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) a(n) integer ( kind = 4 ) ihi integer ( kind = 4 ) ilo character ( len = * ) title if ( 0 < len_trim ( title ) ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) end if do ilo = 1, n, 5 ihi = min ( ilo + 5 - 1, n ) write ( *, '(5i12)' ) a(ilo:ihi) end do return end subroutine legendre_zeros ( n, x ) !*****************************************************************************80 ! !! LEGENDRE_ZEROS computes the zeros of the Legendre polynomial of degree N. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 June 2011 ! ! Author: ! ! Original FORTRAN77 version by Philip Davis, Philip Rabinowitz. ! FORTRAN90 version by John Burkardt. ! ! Reference: ! ! Philip Davis, Philip Rabinowitz, ! Methods of Numerical Integration, ! Second Edition, ! Dover, 2007, ! ISBN: 0486453391, ! LC: QA299.3.D28. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order. ! 0 < N. ! ! Output, real ( kind = 8 ) X(N), the locations of the zeros. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) d1 real ( kind = 8 ) d2pn real ( kind = 8 ) d3pn real ( kind = 8 ) d4pn real ( kind = 8 ) dp real ( kind = 8 ) dpn real ( kind = 8 ) e1 real ( kind = 8 ) fx real ( kind = 8 ) h integer ( kind = 4 ) i integer ( kind = 4 ) iback integer ( kind = 4 ) k integer ( kind = 4 ) m integer ( kind = 4 ) mp1mi integer ( kind = 4 ) ncopy integer ( kind = 4 ) nmove real ( kind = 8 ) p real ( kind = 8 ) pk real ( kind = 8 ) pkm1 real ( kind = 8 ) pkp1 real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) t real ( kind = 8 ) u real ( kind = 8 ) v real ( kind = 8 ) x(n) real ( kind = 8 ) x0 real ( kind = 8 ) xtemp e1 = real ( n * ( n + 1 ), kind = 8 ) m = ( n + 1 ) / 2 do i = 1, m mp1mi = m + 1 - i t = real ( 4 * i - 1, kind = 8 ) * r8_pi & / real ( 4 * n + 2, kind = 8 ) x0 = cos ( t ) * ( 1.0D+00 - ( 1.0D+00 - 1.0D+00 & / real ( n, kind = 8 ) ) & / real ( 8 * n * n, kind = 8 ) ) pkm1 = 1.0D+00 pk = x0 do k = 2, n pkp1 = 2.0D+00 * x0 * pk - pkm1 - ( x0 * pk - pkm1 ) & / real ( k, kind = 8 ) pkm1 = pk pk = pkp1 end do d1 = real ( n, kind = 8 ) * ( pkm1 - x0 * pk ) dpn = d1 / ( 1.0D+00 - x0 ) / ( 1.0D+00 + x0 ) d2pn = ( 2.0D+00 * x0 * dpn - e1 * pk ) / ( 1.0D+00 - x0 ) & / ( 1.0D+00 + x0 ) d3pn = ( 4.0D+00 * x0 * d2pn + ( 2.0D+00 - e1 ) * dpn ) & / ( 1.0D+00 - x0 ) / ( 1.0D+00 + x0 ) d4pn = ( 6.0D+00 * x0 * d3pn + ( 6.0D+00 - e1 ) * d2pn ) & / ( 1.0D+00 - x0 ) / ( 1.0D+00 + x0 ) u = pk / dpn v = d2pn / dpn ! ! Initial approximation H: ! h = - u * ( 1.0D+00 + 0.5D+00 * u * ( v + u * ( v * v - d3pn / & ( 3.0D+00 * dpn ) ) ) ) ! ! Refine H using one step of Newton's method: ! p = pk + h * ( dpn + 0.5D+00 * h * ( d2pn + h / 3.0D+00 & * ( d3pn + 0.25D+00 * h * d4pn ) ) ) dp = dpn + h * ( d2pn + 0.5D+00 * h * ( d3pn + h * d4pn / 3.0D+00 ) ) h = h - p / dp xtemp = x0 + h x(mp1mi) = xtemp fx = d1 - h * e1 * ( pk + 0.5D+00 * h * ( dpn + h / 3.0D+00 & * ( d2pn + 0.25D+00 * h * ( d3pn + 0.2D+00 * h * d4pn ) ) ) ) end do if ( mod ( n, 2 ) == 1 ) then x(1) = 0.0D+00 end if ! ! Shift the data up. ! nmove = ( n + 1 ) / 2 ncopy = n - nmove do i = 1, nmove iback = n + 1 - i x(iback) = x(iback-ncopy) end do ! ! Reflect values for the negative abscissas. ! do i = 1, n - nmove x(i) = - x(n+1-i) end do return end function perm0_check ( n, p ) !*****************************************************************************80 ! !! PERM0_CHECK checks a 0-based permutation. ! ! Discussion: ! ! The routine verifies that each of the integers from 0 to ! to N-1 occurs among the N entries of the permutation. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 May 2015 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries. ! ! Input, integer ( kind = 4 ) P(N), the array to check. ! ! Output, integer ( kind = 4 ) PERM0_CHECK: ! 0, no error. ! 1, P is not a 0-based permutation. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) ierror integer ( kind = 4 ) location integer ( kind = 4 ) p(n) integer ( kind = 4 ) perm0_check integer ( kind = 4 ) value ierror = 0 do value = 0, n - 1 ierror = 1 do location = 1, n if ( p(location) == value ) then ierror = 0 exit end if end do if ( ierror /= 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'PERM0_CHECK - Fatal error!' write ( *, '(a,i4)' ) ' Permutation is missing value ', value exit end if end do perm0_check = ierror return end subroutine perm0_uniform ( n, seed, p ) !*****************************************************************************80 ! !! PERM0_UNIFORM selects a random permutation of objects 0, ..., N-1. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 May 2015 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of objects to be permuted. ! ! Input/output, integer ( kind = 4 ) SEED, a seed for the random ! number generator. ! ! Output, integer ( kind = 4 ) P(N), the permutation. P(I) is the "new" ! location of the object originally at I. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) i4_uniform_ab integer ( kind = 4 ) j integer ( kind = 4 ) k integer ( kind = 4 ) p(n) integer ( kind = 4 ) seed do i = 1, n p(i) = i - 1 end do do i = 1, n - 1 j = i4_uniform_ab ( i, n, seed ) k = p(i) p(i) = p(j) p(j) = k end do return end function perm1_check ( n, p ) !*****************************************************************************80 ! !! PERM1_CHECK checks a 1-based permutation. ! ! Discussion: ! ! The routine verifies that each of the integers from 1 to ! to N occurs among the N entries of the permutation. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 May 2015 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries. ! ! Input, integer ( kind = 4 ) P(N), the array to check. ! ! Output, integer ( kind = 4 ) PERM1_CHECK: ! 0, P is a 1-based permutation. ! 1, P is not a 1-based permutation. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) ierror integer ( kind = 4 ) location integer ( kind = 4 ) p(n) integer ( kind = 4 ) perm1_check integer ( kind = 4 ) value ierror = 0 do value = 1, n ierror = 1 do location = 1, n if ( p(location) == value ) then ierror = 0 exit end if end do if ( ierror /= 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'PERM1_CHECK - Fatal error!' write ( *, '(a,i4)' ) ' Permutation is missing value ', value exit end if end do perm1_check = ierror return end subroutine perm1_uniform ( n, seed, p ) !*****************************************************************************80 ! !! PERM1_UNIFORM selects a random permutation of objects 1, ..., N. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 November 2008 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of objects to be permuted. ! ! Input/output, integer ( kind = 4 ) SEED, a seed for the random ! number generator. ! ! Output, integer ( kind = 4 ) P(N), the permutation. P(I) is the "new" ! location of the object originally at I. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) i4_uniform_ab integer ( kind = 4 ) j integer ( kind = 4 ) k integer ( kind = 4 ) p(n) integer ( kind = 4 ) seed do i = 1, n p(i) = i end do do i = 1, n - 1 j = i4_uniform_ab ( i, n, seed ) k = p(i) p(i) = p(j) p(j) = k end do return end function r8_abs ( x ) !*****************************************************************************80 ! !! R8_ABS returns the absolute value of an R8. ! ! Discussion: ! ! An R8 is a real ( kind = 8 ) value. ! ! FORTRAN90 supplies the ABS function, which should be used instead ! of this function! ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 September 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose absolute value is desired. ! ! Output, real ( kind = 8 ) R8_ABS, the absolute value of X. ! implicit none real ( kind = 8 ) r8_abs real ( kind = 8 ) x if ( 0.0D+00 <= x ) then r8_abs = + x else r8_abs = - x end if return end function r8_acos ( c ) !*****************************************************************************80 ! !! R8_ACOS computes the arc cosine function, with argument truncation. ! ! Discussion: ! ! If you call your system ACOS routine with an input argument that is ! even slightly outside the range [-1.0, 1.0 ], you may get an unpleasant ! surprise (I did). ! ! This routine simply truncates arguments outside the range. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 October 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) C, the argument. ! ! Output, real ( kind = 8 ) R8_ACOS, an angle whose cosine is C. ! implicit none real ( kind = 8 ) c real ( kind = 8 ) c2 real ( kind = 8 ) r8_acos c2 = c c2 = max ( c2, -1.0D+00 ) c2 = min ( c2, +1.0D+00 ) r8_acos = acos ( c2 ) return end function r8_acosh ( x ) !*****************************************************************************80 ! !! R8_ACOSH returns the inverse hyperbolic cosine of a number. ! ! Discussion: ! ! One formula is: ! ! R8_ACOSH = log ( X + SQRT ( X^2 - 1.0 ) ) ! ! but this formula suffers from roundoff and overflow problems. ! The formula used here was recommended by W Kahan, as discussed ! by Moler. ! ! Applying the inverse function ! ! Y = R8_ACOSH ( X ) ! ! implies that ! ! X = COSH(Y) = 0.5 * ( EXP(Y) + EXP(-Y) ). ! ! For every X greater than or equal to 1, there are two possible ! choices Y such that X = COSH(Y), differing only in sign. It ! is usual to resolve this choice by taking the value of ! R8_ACOSH ( X ) to be nonnegative. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 November 2007 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Cleve Moler, ! Trigonometry is a Complex Subject, ! MATLAB News and Notes, ! Summer 1998. ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose inverse hyperbolic ! cosine is desired. X should be greater than or equal to 1. ! ! Output, real ( kind = 8 ) R8_ACOSH, the inverse hyperbolic cosine of ! X. The principal value (that is, the positive value of the two ) ! is returned. ! implicit none real ( kind = 8 ) r8_acosh real ( kind = 8 ) x if ( x < 1.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_ACOSH - Fatal error!' write ( *, '(a)' ) ' Argument X must satisfy 1 <= X.' write ( *, '(a,g14.6)' ) ' The input X = ', x stop 1 end if r8_acosh = 2.0D+00 * log ( & sqrt ( 0.5D+00 * ( x + 1.0D+00 ) ) & + sqrt ( 0.5D+00 * ( x - 1.0D+00 ) ) ) return end function r8_add ( x, y ) !*****************************************************************************80 ! !! R8_ADD returns the sum of two R8's. ! ! Discussion: ! ! An R8 is a real ( kind = 8 ) value. ! ! FORTRAN90 supplies the + operator, which should generally be used instead ! of this function! ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 August 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, Y, the numbers to be added. ! ! Output, real ( kind = 8 ) R8_ADD, the sum. ! implicit none real ( kind = 8 ) r8_add real ( kind = 8 ) x real ( kind = 8 ) y r8_add = x + y return end function r8_agm ( a, b ) !*****************************************************************************80 ! !! R8_AGM computes the arithmetic-geometric mean of A and B. ! ! Discussion: ! ! The AGM is defined for nonnegative A and B. ! ! The AGM of numbers A and B is defined by by an iteration: ! ! A(0) = A ! B(0) = B ! ! A(N+1) = ( A(N) + B(N) ) / 2 ! B(N+1) = sqrt ( A(N) * B(N) ) ! ! The two sequences both converge to AGM(A,B). Convergence can be ! assumed when the two values are sufficiently close. ! ! In Mathematica, the AGM can be evaluated by ! ! ArithmeticGeometricMean [ a, b ] ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 February 2008 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Cambridge University Press, 1999, ! ISBN: 0-521-64314-7, ! LC: QA76.95.W65. ! ! Parameters: ! ! Input, real ( kind = 8 ) A, B, the arguments whose AGM is to be computed. ! 0 <= A, 0 <= B. ! ! Output, real ( kind = 8 ) R8_AGM, the arithmetic-geometric mean of A and B. ! implicit none real ( kind = 8 ) a real ( kind = 8 ) a1 real ( kind = 8 ) a2 real ( kind = 8 ) b real ( kind = 8 ) b1 real ( kind = 8 ) b2 integer ( kind = 4 ) it integer ( kind = 4 ), parameter :: it_max = 1000 real ( kind = 8 ) r8_agm real ( kind = 8 ) tol if ( a < 0.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_AGM - Fatal error!' write ( *, '(a)' ) ' A < 0.' stop 1 end if if ( b < 0.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_AGM - Fatal error!' write ( *, '(a)' ) ' B < 0.' stop 1 end if if ( a == 0.0D+00 .or. b == 0.0D+00 ) then r8_agm = 0.0D+00 return end if if ( a == b ) then r8_agm = a return end if it = 0 tol = 100.0D+00 * epsilon ( tol ) a1 = a b1 = b do it = it + 1 a2 = ( a1 + b1 ) / 2.0D+00 b2 = sqrt ( a1 * b1 ) if ( abs ( a2 - b2 ) <= tol * ( a2 + b2 ) ) then exit end if if ( it_max < it ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_AGM - Fatal error!' write ( *, '(a,i8)' ) ' Exceeded iteration limit ', it_max write ( *, '(a,g14.6)' ) ' Estimated value = ', a2 stop 1 end if a1 = a2 b1 = b2 end do r8_agm = a2 return end function r8_aint ( x ) !****************************************************************************80 ! !! R8_AINT truncates an R8 argument to an integer. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 October 2011 ! ! Author: ! ! John Burkardt. ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument. ! ! Output, real ( kind = 8 ) R8_AINT, the truncated version of X. ! implicit none real ( kind = 8 ) r8_aint real ( kind = 8 ) value real ( kind = 8 ) x if ( x < 0.0D+00 ) then value = - int ( abs ( x ) ) else value = int ( abs ( x ) ) end if r8_aint = value return end function r8_asin ( s ) !*****************************************************************************80 ! !! R8_ASIN computes the arc sine function, with argument truncation. ! ! Discussion: ! ! If you call your system ASIN routine with an input argument that is ! even slightly outside the range [-1.0, 1.0 ], you may get an unpleasant ! surprise (I did). ! ! This routine simply truncates arguments outside the range. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 28 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) S, the argument. ! ! Output, real ( kind = 8 ) R8_ASIN, an angle whose sine is S. ! implicit none real ( kind = 8 ) r8_asin real ( kind = 8 ) s real ( kind = 8 ) s2 real ( kind = 8 ) value s2 = s s2 = max ( s2, -1.0D+00 ) s2 = min ( s2, +1.0D+00 ) value = asin ( s2 ) r8_asin = value return end function r8_asinh ( x ) !*****************************************************************************80 ! !! R8_ASINH returns the inverse hyperbolic sine of a number. ! ! Discussion: ! ! The assertion that: ! ! Y = R8_ASINH ( X ) ! ! implies that ! ! X = SINH(Y) = 0.5 * ( EXP(Y) - EXP(-Y) ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 November 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose inverse hyperbolic ! sine is desired. ! ! Output, real ( kind = 8 ) R8_ASINH, the inverse hyperbolic sine of X. ! implicit none real ( kind = 8 ) r8_asinh real ( kind = 8 ) value real ( kind = 8 ) x value = log ( x + sqrt ( x * x + 1.0D+00 ) ) r8_asinh = value return end function r8_atan ( y, x ) !*****************************************************************************80 ! !! R8_ATAN computes the inverse tangent of the ratio Y / X. ! ! Discussion: ! ! R8_ATAN returns an angle whose tangent is ( Y / X ), a job which ! the built in functions ATAN and ATAN2 already do. ! ! However: ! ! * R8_ATAN always returns a positive angle, between 0 and 2 PI, ! while ATAN and ATAN2 return angles in the interval [-PI/2,+PI/2] ! and [-PI,+PI] respectively; ! ! * R8_ATAN accounts for the signs of X and Y, (as does ATAN2). The ATAN ! function by contrast always returns an angle in the first or fourth ! quadrants. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 April 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) Y, X, two quantities which represent the ! tangent of an angle. If Y is not zero, then the tangent is (Y/X). ! ! Output, real ( kind = 8 ) R8_ATAN, an angle between 0 and 2 * PI, whose ! tangent is (Y/X), and which lies in the appropriate quadrant so that ! the signs of its cosine and sine match those of X and Y. ! implicit none real ( kind = 8 ) abs_x real ( kind = 8 ) abs_y real ( kind = 8 ) r8_atan real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) theta_0 real ( kind = 8 ) value real ( kind = 8 ) x real ( kind = 8 ) y ! ! Special cases: ! if ( x == 0.0D+00 ) then if ( 0.0D+00 < y ) then value = r8_pi / 2.0D+00 else if ( y < 0.0D+00 ) then value = 3.0D+00 * r8_pi / 2.0D+00 else if ( y == 0.0D+00 ) then value = 0.0D+00 end if else if ( y == 0.0D+00 ) then if ( 0.0D+00 < x ) then value = 0.0D+00 else if ( x < 0.0D+00 ) then value = r8_pi end if ! ! We assume that ATAN2 is correct when both arguments are positive. ! else abs_y = abs ( y ) abs_x = abs ( x ) theta_0 = atan2 ( abs_y, abs_x ) if ( 0.0D+00 < x .and. 0.0D+00 < y ) then value = theta_0 else if ( x < 0.0D+00 .and. 0.0D+00 < y ) then value = r8_pi - theta_0 else if ( x < 0.0D+00 .and. y < 0.0D+00 ) then value = r8_pi + theta_0 else if ( 0.0D+00 < x .and. y < 0.0D+00 ) then value = 2.0D+00 * r8_pi - theta_0 end if end if r8_atan = value return end function r8_atanh ( x ) !*****************************************************************************80 ! !! R8_ATANH returns the inverse hyperbolic tangent of a number. ! ! Discussion: ! ! Y = R8_ATANH ( X ) ! ! implies that ! ! X = TANH(Y) = ( EXP(Y) - EXP(-Y) ) / ( EXP(Y) + EXP(-Y) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 November 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose inverse hyperbolic ! tangent is desired. The absolute value of X should be less than ! or equal to 1. ! ! Output, real ( kind = 8 ) R8_ATANH, the inverse hyperbolic tangent of X. ! implicit none real ( kind = 8 ) r8_atanh real ( kind = 8 ) value real ( kind = 8 ) x if ( 1.0D+00 <= abs ( x ) ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_ATANH - Fatal error!' write ( *, '(a)' ) ' ABS(X) must be < 1.' write ( *, '(a,g14.6)' ) ' Your input is X = ', x stop 1 end if value = 0.5D+00 * log ( ( 1.0D+00 + x ) / ( 1.0D+00 - x ) ) r8_atanh = value return end function r8_big ( ) !*****************************************************************************80 ! !! R8_BIG returns a big R8. ! ! Discussion: ! ! The value returned by this function is NOT required to be the ! maximum representable R8. ! We simply want a "very large" but non-infinite number. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 September 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Output, real ( kind = 8 ) R8_BIG, a "big" value. ! implicit none real ( kind = 8 ) r8_big real ( kind = 8 ), parameter :: value = 1.0D+30 r8_big = value return end function r8_cas ( x ) !*****************************************************************************80 ! !! R8_CAS returns the "casine" of an R8. ! ! Discussion: ! ! The "casine", used in the discrete Hartley transform, is abbreviated ! CAS(X), and defined by: ! ! CAS(X) = cos ( X ) + sin( X ) ! = sqrt ( 2 ) * sin ( X + pi/4 ) ! = sqrt ( 2 ) * cos ( X - pi/4 ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 January 2001 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Ralph Hartley, ! A More Symmetrical Fourier Analysis Applied to Transmission Problems, ! Proceedings of the Institute of Radio Engineers, ! Volume 30, pages 144-150, 1942. ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose casine is desired. ! ! Output, real ( kind = 8 ) R8_CAS, the casine of X, which will be between ! plus or minus the square root of 2. ! implicit none real ( kind = 8 ) r8_cas real ( kind = 8 ) value real ( kind = 8 ) x value = cos ( x ) + sin ( x ) r8_cas = value return end function r8_ceiling ( r ) !*****************************************************************************80 ! !! R8_CEILING rounds an R8 "up" (towards +oo) to an integral R8. ! ! Example: ! ! R Value ! ! -1.1 -1.0 ! -1.0 -1.0 ! -0.9 0.0 ! 0.0 0.0 ! 5.0 5.0 ! 5.1 6.0 ! 5.9 6.0 ! 6.0 6.0 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 March 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, the value to be rounded up. ! ! Output, real ( kind = 8 ) R8_CEILING, the rounded value. ! implicit none real ( kind = 8 ) r integer ( kind = 4 ) r_int real ( kind = 8 ) r8_ceiling real ( kind = 8 ) value r_int = int ( r ) value = real ( r_int, kind = 8 ) if ( value < r ) then value = value + 1.0D+00 end if r8_ceiling = value return end function r8_choose ( n, k ) !*****************************************************************************80 ! !! R8_CHOOSE computes the combinatorial coefficient C(N,K). ! ! Discussion: ! ! Real arithmetic is used, and C(N,K) is computed directly, via ! Gamma functions, rather than recursively. ! ! C(N,K) is the number of distinct combinations of K objects ! chosen from a set of N distinct objects. A combination is ! like a set, in that order does not matter. ! ! C(N,K) = N! / ( (N-K)! * K! ) ! ! Example: ! ! The number of combinations of 2 things chosen from 5 is 10. ! ! C(5,2) = ( 5 * 4 * 3 * 2 * 1 ) / ( ( 3 * 2 * 1 ) * ( 2 * 1 ) ) = 10. ! ! The actual combinations may be represented as: ! ! (1,2), (1,3), (1,4), (1,5), (2,3), ! (2,4), (2,5), (3,4), (3,5), (4,5). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 June 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the value of N. ! ! Input, integer ( kind = 4 ) K, the value of K. ! ! Output, real ( kind = 8 ) R8_CHOOSE, the value of C(N,K) ! implicit none real ( kind = 8 ) arg real ( kind = 8 ) fack real ( kind = 8 ) facn real ( kind = 8 ) facnmk integer ( kind = 4 ) k integer ( kind = 4 ) n real ( kind = 8 ) r8_choose real ( kind = 8 ) r8_gamma_log real ( kind = 8 ) value if ( n < 0 ) then value = 0.0D+00 else if ( k == 0 ) then value = 1.0D+00 else if ( k == 1 ) then value = real ( n, kind = 8 ) else if ( 1 < k .and. k < n - 1 ) then arg = real ( n + 1, kind = 8 ) facn = r8_gamma_log ( arg ) arg = real ( k + 1, kind = 8 ) fack = r8_gamma_log ( arg ) arg = real ( n - k + 1, kind = 8 ) facnmk = r8_gamma_log ( arg ) value = real ( nint ( exp ( facn - fack - facnmk ) ), kind = 8 ) else if ( k == n - 1 ) then value = real ( n, kind = 8 ) else if ( k == n ) then value = 1.0D+00 else value = 0.0D+00 end if r8_choose = value return end function r8_chop ( place, x ) !*****************************************************************************80 ! !! R8_CHOP chops an R8 to a given number of binary places. ! ! Example: ! ! 3.875 = 2 + 1 + 1/2 + 1/4 + 1/8. ! ! The following values would be returned for the 'chopped' value of ! 3.875: ! ! PLACE Value ! ! 1 2 ! 2 3 = 2 + 1 ! 3 3.5 = 2 + 1 + 1/2 ! 4 3.75 = 2 + 1 + 1/2 + 1/4 ! 5+ 3.875 = 2 + 1 + 1/2 + 1/4 + 1/8 ! ! PLACE = 0 means return the integer part of X. ! PLACE = 1 means return the value of X, correct to 1/2. ! PLACE = 2 means return the value of X, correct to 1/4. ! PLACE = -1 means return the value of X, correct to 2. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 April 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) PLACE, the number of binary places to preserve. ! PLACE < 32. ! ! Input, real ( kind = 8 ) X, the number to be chopped. ! ! Output, real ( kind = 8 ) R8_CHOP, the chopped number. ! implicit none real ( kind = 8 ) fac integer ( kind = 4 ) place real ( kind = 8 ) r8_chop real ( kind = 8 ) r8_log_2 real ( kind = 8 ) r8_sign real ( kind = 8 ) s integer ( kind = 4 ) temp real ( kind = 8 ) value real ( kind = 8 ) x s = r8_sign ( x ) temp = int ( r8_log_2 ( abs ( x ) ) ) fac = 2.0D+00 ** ( temp - place + 1 ) value = s * real ( int ( abs ( x ) / fac ), kind = 8 ) * fac r8_chop = value return end function r8_cosd ( degrees ) !*****************************************************************************80 ! !! R8_COSD returns the cosine of an angle given in degrees. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 July 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) DEGREES, the angle in degrees. ! ! Output, real ( kind = 8 ) R8_COSD, the cosine of the angle. ! implicit none real ( kind = 8 ) degrees real ( kind = 8 ) r8_cosd real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) radians real ( kind = 8 ) value radians = r8_pi * ( degrees / 180.0D+00 ) value = cos ( radians ) r8_cosd = value return end function r8_cot ( angle ) !*****************************************************************************80 ! !! R8_COT returns the cotangent of an angle. ! ! Discussion: ! ! R8_COT ( THETA ) = COS ( THETA ) / SIN ( THETA ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 July 1998 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) ANGLE, the angle, in radians. ! ! Output, real ( kind = 8 ) R8_COT, the cotangent of the angle. ! implicit none real ( kind = 8 ) angle real ( kind = 8 ) r8_cot real ( kind = 8 ) value value = cos ( angle ) / sin ( angle ) r8_cot = value return end function r8_cotd ( degrees ) !*****************************************************************************80 ! !! R8_COTD returns the cotangent of an angle given in degrees. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 July 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) DEGREES, the angle in degrees. ! ! Output, real ( kind = 8 ) R8_COTD, the cotangent of the angle. ! implicit none real ( kind = 8 ) degrees real ( kind = 8 ) r8_cotd real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) radians real ( kind = 8 ) value radians = r8_pi * ( degrees / 180.0D+00 ) value = cos ( radians ) / sin ( radians ) r8_cotd = value return end function r8_csc ( theta ) !*****************************************************************************80 ! !! R8_CSC returns the cosecant of X. ! ! Discussion: ! ! R8_CSC ( THETA ) = 1.0 / SIN ( THETA ) ! ! The cosecant is not a built-in function in FORTRAN, and occasionally it ! is handier, or more concise, to be able to refer to it directly ! rather than through its definition in terms of the sine function. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 March 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) THETA, the angle, in radians, whose ! cosecant is desired. It must be the case that SIN ( THETA ) is not zero. ! ! Output, real ( kind = 8 ) R8_CSC, the cosecant of THETA. ! implicit none real ( kind = 8 ) r8_csc real ( kind = 8 ) theta real ( kind = 8 ) value value = sin ( theta ) if ( value == 0.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_CSC - Fatal error!' write ( *, '(a,g14.6)' ) ' Cosecant undefined for THETA = ', theta stop 1 end if r8_csc = 1.0D+00 / value return end function r8_cscd ( degrees ) !*****************************************************************************80 ! !! R8_CSCD returns the cosecant of an angle given in degrees. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 July 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) DEGREES, the angle in degrees. ! ! Output, real ( kind = 8 ) R8_CSCD, the cosecant of the angle. ! implicit none real ( kind = 8 ) degrees real ( kind = 8 ) r8_cscd real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) radians radians = r8_pi * ( degrees / 180.0D+00 ) r8_cscd = 1.0D+00 / sin ( radians ) return end function r8_csqrt ( x ) !*****************************************************************************80 ! !! R8_CSQRT returns the complex square root of an R8. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 October 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose square root is desired. ! ! Output, complex ( kind = 8 ) R8_CSQRT, the square root of X: ! implicit none real ( kind = 8 ) argument real ( kind = 8 ) magnitude complex ( kind = 8 ) r8_csqrt real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) x if ( 0.0D+00 < x ) then magnitude = x argument = 0.0D+00 else if ( 0.0D+00 == x ) then magnitude = 0.0D+00 argument = 0.0D+00 else if ( x < 0.0D+00 ) then magnitude = -x argument = r8_pi end if magnitude = sqrt ( magnitude ) argument = argument / 2.0D+00 r8_csqrt = magnitude * cmplx ( cos ( argument ), sin ( argument ), kind = 8 ) return end function r8_cube_root ( x ) !*****************************************************************************80 ! !! R8_CUBE_ROOT returns the cube root of an R8. ! ! Discussion: ! ! This routine is designed to avoid the possible problems that can occur ! when formulas like 0.0^(1/3) or (-1.0)^(1/3) are to be evaluated. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 March 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose cube root is desired. ! ! Output, real ( kind = 8 ) R8_CUBE_ROOT, the cube root of X. ! implicit none real ( kind = 8 ) r8_cube_root real ( kind = 8 ) value real ( kind = 8 ) x if ( 0.0D+00 < x ) then value = x ** ( 1.0D+00 / 3.0D+00 ) else if ( x == 0.0D+00 ) then value = 0.0D+00 else value = -( abs ( x ) ) ** ( 1.0D+00 / 3.0D+00 ) end if r8_cube_root = value return end function r8_degrees ( radians ) !*****************************************************************************80 ! !! R8_DEGREES converts an angle from radian to degree measure. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 May 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) RADIANS, the angle measurement in radians. ! ! Output, real ( kind = 8 ) R8_DEGREES, the angle measurement in degrees. ! implicit none real ( kind = 8 ) r8_degrees real ( kind = 8 ), parameter :: r8_pi = 3.1415926535897932384626434D+00 real ( kind = 8 ) radians r8_degrees = radians * 180.0D+00 / r8_pi return end function r8_diff ( x, y, n ) !*****************************************************************************80 ! !! R8_DIFF computes the difference of two R8's to a specified accuracy. ! ! Discussion: ! ! The user controls how many binary digits of accuracy ! are to be used. ! ! N determines the accuracy of the value of the result. If N = 10, ! for example, only 11 binary places will be used in the arithmetic. ! In general, only N+1 binary places will be used. ! ! N may be zero. However, a negative value of N should ! not be used, since this will cause both X and Y to look like 0. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 November 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, Y, the two values whose difference is desired. ! ! Input, integer ( kind = 4 ) N, the number of binary digits to use. ! ! Output, real ( kind = 8 ) R8_DIFF, the value of X-Y. ! implicit none real ( kind = 8 ) cx real ( kind = 8 ) cy integer ( kind = 4 ) n real ( kind = 8 ) pow2 real ( kind = 8 ) r8_diff real ( kind = 8 ) size real ( kind = 8 ) x real ( kind = 8 ) y if ( x == y ) then r8_diff = 0.0D+00 return end if pow2 = 2.0D+00 ** n ! ! Compute the magnitude of X and Y, and take the larger of the ! two. At least one of the two values is not zero! ! size = max ( abs ( x ), abs ( y ) ) ! ! Make normalized copies of X and Y. One of the two values will ! actually be equal to 1. ! cx = x / size cy = y / size ! ! Here's where rounding comes in. We know that the larger of the ! the two values equals 1. We multiply both values by 2^N, ! where N+1 is the number of binary digits of accuracy we want ! to use, truncate the values, and divide back by 2^N. ! cx = real ( int ( cx * pow2 + sign ( 0.5D+00, cx ) ), kind = 8 ) / pow2 cy = real ( int ( cy * pow2 + sign ( 0.5D+00, cy ) ), kind = 8 ) / pow2 ! ! Take the difference now. ! r8_diff = cx - cy ! ! Undo the scaling. ! r8_diff = r8_diff * size return end subroutine r8_digit ( x, idigit, digit ) !*****************************************************************************80 ! !! R8_DIGIT returns a particular decimal digit of an R8. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 November 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose NDIG-th decimal digit ! is desired. If X is zero, all digits will be returned as 0. ! ! Input, integer ( kind = 4 ) IDIGIT, the position of the desired decimal ! digit. A value of 1 means the leading digit, a value of 2 the second digit ! and so on. ! ! Output, integer ( kind = 4 ) DIGIT, the value of the IDIGIT-th decimal ! digit of X. ! implicit none integer ( kind = 4 ) digit integer ( kind = 4 ) i integer ( kind = 4 ) idigit integer ( kind = 4 ) ival real ( kind = 8 ) x real ( kind = 8 ) xcopy if ( x == 0.0D+00 ) then digit = 0 return end if if ( idigit <= 0 ) then digit = 0 return end if ! ! Set XCOPY = X, and then force XCOPY to lie between 1 and 10. ! xcopy = abs ( x ) do while ( xcopy < 1.0D+00 ) xcopy = xcopy * 10.0D+00 end do do while ( 10.0D+00 <= xcopy ) xcopy = xcopy / 10.0D+00 end do do i = 1, idigit ival = int ( xcopy ) xcopy = ( xcopy - ival ) * 10.0D+00 end do digit = ival return end function r8_divide_i4 ( i, j ) !*****************************************************************************80 ! !! R8_DIVIDE_I4 returns an I4 fraction as an R8. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 June 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) I, J, the numerator and denominator. ! ! Output, real ( kind = 8 ) R8_DIVIDE_I4, the value of (I/J). ! implicit none integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) r8_divide_i4 r8_divide_i4 = real ( i, kind = 8 ) / real ( j, kind = 8 ) return end function r8_e ( ) !*****************************************************************************80 ! !! R8_E returns the value of the base of the natural logarithm system. ! ! Discussion: ! ! E = Limit ( N -> oo ) ( 1 + 1 / N )^N ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 28 April 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Output, real ( kind = 8 ) R8_E, the base of the natural ! logarithm system. ! implicit none real ( kind = 8 ) r8_e r8_e = 2.718281828459045235360287D+00 return end function r8_epsilon ( ) !*****************************************************************************80 ! !! R8_EPSILON returns the R8 roundoff unit. ! ! Discussion: ! ! The roundoff unit is a number R which is a power of 2 with the ! property that, to the precision of the computer's arithmetic, ! 1 < 1 + R ! but ! 1 = ( 1 + R / 2 ) ! ! FORTRAN90 provides the superior library routine ! ! EPSILON ( X ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 September 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Output, real ( kind = 8 ) R8_EPSILON, the round-off unit. ! implicit none real ( kind = 8 ) r8_epsilon r8_epsilon = 2.220446049250313D-016 return end function r8_epsilon_compute ( ) !*****************************************************************************80 ! !! R8_EPSILON_COMPUTE computes the R8 roundoff unit. ! ! Discussion: ! ! The roundoff unit is a number R which is a power of 2 with the ! property that, to the precision of the computer's arithmetic, ! 1 < 1 + R ! but ! 1 = ( 1 + R / 2 ) ! ! FORTRAN90 provides the superior library routine ! ! EPSILON ( X ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 31 August 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Output, real ( kind = 8 ) R8_EPSILON_COMPUTE, the computed round-off unit. ! implicit none real ( kind = 8 ) one real ( kind = 8 ) r8_add real ( kind = 8 ) r8_epsilon_compute real ( kind = 8 ) temp real ( kind = 8 ) test real ( kind = 8 ) value one = real ( 1, kind = 8 ) value = one temp = value / 2.0D+00 test = r8_add ( one, temp ) do while ( one < test ) value = temp temp = value / 2.0D+00 test = r8_add ( one, temp ) end do r8_epsilon_compute = value return end function r8_exp ( x ) !*****************************************************************************80 ! !! R8_EXP computes the exponential of an R8, avoiding overflow and underflow. ! ! Discussion: ! ! For arguments of very large magnitude, the evaluation of the ! exponential function can cause computational problems. Some languages ! and compilers may return an infinite value or a "Not-a-Number". ! An alternative, when dealing with a wide range of inputs, is simply ! to truncate the calculation for arguments whose magnitude is too large. ! Whether this is the right or convenient approach depends on the problem ! you are dealing with, and whether or not you really need accurate ! results for large magnitude inputs, or you just want your code to ! stop crashing. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 September 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument of the exponential function. ! ! Output, real ( kind = 8 ) R8_EXP, the value of exp ( X ). ! implicit none real ( kind = 8 ) r8_exp real ( kind = 8 ), parameter :: r8_huge = 1.79769313486231571D+308 real ( kind = 8 ), parameter :: r8_log_max = +69.0776D+00 real ( kind = 8 ), parameter :: r8_log_min = -69.0776D+00 real ( kind = 8 ) value real ( kind = 8 ) x if ( x <= r8_log_min ) then value = 0.0D+00 else if ( x < r8_log_max ) then value = exp ( x ) else value = r8_huge end if r8_exp = value return end function r8_factorial ( n ) !*****************************************************************************80 ! !! R8_FACTORIAL computes the factorial of N. ! ! Discussion: ! ! factorial ( N ) = product ( 1 <= I <= N ) I ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 January 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the argument of the factorial function. ! If N is less than 1, the function value is returned as 1. ! ! Output, real ( kind = 8 ) R8_FACTORIAL, the factorial of N. ! implicit none real ( kind = 8 ) r8_factorial integer ( kind = 4 ) i integer ( kind = 4 ) n real ( kind = 8 ) value value = 1.0D+00 do i = 1, n value = value * real ( i, kind = 8 ) end do r8_factorial = value return end function r8_factorial_stirling ( n ) !*****************************************************************************80 ! !! R8_FACTORIAL_STIRLING computes Stirling's approximation to N!. ! ! Discussion: ! ! This routine returns the raw approximation for all nonnegative ! values of N. If N is less than 0, the value is returned as 0, ! and if N is 0, the value of 1 is returned. In all other cases, ! Stirling's formula is used. ! ! The factorial function N! is defined by ! ! N! = Product ( 1 <= I <= N ) I ! ! Stirling's approximation to N! is ! ! Stirling ( N ) = sqrt ( 2 * PI * N ) * ( N / E )^N * E^(1/(12*N) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 April 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the argument of the function. ! ! Output, real ( kind = 8 ) R8_FACTORIAL_STIRLING, an approximation to N!. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ), parameter :: r8_e = 2.71828182845904523D+00 real ( kind = 8 ) r8_factorial_stirling real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) value if ( n < 0 ) then value = 0.0D+00 else if ( n == 0 ) then value = 1.0D+00 else value = sqrt ( 2.0D+00 * r8_pi * real ( n, kind = 8 ) ) & * ( real ( n, kind = 8 ) / r8_e ) ** n & * exp ( 1.0D+00 / real ( 12 * n, kind = 8 ) ) end if r8_factorial_stirling = value return end subroutine r8_factorial_values ( n_data, n, fn ) !*****************************************************************************80 ! !! R8_FACTORIAL_VALUES returns values of the real factorial function. ! ! Discussion: ! ! 0! = 1 ! I! = Product ( 1 <= J <= I ) J ! ! Although the factorial is an integer valued function, it quickly ! becomes too large for an integer to hold. This routine still accepts ! an integer as the input argument, but returns the function value ! as a real number. ! ! In Mathematica, the function can be evaluated by: ! ! n! ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! National Bureau of Standards, 1964, ! ISBN: 0-486-61272-4, ! LC: QA47.A34. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Cambridge University Press, 1999, ! ISBN: 0-521-64314-7, ! LC: QA76.95.W65. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 ! before the first call. On each call, the routine increments N_DATA by 1, ! and returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, integer ( kind = 4 ) N, the argument of the function. ! ! Output, real ( kind = 8 ) FN, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 25 real ( kind = 8 ) fn real ( kind = 8 ), save, dimension ( n_max ) :: fn_vec = (/ & 0.1000000000000000D+01, & 0.1000000000000000D+01, & 0.2000000000000000D+01, & 0.6000000000000000D+01, & 0.2400000000000000D+02, & 0.1200000000000000D+03, & 0.7200000000000000D+03, & 0.5040000000000000D+04, & 0.4032000000000000D+05, & 0.3628800000000000D+06, & 0.3628800000000000D+07, & 0.3991680000000000D+08, & 0.4790016000000000D+09, & 0.6227020800000000D+10, & 0.8717829120000000D+11, & 0.1307674368000000D+13, & 0.2092278988800000D+14, & 0.3556874280960000D+15, & 0.6402373705728000D+16, & 0.1216451004088320D+18, & 0.2432902008176640D+19, & 0.1551121004333099D+26, & 0.3041409320171338D+65, & 0.9332621544394415D+158, & 0.5713383956445855D+263 /) integer ( kind = 4 ) n integer ( kind = 4 ) n_data integer ( kind = 4 ), save, dimension ( n_max ) :: n_vec = (/ & 0, & 1, & 2, & 3, & 4, & 5, & 6, & 7, & 8, & 9, & 10, & 11, & 12, & 13, & 14, & 15, & 16, & 17, & 18, & 19, & 20, & 25, & 50, & 100, & 150 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 n = 0 fn = 0.0D+00 else n = n_vec(n_data) fn = fn_vec(n_data) end if return end function r8_factorial2 ( n ) !*****************************************************************************80 ! !! R8_FACTORIAL2 computes the double factorial function. ! ! Discussion: ! ! FACTORIAL2( N ) = Product ( N * (N-2) * (N-4) * ... * 2 ) (N even) ! = Product ( N * (N-2) * (N-4) * ... * 1 ) (N odd) ! ! Example: ! ! N Value ! ! 0 1 ! 1 1 ! 2 2 ! 3 3 ! 4 8 ! 5 15 ! 6 48 ! 7 105 ! 8 384 ! 9 945 ! 10 3840 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 September 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the argument of the double factorial ! function. If N is less than 1, the value is returned as 1.0. ! ! Output, real ( kind = 8 ) R8_FACTORIAL2, the value. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) r8_factorial2 real ( kind = 8 ) r8_n real ( kind = 8 ) value value = 1.0D+00 if ( 1 <= n ) then r8_n = real ( n, kind = 8 ) do while ( 1.0D+00 < r8_n ) value = value * r8_n r8_n = r8_n - 2.0D+00 end do end if r8_factorial2 = value return end subroutine r8_factorial2_values ( n_data, n, f ) !*****************************************************************************80 ! !! R8_FACTORIAL2_VALUES returns values of the double factorial function. ! ! Discussion: ! ! FACTORIAL2( N ) = Product ( N * (N-2) * (N-4) * ... * 2 ) (N even) ! = Product ( N * (N-2) * (N-4) * ... * 1 ) (N odd) ! ! In Mathematica, the function can be evaluated by: ! ! n!! ! ! Example: ! ! N N!! ! ! 0 1 ! 1 1 ! 2 2 ! 3 3 ! 4 8 ! 5 15 ! 6 48 ! 7 105 ! 8 384 ! 9 945 ! 10 3840 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! National Bureau of Standards, 1964, ! ISBN: 0-486-61272-4, ! LC: QA47.A34. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Cambridge University Press, 1999, ! ISBN: 0-521-64314-7, ! LC: QA76.95.W65. ! ! Daniel Zwillinger, editor, ! CRC Standard Mathematical Tables and Formulae, ! 30th Edition, ! CRC Press, 1996, ! ISBN: 0-8493-2479-3, ! LC: QA47.M315. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 ! before the first call. On each call, the routine increments N_DATA by 1, ! and returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, integer ( kind = 4 ) N, the argument of the function. ! ! Output, real ( kind = 8 ) F, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 16 real ( kind = 8 ), save, dimension ( n_max ) :: f_vec = (/ & 1.0D+00, & 1.0D+00, & 2.0D+00, & 3.0D+00, & 8.0D+00, & 15.0D+00, & 48.0D+00, & 105.0D+00, & 384.0D+00, & 945.0D+00, & 3840.0D+00, & 10395.0D+00, & 46080.0D+00, & 135135.0D+00, & 645120.0D+00, & 2027025.0D+00 /) real ( kind = 8 ) f integer ( kind = 4 ) n_data integer ( kind = 4 ) n integer ( kind = 4 ), save, dimension ( n_max ) :: n_vec = (/ & 0, & 1, 2, 3, 4, 5, & 6, 7, 8, 9, 10, & 11, 12, 13, 14, 15 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 n = 0 f = 0.0D+00 else n = n_vec(n_data) f = f_vec(n_data) end if return end function r8_fall ( x, n ) !*****************************************************************************80 ! !! R8_FALL computes the falling factorial function [X]_N. ! ! Discussion: ! ! Note that the number of "injections" or 1-to-1 mappings from ! a set of N elements to a set of M elements is [M]_N. ! ! The number of permutations of N objects out of M is [M]_N. ! ! Moreover, the Stirling numbers of the first kind can be used ! to convert a falling factorial into a polynomial, as follows: ! ! [X]_N = S^0_N + S^1_N * X + S^2_N * X^2 + ... + S^N_N X^N. ! ! The formula used is: ! ! [X]_N = X * ( X - 1 ) * ( X - 2 ) * ... * ( X - N + 1 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 May 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument of the falling factorial function. ! ! Input, integer ( kind = 4 ) N, the order of the falling factorial function. ! If N = 0, FALL = 1, if N = 1, FALL = X. Note that if N is ! negative, a "rising" factorial will be computed. ! ! Output, real ( kind = 8 ) R8_FALL, the value of the falling ! factorial function. ! implicit none real ( kind = 8 ) arg integer ( kind = 4 ) i integer ( kind = 4 ) n real ( kind = 8 ) r8_fall real ( kind = 8 ) value real ( kind = 8 ) x value = 1.0D+00 arg = x if ( 0 < n ) then do i = 1, n value = value * arg arg = arg - 1.0D+00 end do else if ( n < 0 ) then do i = -1, n, -1 value = value * arg arg = arg + 1.0D+00 end do end if r8_fall = value return end subroutine r8_fall_values ( n_data, x, n, f ) !*****************************************************************************80 ! !! R8_FALL_VALUES returns some values of the falling factorial function. ! ! Discussion: ! ! In Mathematica, the function can be evaluated by: ! ! FactorialPower[X,Y] ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 December 2014 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! National Bureau of Standards, 1964, ! ISBN: 0-486-61272-4, ! LC: QA47.A34. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Cambridge University Press, 1999, ! ISBN: 0-521-64314-7, ! LC: QA76.95.W65. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 ! before the first call. On each call, the routine increments N_DATA by 1, ! and returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, integer ( kind = 4 ) N, the arguments. ! ! Output, real ( kind = 8 ) F, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 15 real ( kind = 8 ), save, dimension ( n_max ) :: f_vec = (/ & 120.0000000000000D+00, & 163.1601562500000D+00, & 216.5625000000000D+00, & 281.6601562500000D+00, & 360.0000000000000D+00, & 1.000000000000000D+00, & 7.500000000000000D+00, & 48.75000000000000D+00, & 268.1250000000000D+00, & 1206.562500000000D+00, & 4222.968750000000D+00, & 10557.42187500000D+00, & 15836.13281250000D+00, & 7918.066406250000D+00, & -3959.03320312500D+00 /) real ( kind = 8 ) f integer ( kind = 4 ) n integer ( kind = 4 ) n_data integer ( kind = 4 ), save, dimension ( n_max ) :: n_vec = (/ & 4, & 4, & 4, & 4, & 4, & 0, & 1, & 2, & 3, & 4, & 5, & 6, & 7, & 8, & 9 /) real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 5.00D+00, & 5.25D+00, & 5.50D+00, & 5.75D+00, & 6.00D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 n = 0 f = 0.0D+00 else x = x_vec(n_data) n = n_vec(n_data) f = f_vec(n_data) end if return end function r8_floor ( r ) !*****************************************************************************80 ! !! R8_FLOOR rounds an R8 "down" (towards -oo) to the nearest integral R8. ! ! Example: ! ! R Value ! ! -1.1 -2.0 ! -1.0 -1.0 ! -0.9 -1.0 ! 0.0 0.0 ! 5.0 5.0 ! 5.1 5.0 ! 5.9 5.0 ! 6.0 6.0 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, the value to be rounded down. ! ! Output, real ( kind = 8 ) R8_FLOOR, the rounded value. ! implicit none real ( kind = 8 ) r real ( kind = 8 ) r8_floor real ( kind = 8 ) value value = real ( int ( r ), kind = 8 ) if ( r < value ) then value = value - 1.0D+00 end if r8_floor = value return end function r8_fraction ( i, j ) !*****************************************************************************80 ! !! R8_FRACTION uses real arithmetic on an integer ratio. ! ! Discussion: ! ! Given integer variables I and J, both FORTRAN and C will evaluate ! an expression such as "I/J" using what is called "integer division", ! with the result being an integer. It is often convenient to express ! the parts of a fraction as integers but expect the result to be computed ! using real arithmetic. This function carries out that operation. ! ! Example: ! ! I J I/J R8_FRACTION ! ! 1 2 0 0.5 ! 7 4 1 1.75 ! 8 4 2 2.00 ! 9 4 2 2.25 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) I, J, the arguments. ! ! Output, real ( kind = 8 ) R8_FRACTION, the value of the ratio. ! implicit none integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) r8_fraction r8_fraction = real ( i, kind = 8 ) / real ( j, kind = 8 ) return end function r8_fractional ( x ) !*****************************************************************************80 ! !! R8_FRACTIONAL returns the fractional part of an R8. ! ! Discussion: ! ! If we regard a real number as ! ! R = SIGN * ( WHOLE + FRACTION ) ! ! where ! ! SIGN is +1 or -1, ! WHOLE is a nonnegative integer ! FRACTION is a nonnegative real number strictly less than 1, ! ! then this routine returns the value of FRACTION. ! ! Example: ! ! R FRACTION ! ! 0.00 0.00 ! 1.01 0.01 ! 2.02 0.02 ! 19.73 0.73 ! -4.34 0.34 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 January 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument. ! ! Output, real ( kind = 8 ) R8_FRACTIONAL, the fractional part of X. ! implicit none real ( kind = 8 ) r8_fractional real ( kind = 8 ) x r8_fractional = abs ( x ) - real ( int ( abs ( x ) ), kind = 8 ) return end function r8_gamma ( x ) !*****************************************************************************80 ! !! R8_GAMMA evaluates Gamma(X) for a real argument. ! ! Discussion: ! ! This routine calculates the gamma function for a real argument X. ! ! Computation is based on an algorithm outlined in reference 1. ! The program uses rational functions that approximate the gamma ! function to at least 20 significant decimal digits. Coefficients ! for the approximation over the interval (1,2) are unpublished. ! Those for the approximation for 12 <= X are from reference 2. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 April 2013 ! ! Author: ! ! Original FORTRAN77 version by William Cody, Laura Stoltz. ! FORTRAN90 version by John Burkardt. ! ! Reference: ! ! William Cody, ! An Overview of Software Development for Special Functions, ! in Numerical Analysis Dundee, 1975, ! edited by GA Watson, ! Lecture Notes in Mathematics 506, ! Springer, 1976. ! ! John Hart, Ward Cheney, Charles Lawson, Hans Maehly, ! Charles Mesztenyi, John Rice, Henry Thatcher, ! Christoph Witzgall, ! Computer Approximations, ! Wiley, 1968, ! LC: QA297.C64. ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) R8_GAMMA, the value of the function. ! implicit none real ( kind = 8 ), dimension ( 7 ) :: c = (/ & -1.910444077728D-03, & 8.4171387781295D-04, & -5.952379913043012D-04, & 7.93650793500350248D-04, & -2.777777777777681622553D-03, & 8.333333333333333331554247D-02, & 5.7083835261D-03 /) real ( kind = 8 ) fact integer ( kind = 4 ) i integer ( kind = 4 ) n real ( kind = 8 ), dimension ( 8 ) :: p = (/ & -1.71618513886549492533811D+00, & 2.47656508055759199108314D+01, & -3.79804256470945635097577D+02, & 6.29331155312818442661052D+02, & 8.66966202790413211295064D+02, & -3.14512729688483675254357D+04, & -3.61444134186911729807069D+04, & 6.64561438202405440627855D+04 /) logical ( kind = 4 ) parity real ( kind = 8 ), dimension ( 8 ) :: q = (/ & -3.08402300119738975254353D+01, & 3.15350626979604161529144D+02, & -1.01515636749021914166146D+03, & -3.10777167157231109440444D+03, & 2.25381184209801510330112D+04, & 4.75584627752788110767815D+03, & -1.34659959864969306392456D+05, & -1.15132259675553483497211D+05 /) real ( kind = 8 ) r8_epsilon real ( kind = 8 ) r8_gamma real ( kind = 8 ), parameter :: r8_pi = 3.1415926535897932384626434D+00 real ( kind = 8 ) res real ( kind = 8 ), parameter :: sqrtpi = 0.9189385332046727417803297D+00 real ( kind = 8 ) sum real ( kind = 8 ) x real ( kind = 8 ), parameter :: xbig = 171.624D+00 real ( kind = 8 ) xden real ( kind = 8 ), parameter :: xinf = 1.79D+308 real ( kind = 8 ), parameter :: xminin = 2.23D-308 real ( kind = 8 ) xnum real ( kind = 8 ) y real ( kind = 8 ) y1 real ( kind = 8 ) ysq real ( kind = 8 ) z parity = .false. fact = 1.0D+00 n = 0 y = x ! ! Argument is negative. ! if ( y <= 0.0D+00 ) then y = - x y1 = aint ( y ) res = y - y1 if ( res /= 0.0D+00 ) then if ( y1 /= aint ( y1 * 0.5D+00 ) * 2.0D+00 ) then parity = .true. end if fact = - r8_pi / sin ( r8_pi * res ) y = y + 1.0D+00 else res = xinf r8_gamma = res return end if end if ! ! Argument is positive. ! if ( y < r8_epsilon ( ) ) then ! ! Argument < EPS. ! if ( xminin <= y ) then res = 1.0D+00 / y else res = xinf r8_gamma = res return end if else if ( y < 12.0D+00 ) then y1 = y ! ! 0.0 < argument < 1.0. ! if ( y < 1.0D+00 ) then z = y y = y + 1.0D+00 ! ! 1.0 < argument < 12.0. ! Reduce argument if necessary. ! else n = int ( y ) - 1 y = y - real ( n, kind = 8 ) z = y - 1.0D+00 end if ! ! Evaluate approximation for 1.0 < argument < 2.0. ! xnum = 0.0D+00 xden = 1.0D+00 do i = 1, 8 xnum = ( xnum + p(i) ) * z xden = xden * z + q(i) end do res = xnum / xden + 1.0D+00 ! ! Adjust result for case 0.0 < argument < 1.0. ! if ( y1 < y ) then res = res / y1 ! ! Adjust result for case 2.0 < argument < 12.0. ! else if ( y < y1 ) then do i = 1, n res = res * y y = y + 1.0D+00 end do end if else ! ! Evaluate for 12.0 <= argument. ! if ( y <= xbig ) then ysq = y * y sum = c(7) do i = 1, 6 sum = sum / ysq + c(i) end do sum = sum / y - y + sqrtpi sum = sum + ( y - 0.5D+00 ) * log ( y ) res = exp ( sum ) else res = xinf r8_gamma = res return end if end if ! ! Final adjustments and return. ! if ( parity ) then res = - res end if if ( fact /= 1.0D+00 ) then res = fact / res end if r8_gamma = res return end function r8_gamma_log ( x ) !*****************************************************************************80 ! !! R8_GAMMA_LOG evaluates the logarithm of the gamma function. ! ! Discussion: ! ! This routine calculates the LOG(GAMMA) function for a positive real ! argument X. Computation is based on an algorithm outlined in ! references 1 and 2. The program uses rational functions that ! theoretically approximate LOG(GAMMA) to at least 18 significant ! decimal digits. The approximation for X > 12 is from reference ! 3, while approximations for X < 12.0 are similar to those in ! reference 1, but are unpublished. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 April 2013 ! ! Author: ! ! Original FORTRAN77 version by William Cody, Laura Stoltz. ! FORTRAN90 version by John Burkardt. ! ! Reference: ! ! William Cody, Kenneth Hillstrom, ! Chebyshev Approximations for the Natural Logarithm of the ! Gamma Function, ! Mathematics of Computation, ! Volume 21, Number 98, April 1967, pages 198-203. ! ! Kenneth Hillstrom, ! ANL/AMD Program ANLC366S, DGAMMA/DLGAMA, ! May 1969. ! ! John Hart, Ward Cheney, Charles Lawson, Hans Maehly, ! Charles Mesztenyi, John Rice, Henry Thatcher, ! Christoph Witzgall, ! Computer Approximations, ! Wiley, 1968, ! LC: QA297.C64. ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) R8_GAMMA_LOG, the value of the function. ! implicit none real ( kind = 8 ), dimension ( 7 ) :: c = (/ & -1.910444077728D-03, & 8.4171387781295D-04, & -5.952379913043012D-04, & 7.93650793500350248D-04, & -2.777777777777681622553D-03, & 8.333333333333333331554247D-02, & 5.7083835261D-03 /) real ( kind = 8 ) corr real ( kind = 8 ) :: d1 = -5.772156649015328605195174D-01 real ( kind = 8 ) :: d2 = 4.227843350984671393993777D-01 real ( kind = 8 ) :: d4 = 1.791759469228055000094023D+00 real ( kind = 8 ), parameter :: frtbig = 2.25D+76 integer ( kind = 4 ) i real ( kind = 8 ), dimension ( 8 ) :: p1 = (/ & 4.945235359296727046734888D+00, & 2.018112620856775083915565D+02, & 2.290838373831346393026739D+03, & 1.131967205903380828685045D+04, & 2.855724635671635335736389D+04, & 3.848496228443793359990269D+04, & 2.637748787624195437963534D+04, & 7.225813979700288197698961D+03 /) real ( kind = 8 ), dimension ( 8 ) :: p2 = (/ & 4.974607845568932035012064D+00, & 5.424138599891070494101986D+02, & 1.550693864978364947665077D+04, & 1.847932904445632425417223D+05, & 1.088204769468828767498470D+06, & 3.338152967987029735917223D+06, & 5.106661678927352456275255D+06, & 3.074109054850539556250927D+06 /) real ( kind = 8 ), dimension ( 8 ) :: p4 = (/ & 1.474502166059939948905062D+04, & 2.426813369486704502836312D+06, & 1.214755574045093227939592D+08, & 2.663432449630976949898078D+09, & 2.940378956634553899906876D+10, & 1.702665737765398868392998D+11, & 4.926125793377430887588120D+11, & 5.606251856223951465078242D+11 /) real ( kind = 8 ), dimension ( 8 ) :: q1 = (/ & 6.748212550303777196073036D+01, & 1.113332393857199323513008D+03, & 7.738757056935398733233834D+03, & 2.763987074403340708898585D+04, & 5.499310206226157329794414D+04, & 6.161122180066002127833352D+04, & 3.635127591501940507276287D+04, & 8.785536302431013170870835D+03 /) real ( kind = 8 ), dimension ( 8 ) :: q2 = (/ & 1.830328399370592604055942D+02, & 7.765049321445005871323047D+03, & 1.331903827966074194402448D+05, & 1.136705821321969608938755D+06, & 5.267964117437946917577538D+06, & 1.346701454311101692290052D+07, & 1.782736530353274213975932D+07, & 9.533095591844353613395747D+06 /) real ( kind = 8 ), dimension ( 8 ) :: q4 = (/ & 2.690530175870899333379843D+03, & 6.393885654300092398984238D+05, & 4.135599930241388052042842D+07, & 1.120872109616147941376570D+09, & 1.488613728678813811542398D+10, & 1.016803586272438228077304D+11, & 3.417476345507377132798597D+11, & 4.463158187419713286462081D+11 /) real ( kind = 8 ) r8_gamma_log real ( kind = 8 ) res real ( kind = 8 ), parameter :: sqrtpi = 0.9189385332046727417803297D+00 real ( kind = 8 ) x real ( kind = 8 ), parameter :: xbig = 2.55D+305 real ( kind = 8 ) xden real ( kind = 8 ), parameter :: xinf = 1.79D+308 real ( kind = 8 ) xm1 real ( kind = 8 ) xm2 real ( kind = 8 ) xm4 real ( kind = 8 ) xnum real ( kind = 8 ) y real ( kind = 8 ) ysq y = x if ( 0.0D+00 < y .and. y <= xbig ) then if ( y <= epsilon ( y ) ) then res = - log ( y ) ! ! EPS < X <= 1.5. ! else if ( y <= 1.5D+00 ) then if ( y < 0.6796875D+00 ) then corr = -log ( y ) xm1 = y else corr = 0.0D+00 xm1 = ( y - 0.5D+00 ) - 0.5D+00 end if if ( y <= 0.5D+00 .or. 0.6796875D+00 <= y ) then xden = 1.0D+00 xnum = 0.0D+00 do i = 1, 8 xnum = xnum * xm1 + p1(i) xden = xden * xm1 + q1(i) end do res = corr + ( xm1 * ( d1 + xm1 * ( xnum / xden ) ) ) else xm2 = ( y - 0.5D+00 ) - 0.5D+00 xden = 1.0D+00 xnum = 0.0D+00 do i = 1, 8 xnum = xnum * xm2 + p2(i) xden = xden * xm2 + q2(i) end do res = corr + xm2 * ( d2 + xm2 * ( xnum / xden ) ) end if ! ! 1.5 < X <= 4.0. ! else if ( y <= 4.0D+00 ) then xm2 = y - 2.0D+00 xden = 1.0D+00 xnum = 0.0D+00 do i = 1, 8 xnum = xnum * xm2 + p2(i) xden = xden * xm2 + q2(i) end do res = xm2 * ( d2 + xm2 * ( xnum / xden ) ) ! ! 4.0 < X <= 12.0. ! else if ( y <= 12.0D+00 ) then xm4 = y - 4.0D+00 xden = -1.0D+00 xnum = 0.0D+00 do i = 1, 8 xnum = xnum * xm4 + p4(i) xden = xden * xm4 + q4(i) end do res = d4 + xm4 * ( xnum / xden ) ! ! Evaluate for 12 <= argument. ! else res = 0.0D+00 if ( y <= frtbig ) then res = c(7) ysq = y * y do i = 1, 6 res = res / ysq + c(i) end do end if res = res / y corr = log ( y ) res = res + sqrtpi - 0.5D+00 * corr res = res + y * ( corr - 1.0D+00 ) end if ! ! Return for bad arguments. ! else res = xinf end if ! ! Final adjustments and return. ! r8_gamma_log = res return end function r8_haversine ( a ) !******************************************************************************/ ! !! R8_HAVERSINE computes the haversine of an angle. ! ! Discussion: ! ! haversine(A) = ( 1 - cos ( A ) ) / 2 ! ! The haversine is useful in spherical trigonometry. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 November 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A, the angle. ! ! Output, double R8_HAVERSINE, the haversine of the angle. ! implicit none real ( kind = 8 ) a real ( kind = 8 ) r8_haversine real ( kind = 8 ) value value = ( 1.0D+00 - cos ( a ) ) / 2.0 r8_haversine = value return end function r8_heaviside ( x ) !*****************************************************************************80 ! !! R8_HEAVISIDE evaluates the Heaviside function. ! ! Discussion: ! ! The Heaviside function is 0 for x < 0, 1 for x > 0, and 1/2 for x = 0. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 November 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument. ! ! Output, real ( kind = 8 ) R8_HEAVISIDE, the value. ! implicit none real ( kind = 8 ) r8_heaviside real ( kind = 8 ) value real ( kind = 8 ) x if ( x < 0.0D+00 ) then value = 0.0D+00 else if ( x == 0.0D+00 ) then value = 0.5D+00 else value = 1.0D+00 end if r8_heaviside = value return end function r8_huge ( ) !*****************************************************************************80 ! !! R8_HUGE returns a very large R8. ! ! Discussion: ! ! The value returned by this function is intended to be the largest ! representable real value. ! ! FORTRAN90 provides a built-in routine HUGE ( X ) that ! can return the maximum representable number of the same datatype ! as X, if that is what is really desired. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 September 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Output, real ( kind = 8 ) R8_HUGE, a "huge" value. ! implicit none real ( kind = 8 ) r8_huge r8_huge = 1.79769313486231571D+308 return end function r8_hypot ( x, y ) !*****************************************************************************80 ! !! R8_HYPOT returns the value of sqrt ( X^2 + Y^2 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 22 March 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, Y, the arguments. ! ! Output, real ( kind = 8 ) R8_HYPOT, the value of sqrt ( X^2 + Y^2 ). ! implicit none real ( kind = 8 ) a real ( kind = 8 ) b real ( kind = 8 ) c real ( kind = 8 ) r8_hypot real ( kind = 8 ) x real ( kind = 8 ) y if ( abs ( x ) < abs ( y ) ) then a = abs ( y ) b = abs ( x ) else a = abs ( x ) b = abs ( y ) end if ! ! A contains the larger value. ! if ( a == 0.0D+00 ) then c = 0.0D+00 else c = a * sqrt ( 1.0D+00 + ( b / a ) ** 2 ) end if r8_hypot = c return end function r8_is_in_01 ( a ) !*****************************************************************************80 ! !! R8_IN_01 is TRUE if an R8 is in the range [0,1]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 October 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A, the value. ! ! Output, logical ( kind = 4 ) R8_IS_IN_01, is TRUE if 0 <= A <= 1. ! implicit none real ( kind = 8 ) a logical ( kind = 4 ) r8_is_in_01 logical ( kind = 4 ) value if ( a < 0.0D+00 .or. 1.0D+00 < a ) then value = .false. else value = .true. end if r8_is_in_01 = value return end function r8_is_inf ( r ) !*****************************************************************************80 ! !! R8_IS_INF determines if an R8 represents an infinite value. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 May 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, the number to be checked. ! ! Output, logical ( kind = 4 ) R8_IS_INF, is TRUE if R is an infinite value. ! implicit none real ( kind = 8 ) r logical ( kind = 4 ) r8_is_inf if ( r < 0.0D+00 ) then r8_is_inf = ( r < - huge ( r ) ) else r8_is_inf = ( huge ( r ) < r ) end if return end function r8_is_insignificant ( r, s ) !*****************************************************************************80 ! !! R8_IS_INSIGNIFICANT determines if an R8 is insignificant. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, the number to be compared against. ! ! Input, real ( kind = 8 ) S, the number to be compared. ! ! Output, logical ( kind = 4 ) R8_IS_INSIGNIFICANT, is TRUE if S is ! insignificant compared to R. ! implicit none real ( kind = 8 ) r logical ( kind = 4 ) r8_is_insignificant real ( kind = 8 ) s real ( kind = 8 ) t real ( kind = 8 ) tol logical ( kind = 4 ) value value = .true. t = r + s tol = epsilon ( r ) * abs ( r ) if ( tol < abs ( r - t ) ) then value = .false. end if r8_is_insignificant = value return end function r8_is_integer ( r ) !*****************************************************************************80 ! !! R8_IS_INTEGER determines if an R8 represents an integer value. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 September 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, the number to be checked. ! ! Output, logical ( kind = 4 ) R8_IS_INTEGER, is TRUE if R is an integer. ! implicit none integer ( kind = 4 ), parameter :: i4_huge = 2147483647 real ( kind = 8 ) r logical ( kind = 4 ) r8_is_integer logical ( kind = 4 ) value if ( real ( i4_huge, kind = 8 ) < r ) then value = .false. else if ( r < - real ( i4_huge, kind = 8 ) ) then value = .false. else if ( r == real ( int ( r ), kind = 8 ) ) then value = .true. else value = .false. end if r8_is_integer = value return end function r8_is_nan ( r ) !*****************************************************************************80 ! !! R8_IS_NAN determines if an R8 represents a NaN value. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 May 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, the number to be checked. ! ! Output, logical ( kind = 4 ) R8_IS_NAN, is TRUE if R is a NaN ! implicit none real ( kind = 8 ) r logical ( kind = 4 ) r8_is_nan r8_is_nan = ( r /= r ) return end function r8_log_2 ( x ) !*****************************************************************************80 ! !! R8_LOG_2 returns the logarithm base 2 of an R8. ! ! Discussion: ! ! value = Log ( |X| ) / Log ( 2.0 ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 August 2002 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose base 2 logarithm is desired. ! X should not be 0. ! ! Output, real ( kind = 8 ) R8_LOG_2, the logarithm base 2 of the absolute ! value of X. It should be true that |X| = 2^R8_LOG_2. ! implicit none real ( kind = 8 ) r8_log_2 real ( kind = 8 ) x if ( x == 0.0D+00 ) then r8_log_2 = - huge ( x ) else r8_log_2 = log ( abs ( x ) ) / log ( 2.0D+00 ) end if return end function r8_log_10 ( x ) !*****************************************************************************80 ! !! R8_LOG_10 returns the logarithm base 10 of an R8. ! ! Discussion: ! ! value = Log10 ( |X| ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 August 2002 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose base 2 logarithm is desired. ! X should not be 0. ! ! Output, real ( kind = 8 ) R8_LOG_10, the logarithm base 10 of the absolute ! value of X. It should be true that |X| = 10^R8_LOG_10. ! implicit none real ( kind = 8 ) r8_log_10 real ( kind = 8 ) x if ( x == 0.0D+00 ) then r8_log_10 = - huge ( x ) else r8_log_10 = log10 ( abs ( x ) ) end if return end function r8_log_b ( x, b ) !*****************************************************************************80 ! !! R8_LOG_B returns the logarithm base B of an R8. ! ! Discussion: ! ! value = log ( |X| ) / log ( |B| ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 August 2002 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose base B logarithm is desired. ! X should not be 0. ! ! Input, real ( kind = 8 ) B, the base, which should not be 0, 1 or -1. ! ! Output, real ( kind = 8 ) R8_LOG_B, the logarithm base B of the absolute ! value of X. It should be true that |X| = |B|^R8_LOG_B. ! implicit none real ( kind = 8 ) b real ( kind = 8 ) r8_log_b real ( kind = 8 ) x if ( b == 0.0D+00 .or. b == 1.0D+00 .or. b == - 1.0D+00 ) then r8_log_b = - huge ( x ) else if ( abs ( x ) == 0.0D+00 ) then r8_log_b = - huge ( x ) else r8_log_b = log ( abs ( x ) ) / log ( abs ( b ) ) end if return end subroutine r8_mant ( x, s, r, l ) !*****************************************************************************80 ! !! R8_MANT computes the "mantissa" or "fraction part" of an R8. ! ! Discussion: ! ! X = S * R * 2^L ! ! S is +1 or -1, ! R is an real value between 1.0 and 2.0, ! L is an integer. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 June 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number to be decomposed. ! ! Output, integer ( kind = 4 ) S, the "sign" of the number. ! S will be -1 if X is less than 0, and +1 if X is greater ! than or equal to zero. ! ! Output, real ( kind = 8 ) R, the mantissa of X. R will be greater ! than or equal to 1, and strictly less than 2. The one ! exception occurs if X is zero, in which case R will also ! be zero. ! ! Output, integer ( kind = 4 ) L, the integer part of the logarithm ! (base 2) of X. ! implicit none integer ( kind = 4 ) l real ( kind = 8 ) r integer ( kind = 4 ) s real ( kind = 8 ) x ! ! Determine the sign. ! if ( x < 0.0D+00 ) then s = -1 else s = + 1 end if ! ! Set R to the absolute value of X, and L to zero. ! Then force R to lie between 1 and 2. ! if ( x < 0.0D+00 ) then r = - x else r = + x end if l = 0 ! ! Time to bail out if X is zero. ! if ( x == 0.0D+00 ) then return end if do while ( 2.0D+00 <= r ) r = r / 2.0D+00 l = l + 1 end do do while ( r < 1.0D+00 ) r = r * 2.0D+00 l = l - 1 end do return end function r8_max ( x, y ) !*****************************************************************************80 ! !! R8_MAX returns the maximum of two R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 May 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, Y, the numbers to compare. ! ! Output, real ( kind = 8 ) R8_MAX, the maximum of X and Y. ! implicit none real ( kind = 8 ) r8_max real ( kind = 8 ) x real ( kind = 8 ) y if ( x < y ) then r8_max = y else r8_max = x end if return end function r8_min ( x, y ) !*****************************************************************************80 ! !! R8_MIN returns the minimum of two R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 May 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, Y, the numbers to compare. ! ! Output, real ( kind = 8 ) R8_MIN, the minimum of X and Y. ! implicit none real ( kind = 8 ) r8_min real ( kind = 8 ) x real ( kind = 8 ) y if ( x < y ) then r8_min = x else r8_min = y end if return end function r8_mod ( x, y ) !*****************************************************************************80 ! !! R8_MOD returns the remainder of R8 division. ! ! Discussion: ! ! If ! REM = R8_MOD ( X, Y ) ! RMULT = ( X - REM ) / Y ! then ! X = Y * RMULT + REM ! where REM has the same sign as X, and abs ( REM ) < Y. ! ! Example: ! ! X Y R8_MOD R8_MOD Factorization ! ! 107 50 7 107 = 2 * 50 + 7 ! 107 -50 7 107 = -2 * -50 + 7 ! -107 50 -7 -107 = -2 * 50 - 7 ! -107 -50 -7 -107 = 2 * -50 - 7 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 June 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number to be divided. ! ! Input, real ( kind = 8 ) Y, the number that divides X. ! ! Output, real ( kind = 8 ) R8_MOD, the remainder when X is divided by Y. ! implicit none real ( kind = 8 ) r8_mod real ( kind = 8 ) x real ( kind = 8 ) y if ( y == 0.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_MOD - Fatal error!' write ( *, '(a,g14.6)' ) ' R8_MOD ( X, Y ) called with Y = ', y stop 1 end if r8_mod = x - real ( int ( x / y ), kind = 8 ) * y if ( x < 0.0D+00 .and. 0.0D+00 < r8_mod ) then r8_mod = r8_mod - abs ( y ) else if ( 0.0D+00 < x .and. r8_mod < 0.0D+00 ) then r8_mod = r8_mod + abs ( y ) end if return end function r8_modp ( x, y ) !*****************************************************************************80 ! !! R8_MODP returns the nonnegative remainder of R8 division. ! ! Discussion: ! ! If ! REM = R8_MODP ( X, Y ) ! RMULT = ( X - REM ) / Y ! then ! X = Y * RMULT + REM ! where REM is always nonnegative. ! ! The MOD function computes a result with the same sign as the ! quantity being divided. Thus, suppose you had an angle A, ! and you wanted to ensure that it was between 0 and 360. ! Then mod(A,360.0) would do, if A was positive, but if A ! was negative, your result would be between -360 and 0. ! ! On the other hand, R8_MODP(A,360.0) is between 0 and 360, always. ! ! Example: ! ! X Y MOD R8_MODP R8_MODP Factorization ! ! 107 50 7 7 107 = 2 * 50 + 7 ! 107 -50 7 7 107 = -2 * -50 + 7 ! -107 50 -7 43 -107 = -3 * 50 + 43 ! -107 -50 -7 43 -107 = 3 * -50 + 43 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 October 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number to be divided. ! ! Input, real ( kind = 8 ) Y, the number that divides X. ! ! Output, real ( kind = 8 ) R8_MODP, the nonnegative remainder ! when X is divided by Y. ! implicit none real ( kind = 8 ) r8_modp real ( kind = 8 ) x real ( kind = 8 ) y if ( y == 0.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_MODP - Fatal error!' write ( *, '(a,g14.6)' ) ' R8_MODP ( X, Y ) called with Y = ', y stop 1 end if r8_modp = mod ( x, y ) if ( r8_modp < 0.0D+00 ) then r8_modp = r8_modp + abs ( y ) end if return end function r8_mop ( i ) !*****************************************************************************80 ! !! R8_MOP returns the I-th power of -1 as an R8. ! ! Discussion: ! ! An R8 is a real ( kind = 8 ) value. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 November 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) I, the power of -1. ! ! Output, real ( kind = 8 ) R8_MOP, the I-th power of -1. ! implicit none integer ( kind = 4 ) i real ( kind = 8 ) r8_mop real ( kind = 8 ) value if ( mod ( i, 2 ) == 0 ) then value = + 1.0D+00 else value = - 1.0D+00 end if r8_mop = value return end function r8_nan ( ) !*****************************************************************************80 ! !! R8_NAN returns the value of NaN as an R8. ! ! Discussion: ! ! Some compilers have a NaN function built in. ! ! So far I haven't found a simple portable way of implementing ! this function! ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 August 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Output, real ( kind = 8 ) R8_NAN, a NaN value. ! ! use, intrinsic :: iso_fortran_env ! use, intrinsic :: ieee_arithmetic real ( kind = 8 ) r8_nan integer ( kind = 4 ) r8_nint ! ! My "clever" compiler won't compile: ! ! r8_nan = 0.0D+00 / 0.0D+00 ! ! Maybe something stupid like this will work. ! r8_nan = r8_nint ( 0.1D+00) / r8_nint ( 0.2D+00 ) ! ! Something smart like this is just bound not to work. ! ! value = ieee_value ( value, ieee_quiet_nan ) ! r8_nan = value return end function r8_nint ( x ) !*****************************************************************************80 ! !! R8_NINT returns the nearest integer to an R8. ! ! Example: ! ! X R8_NINT ! ! 1.3 1 ! 1.4 1 ! 1.5 1 or 2 ! 1.6 2 ! 0.0 0 ! -0.7 -1 ! -1.1 -1 ! -1.6 -2 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 September 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the value. ! ! Output, integer ( kind = 4 ) R8_NINT, the nearest integer to X. ! implicit none integer ( kind = 4 ) r8_nint integer ( kind = 4 ) s real ( kind = 8 ) x if ( x < 0.0D+00 ) then s = - 1 else s = + 1 end if r8_nint = s * int ( abs ( x ) + 0.5D+00 ) return end function r8_normal_01 ( seed ) !*****************************************************************************80 ! !! R8_NORMAL_01 returns a unit pseudonormal R8. ! ! Discussion: ! ! The standard normal probability distribution function (PDF) has ! mean 0 and standard deviation 1. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 August 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) SEED, a seed for the random number ! generator. ! ! Output, real ( kind = 8 ) R8_NORMAL_01, a sample of the standard ! normal PDF. ! implicit none real ( kind = 8 ) r1 real ( kind = 8 ) r2 real ( kind = 8 ) r8_normal_01 real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) r8_uniform_01 integer ( kind = 4 ) seed real ( kind = 8 ) x r1 = r8_uniform_01 ( seed ) r2 = r8_uniform_01 ( seed ) x = sqrt ( - 2.0D+00 * log ( r1 ) ) * cos ( 2.0D+00 * r8_pi * r2 ) r8_normal_01 = x return end function r8_normal_ab ( a, b, seed ) !*****************************************************************************80 ! !! R8_NORMAL_AB returns a scaled pseudonormal R8. ! ! Discussion: ! ! The normal probability distribution function (PDF) is sampled, ! with mean A and standard deviation B. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 August 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A, the mean of the PDF. ! ! Input, real ( kind = 8 ) B, the standard deviation of the PDF. ! ! Input/output, integer ( kind = 4 ) SEED, a seed for the random number ! generator. ! ! Output, real ( kind = 8 ) R8_NORMAL_AB, a sample of the normal PDF. ! implicit none real ( kind = 8 ) a real ( kind = 8 ) b real ( kind = 8 ) r1 real ( kind = 8 ) r2 real ( kind = 8 ) r8_normal_ab real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) r8_uniform_01 integer ( kind = 4 ) seed real ( kind = 8 ) x r1 = r8_uniform_01 ( seed ) r2 = r8_uniform_01 ( seed ) x = sqrt ( - 2.0D+00 * log ( r1 ) ) * cos ( 2.0D+00 * r8_pi * r2 ) r8_normal_ab = a + b * x return end function r8_nth_root ( x, n ) !*****************************************************************************80 ! !! R8_NTH_ROOT returns the nth-root of an R8. ! ! Discussion: ! ! The nth root of X is x^(1/n) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 August 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose Nth root is desired. ! ! Input, integer ( kind = 4 ) N, the index of the root. ! ! Output, real ( kind = 8 ) R8_NTH_ROOT, the Nth root of X. ! implicit none real ( kind = 8 ) e integer ( kind = 4 ) n ! real ( kind = 8 ) r8_nan real ( kind = 8 ) r8_nth_root real ( kind = 8 ) value real ( kind = 8 ) x ! ! Potential Error 1: 0^0 ! But we will use it as 1. ! if ( x == 0.0D+00 .and. n == 0 ) then value = 1.0; ! ! Error 2: 0^(negative power) ! Returned as Inf on my system. ! else if ( x == 0.0D+00 .and. n < 0 ) then value = x ** ( 1.0 / real ( n, kind = 8 ) ) ! ! Error 3: (negative)^(even strictly positive root) ! Returned as NaN on my system. ! else if ( x < 0.0D+00 .and. mod ( n, 2 ) == 0 .and. 0 < n ) then value = x ** ( 1.0 / real ( n, kind = 8 ) ) ! ! X^0 = 1 ! else if ( n == 0 ) then value = 1.0D+00 ! ! X^1 = X ! else if ( n == 1 ) then value = x ! ! X^(-1) = 1/X ! else if ( n == -1 ) then value = 1.0D+00 / x else e = 1.0D+00 / abs ( n ) if ( 0.0 < x ) then value = x ** e else if ( x == 0.0D+00 ) then value = 0.0D+00 else value = - ( abs ( x ) ) ** e end if if ( n < 0 ) then value = 1.0D+00 / value end if end if r8_nth_root = value return end function r8_pi ( ) !*****************************************************************************80 ! !! R8_PI returns the value of pi as an R8. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 December 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Output, real ( kind = 8 ) R8_PI, the value of pi. ! implicit none real ( kind = 8 ) r8_pi r8_pi = 3.141592653589793D+00 return end function r8_pi_sqrt ( ) !*****************************************************************************80 ! !! R8_PI_SQRT returns the square root of pi as an R8. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 September 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Output, real ( kind = 8 ) R8_PI_SQRT, the square root of pi. ! implicit none real ( kind = 8 ) r8_pi_sqrt r8_pi_sqrt = 1.7724538509055160273D+00 return end function r8_power ( r, p ) !*****************************************************************************80 ! !! R8_POWER computes the P-th power of an R8. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 October 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, the base. ! ! Input, integer ( kind = 4 ) P, the power, which may be negative. ! ! Output, real ( kind = 8 ) R8_POWER, the value of the P-th power of R. ! implicit none integer ( kind = 4 ) p real ( kind = 8 ) r real ( kind = 8 ) r8_power real ( kind = 8 ) value ! ! Special case. R^0 = 1. ! if ( p == 0 ) then value = 1.0D+00 ! ! Special case. Positive powers of 0 are 0. ! For negative powers of 0, we go ahead and compute R^P, ! relying on the software to complain. ! else if ( r == 0.0D+00 ) then if ( 0 < p ) then value = 0.0D+00 else value = r ** p end if else if ( 1 <= p ) then value = r ** p else value = 1.0D+00 / r ** ( - p ) end if r8_power = value return end subroutine r8_power_fast ( r, p, rp, mults ) !*****************************************************************************80 ! !! R8_POWER_FAST computes an integer power of an R8. ! ! Discussion: ! ! Obviously, R^P can be computed using P-1 multiplications. ! ! However, R^P can also be computed using at most 2*LOG2(P) multiplications. ! To do the calculation this way, let N = LOG2(P). ! Compute A, A^2, A^4, ..., A^N by N-1 successive squarings. ! Start the value of R^P at A, and each time that there is a 1 in ! the binary expansion of P, multiply by the current result of the squarings. ! ! This algorithm is not optimal. For small exponents, and for special ! cases, the result can be computed even more quickly. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 March 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, the base. ! ! Input, integer ( kind = 4 ) P, the power, which may be negative. ! ! Output, real ( kind = 8 ) RP, the value of R^P. ! ! Output, integer ( kind = 4 ) MULTS, the number of multiplications ! and divisions. ! implicit none integer ( kind = 4 ) mults integer ( kind = 4 ) p integer ( kind = 4 ) p_mag integer ( kind = 4 ) p_sign real ( kind = 8 ) r real ( kind = 8 ) r2 real ( kind = 8 ) rp mults = 0 ! ! Special bases. ! if ( r == 1.0D+00 ) then rp = 1.0D+00 return end if if ( r == -1.0D+00 ) then if ( mod ( p, 2 ) == 1 ) then rp = -1.0D+00 else rp = 1.0D+00 end if return end if if ( r == 0.0D+00 ) then if ( p <= 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_POWER_FAST - Fatal error!' write ( *, '(a)' ) ' Base R is zero, and exponent is negative.' write ( *, '(a,i8)' ) ' Exponent P = ', p stop 1 end if rp = 0.0D+00 return end if ! ! Special powers. ! if ( p == -1 ) then rp = 1.0D+00 / r mults = mults + 1 return else if ( p == 0 ) then rp = 1.0D+00 return else if ( p == 1 ) then rp = r return end if ! ! Some work to do. ! p_mag = abs ( p ) p_sign = sign ( 1, p ) rp = 1.0D+00 r2 = r do while ( 0 < p_mag ) if ( mod ( p_mag, 2 ) == 1 ) then rp = rp * r2 mults = mults + 1 end if p_mag = p_mag / 2 r2 = r2 * r2 mults = mults + 1 end do if ( p_sign == -1 ) then rp = 1.0D+00 / rp mults = mults + 1 end if return end subroutine r8_print ( r, title ) !*****************************************************************************80 ! !! R8_PRINT prints an R8. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 August 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, the value. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none real ( kind = 8 ) r character ( len = * ) title write ( *, '(a,2x,g14.6)' ) trim ( title ), r return end function r8_pythag ( a, b ) !*****************************************************************************80 ! !! R8_PYTHAG computes sqrt ( A * A + B * B ) as an R8. ! ! Discussion: ! ! The computation avoids unnecessary overflow and underflow. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 March 2017 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A, B, the values for which sqrt ( A * A + B * B ) ! is desired. ! ! Output, real ( kind = 8 ) R8_PYTHAG, the value of sqrt ( A * A + B * B ). ! implicit none real ( kind = 8 ) a real ( kind = 8 ) a_abs real ( kind = 8 ) b real ( kind = 8 ) b_abs real ( kind = 8 ) r8_pythag a_abs = abs ( a ) b_abs = abs ( b ) if ( b_abs .lt. a_abs ) then r8_pythag = a_abs * sqrt ( 1.0D+00 + ( b_abs / a_abs ) * ( b_abs / a_abs ) ) else if ( b_abs .eq. 0.0D+00 ) then r8_pythag = 0.0D+00 else if ( a_abs .le. b_abs ) then r8_pythag = b_abs * sqrt ( 1.0D+00 + ( a_abs / b_abs ) * ( a_abs / b_abs ) ) end if return end function r8_radians ( degrees ) !*****************************************************************************80 ! !! R8_RADIANS converts an angle from degree to radian measure. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 May 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) DEGREES, the angle measurement in degrees. ! ! Output, real ( kind = 8 ) R8_RADIANS, the angle measurement in radians. ! implicit none real ( kind = 8 ) degrees real ( kind = 8 ), parameter :: r8_pi = 3.1415926535897932384626434D+00 real ( kind = 8 ) r8_radians r8_radians = degrees * r8_pi / 180.0D+00 return end function r8_relu ( x ) !*****************************************************************************80 ! !! R8_RELU evaluates the ReLU function of an R8. ! ! Discussion: ! ! An R8 is a double precision real value. ! ! The ReLU function is a max(x,0.0). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 January 2019 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument. ! ! Output, real ( kind = 8 ) R8_RELU, the function value. ! implicit none real ( kind = 8 ) r8_relu real ( kind = 8 ) x r8_relu = max ( x, 0.0D+00 ); return end function r8_rise ( x, n ) !*****************************************************************************80 ! !! R8_RISE computes the rising factorial function [X]^N. ! ! Discussion: ! ! [X]^N = X * ( X + 1 ) * ( X + 2 ) * ... * ( X + N - 1 ). ! ! Note that the number of ways of arranging N objects in M ordered ! boxes is [M]^N. (Here, the ordering of the objects in each box matters). ! Thus, 2 objects in 2 boxes have the following 6 possible arrangements: ! ! -|12, 1|2, 12|-, -|21, 2|1, 21|-. ! ! Moreover, the number of non-decreasing maps from a set of ! N to a set of M ordered elements is [M]^N / N!. Thus the set of ! nondecreasing maps from (1,2,3) to (a,b,c,d) is the 20 elements: ! ! aaa, abb, acc, add, aab, abc, acd, aac, abd, aad ! bbb, bcc, bdd, bbc, bcd, bbd, ccc, cdd, ccd, ddd. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 May 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument of the rising factorial function. ! ! Input, integer ( kind = 4 ) N, the order of the rising factorial function. ! If N = 0, RISE = 1, if N = 1, RISE = X. Note that if N is ! negative, a "falling" factorial will be computed. ! ! Output, real ( kind = 8 ) R8_RISE, the value of the rising factorial ! function. ! implicit none real ( kind = 8 ) arg integer ( kind = 4 ) i integer ( kind = 4 ) n real ( kind = 8 ) r8_rise real ( kind = 8 ) value real ( kind = 8 ) x value = 1.0D+00 arg = x if ( 0 < n ) then do i = 1, n value = value * arg arg = arg + 1.0D+00 end do else if ( n < 0 ) then do i = -1, n, -1 value = value * arg arg = arg - 1.0D+00 end do end if r8_rise = value return end subroutine r8_rise_values ( n_data, x, n, f ) !*****************************************************************************80 ! !! R8_RISE_VALUES returns some values of the rising factorial function. ! ! Discussion: ! ! Pochhammer(X,Y) = Gamma(X+Y) / Gamma(X) ! ! For integer arguments, Pochhammer(M,N) = ( M + N - 1 )! / ( N - 1 )! ! ! In Mathematica, the function can be evaluated by: ! ! Pochhammer[X,Y] ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 December 2014 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz, Irene Stegun, ! Handbook of Mathematical Functions, ! National Bureau of Standards, 1964, ! ISBN: 0-486-61272-4, ! LC: QA47.A34. ! ! Stephen Wolfram, ! The Mathematica Book, ! Fourth Edition, ! Cambridge University Press, 1999, ! ISBN: 0-521-64314-7, ! LC: QA76.95.W65. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N_DATA. The user sets N_DATA to 0 ! before the first call. On each call, the routine increments N_DATA by 1, ! and returns the corresponding data; when there is no more data, the ! output value of N_DATA will be 0 again. ! ! Output, real ( kind = 8 ) X, integer ( kind = 4 ) N, the arguments. ! ! Output, real ( kind = 8 ) F, the value of the function. ! implicit none integer ( kind = 4 ), parameter :: n_max = 15 real ( kind = 8 ), save, dimension ( n_max ) :: f_vec = (/ & 1680.000000000000D+00, & 1962.597656250000D+00, & 2279.062500000000D+00, & 2631.972656250000D+00, & 3024.000000000000D+00, & 1.000000000000000D+00, & 7.500000000000000D+00, & 63.75000000000000D+00, & 605.6250000000000D+00, & 6359.062500000000D+00, & 73129.21875000000D+00, & 914115.2343750000D+00, & 1.234055566406250D+07, & 1.789380571289063D+08, & 2.773539885498047D+09 /) real ( kind = 8 ) f integer ( kind = 4 ) n integer ( kind = 4 ) n_data integer ( kind = 4 ), save, dimension ( n_max ) :: n_vec = (/ & 4, & 4, & 4, & 4, & 4, & 0, & 1, & 2, & 3, & 4, & 5, & 6, & 7, & 8, & 9 /) real ( kind = 8 ) x real ( kind = 8 ), save, dimension ( n_max ) :: x_vec = (/ & 5.00D+00, & 5.25D+00, & 5.50D+00, & 5.75D+00, & 6.00D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00, & 7.50D+00 /) if ( n_data < 0 ) then n_data = 0 end if n_data = n_data + 1 if ( n_max < n_data ) then n_data = 0 x = 0.0D+00 n = 0 f = 0.0D+00 else x = x_vec(n_data) n = n_vec(n_data) f = f_vec(n_data) end if return end function r8_round ( x ) !*****************************************************************************80 ! !! R8_ROUND sets an R8 to the nearest integral value. ! ! Example: ! ! X R8_ROUND ! ! 1.3 1.0 ! 1.4 1.0 ! 1.5 1.0 or 2.0 ! 1.6 2.0 ! 0.0 0.0 ! -0.7 -1.0 ! -1.1 -1.0 ! -1.6 -2.0 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 October 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the value. ! ! Output, real ( kind = 8 ) R8_ROUND, the rounded value. ! implicit none real ( kind = 8 ) r8_round real ( kind = 8 ) value real ( kind = 8 ) x if ( x < 0.0D+00 ) then value = - real ( int ( - x + 0.5D+00 ), kind = 8 ) else value = real ( int ( + x + 0.5D+00 ), kind = 8 ) end if r8_round = value return end function r8_round_i4 ( x ) !*****************************************************************************80 ! !! R8_ROUND_I4 sets an R8 to the nearest integral value, returning an I4 ! ! Example: ! ! X R8_ROUND_I4 ! ! 1.3 1 ! 1.4 1 ! 1.5 1 or 2 ! 1.6 2 ! 0.0 0 ! -0.7 -1 ! -1.1 -1 ! -1.6 -2 ! ! Discussion: ! ! In FORTRAN90, we rely on the fact that, for positive X, int ( X ) ! is the "floor" function, returning the largest integer less than ! or equal to X. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 March 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the value. ! ! Output, integer ( kind = 4 ) R8_ROUND_I4, the rounded value. ! implicit none integer ( kind = 4 ) r8_round_i4 integer ( kind = 4 ) value real ( kind = 8 ) x if ( x < 0.0D+00 ) then value = - int ( - x + 0.5D+00 ) else value = int ( + x + 0.5D+00 ) end if r8_round_i4 = value return end subroutine r8_round2 ( nplace, x, xround ) !*****************************************************************************80 ! !! R8_ROUND2 rounds an R8 in base 2. ! ! Discussion: ! ! Assume that the input quantity X has the form ! ! X = S * J * 2^L ! ! where S is plus or minus 1, L is an integer, and J is a binary ! mantissa which is either exactly zero, or greater than or equal ! to 0.5 and strictly less than 1.0. ! ! Then on return, XROUND will satisfy ! ! XROUND = S * K * 2^L ! ! where S and L are unchanged, and K is a binary mantissa which ! agrees with J in the first NPLACE binary digits and is zero ! thereafter. ! ! If NPLACE is 0, XROUND will always be zero. ! ! If NPLACE is 1, the mantissa of XROUND will be 0 or 0.5. ! ! If NPLACE is 2, the mantissa of XROUND will be 0, 0.25, 0.50, ! or 0.75. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 March 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) NPLACE, the number of binary digits to ! preserve. NPLACE should be 0 or positive. ! ! Input, real ( kind = 8 ) X, the number to be decomposed. ! ! Output, real ( kind = 8 ) XROUND, the rounded value of X. ! implicit none integer ( kind = 4 ) iplace integer ( kind = 4 ) l integer ( kind = 4 ) nplace integer ( kind = 4 ) s real ( kind = 8 ) x real ( kind = 8 ) xmant real ( kind = 8 ) xround real ( kind = 8 ) xtemp xround = 0.0D+00 ! ! 1: Handle the special case of 0. ! if ( x == 0.0D+00 ) then return end if if ( nplace <= 0 ) then return end if ! ! 2: Determine the sign S. ! if ( 0.0D+00 < x ) then s = 1 xtemp = x else s = -1 xtemp = -x end if ! ! 3: Force XTEMP to lie between 1 and 2, and compute the ! logarithm L. ! l = 0 do while ( 2.0D+00 <= xtemp ) xtemp = xtemp / 2.0D+00 l = l + 1 end do do while ( xtemp < 1.0D+00 ) xtemp = xtemp * 2.0D+00 l = l - 1 end do ! ! 4: Strip out the digits of the mantissa as XMANT, and decrease L. ! xmant = 0.0D+00 iplace = 0 do xmant = 2.0D+00 * xmant if ( 1.0D+00 <= xtemp ) then xmant = xmant + 1.0D+00 xtemp = xtemp - 1.0D+00 end if iplace = iplace + 1 if ( xtemp == 0.0D+00 .or. nplace <= iplace ) then xround = s * xmant * 2.0D+00**l exit end if l = l - 1 xtemp = xtemp * 2.0D+00 end do return end subroutine r8_roundb ( base, nplace, x, xround ) !*****************************************************************************80 ! !! R8_ROUNDB rounds an R8 in a given base. ! ! Discussion: ! ! The code does not seem to do a good job of rounding when ! the base is negative. ! ! Assume that the input quantity X has the form ! ! X = S * J * BASE^L ! ! where S is plus or minus 1, L is an integer, and J is a ! mantissa base BASE which is either exactly zero, or greater ! than or equal to (1/BASE) and less than 1.0. ! ! Then on return, XROUND will satisfy ! ! XROUND = S * K * BASE^L ! ! where S and L are unchanged, and K is a mantissa base BASE ! which agrees with J in the first NPLACE digits and is zero ! thereafter. ! ! Note that because of rounding, for most bases, most numbers ! with a fractional quantities cannot be stored exactly in the ! computer, and hence will have trailing "bogus" digits. ! ! If NPLACE is 0, XROUND will always be zero. ! ! If NPLACE is 1, the mantissa of XROUND will be 0, ! 1/BASE, 2/BASE, ..., (BASE-1)/BASE. ! ! If NPLACE is 2, the mantissa of XROUND will be 0, ! BASE/BASE^2, (BASE+1)/BASE^2, ..., ! BASE^2-2/BASE^2, BASE^2-1/BASE^2. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 March 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) BASE, the base of the arithmetic. ! BASE must not be zero. Theoretically, BASE may be negative. ! ! Input, integer ( kind = 4 ) NPLACE, the number of digits base BASE to ! preserve. NPLACE should be 0 or positive. ! ! Input, real ( kind = 8 ) X, the number to be decomposed. ! ! Output, real ( kind = 8 ) XROUND, the rounded value of X. ! implicit none integer ( kind = 4 ) base integer ( kind = 4 ) iplace integer ( kind = 4 ) is integer ( kind = 4 ) js integer ( kind = 4 ) l integer ( kind = 4 ) nplace real ( kind = 8 ) x real ( kind = 8 ) xmant real ( kind = 8 ) xround real ( kind = 8 ) xtemp xround = 0.0D+00 ! ! 0: Error checks. ! if ( base == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_ROUNDB - Fatal error!' write ( *, '(a)' ) ' The base BASE cannot be zero.' stop 1 end if ! ! 1: Handle the special case of 0. ! if ( x == 0.0D+00 ) then return end if if ( nplace <= 0 ) then return end if ! ! 2: Determine the sign IS. ! if ( 0.0D+00 < x ) then is = 1 xtemp = x else is = -1 xtemp = -x end if ! ! 3: Force XTEMP to lie between 1 and ABS(BASE), and compute the ! logarithm L. ! l = 0 do while ( abs ( base ) <= abs ( xtemp ) ) xtemp = xtemp / real ( base, kind = 8 ) if ( xtemp < 0.0D+00 ) then is = -is xtemp = -xtemp end if l = l + 1 end do do while ( abs ( xtemp ) < 1.0D+00 ) xtemp = xtemp * base if ( xtemp < 0.0D+00 ) then is = -is xtemp = -xtemp end if l = l - 1 end do ! ! 4: Now strip out the digits of the mantissa as XMANT, and ! decrease L. ! xmant = 0.0D+00 iplace = 0 js = is do xmant = base * xmant if ( xmant < 0.0D+00 ) then js = -js xmant = -xmant end if if ( 1.0D+00 <= xtemp ) then xmant = xmant + int ( xtemp ) xtemp = xtemp - int ( xtemp ) end if iplace = iplace + 1 if ( xtemp == 0.0D+00 .or. nplace <= iplace ) then xround = js * xmant * ( real ( base, kind = 8 ) ) ** l exit end if l = l - 1 xtemp = xtemp * base if ( xtemp < 0.0D+00 ) then is = -is xtemp = -xtemp end if end do return end subroutine r8_roundx ( nplace, x, xround ) !*****************************************************************************80 ! !! R8_ROUNDX rounds an R8 in base 10. ! ! Discussion: ! ! Assume that the input quantity X has the form ! ! X = S * J * 10^L ! ! where S is plus or minus 1, L is an integer, and J is a decimal ! mantissa which is either exactly zero, or greater than or equal ! to 0.1 and less than 1.0. ! ! Then on return, XROUND will satisfy ! ! XROUND = S * K * 10^L ! ! where S and L are unchanged, and K is a decimal mantissa which ! agrees with J in the first NPLACE decimal digits and is zero ! thereafter. ! ! Note that because of rounding, most decimal fraction quantities ! cannot be stored exactly in the computer, and hence will have ! trailing "bogus" digits. ! ! If NPLACE is 0, XROUND will always be zero. ! ! If NPLACE is 1, the mantissa of XROUND will be 0, 0.1, ! 0.2, ..., or 0.9. ! ! If NPLACE is 2, the mantissa of XROUND will be 0, 0.01, 0.02, ! 0.03, ..., 0.98, 0.99. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 March 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) NPLACE, the number of decimal digits to ! preserve. NPLACE should be 0 or positive. ! ! Input, real ( kind = 8 ) X, the number to be decomposed. ! ! Output, real ( kind = 8 ) XROUND, the rounded value of X. ! implicit none integer ( kind = 4 ) iplace integer ( kind = 4 ) is integer ( kind = 4 ) l integer ( kind = 4 ) nplace real ( kind = 8 ) x real ( kind = 8 ) xmant real ( kind = 8 ) xround real ( kind = 8 ) xtemp xround = 0.0D+00 ! ! 1: Handle the special case of 0. ! if ( x == 0.0D+00 ) then return end if if ( nplace <= 0 ) then return end if ! ! 2: Determine the sign IS. ! if ( 0.0D+00 < x ) then is = 1 xtemp = x else is = -1 xtemp = -x end if ! ! 3: Force XTEMP to lie between 1 and 10, and compute the ! logarithm L. ! l = 0 do while ( 10.0D+00 <= x ) xtemp = xtemp / 10.0D+00 l = l + 1 end do do while ( xtemp < 1.0D+00 ) xtemp = xtemp * 10.0D+00 l = l - 1 end do ! ! 4: Now strip out the digits of the mantissa as XMANT, and ! decrease L. ! xmant = 0.0D+00 iplace = 0 do xmant = 10.0D+00 * xmant if ( 1.0D+00 <= xtemp ) then xmant = xmant + int ( xtemp ) xtemp = xtemp - int ( xtemp ) end if iplace = iplace + 1 if ( xtemp == 0.0D+00 .or. nplace <= iplace ) then xround = is * xmant * ( 10.0D+00**l ) exit end if l = l - 1 xtemp = xtemp * 10.0D+00 end do return end function r8_secd ( degrees ) !*****************************************************************************80 ! !! R8_SECD returns the secant of an angle given in degrees. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 July 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) DEGREES, the angle in degrees. ! ! Output, real ( kind = 8 ) R8_SECD, the secant of the angle. ! implicit none real ( kind = 8 ) degrees real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) r8_secd real ( kind = 8 ) radians radians = r8_pi * ( degrees / 180.0D+00 ) r8_secd = 1.0D+00 / cos ( radians ) return end function r8_sech ( x ) !*****************************************************************************80 ! !! R8_SECH evaluates the hyperbolic secant, while avoiding COSH overflow. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 August 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument of the function. ! ! Output, real ( kind = 8 ) R8_SECH, the value of the function. ! implicit none real ( kind = 8 ), parameter :: log_huge = 80.0D+00 real ( kind = 8 ) r8_sech real ( kind = 8 ) x if ( log_huge < abs ( x ) ) then r8_sech = 0.0D+00 else r8_sech = 1.0D+00 / cosh ( x ) end if return end function r8_sigmoid ( l, b, m, x ) !*****************************************************************************80 ! !! r8_sigmoid evaluates the sigmoid or logistic function. ! ! Discussion: ! ! An R8 is a real ( kind = 8 ) value. ! ! The sigmoid function is useful for classification problems in ! machine learning. Its value is always between 0 and 1. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 October 2019 ! ! Author: ! ! John Burkardt ! ! Input: ! ! real ( kind = 8 ) l, the maximum value of the function. This is often 1. ! ! real ( kind = 8 ) b, the cutoff value, where the function equals l/2. ! This is often 0. ! ! real ( kind = 8 ) m, the slope, which determines the steepness of the curve ! and the width of the uncertainty interval. This is often 1. ! ! real ( kind = 8 ) x, the argument. ! ! Output: ! ! real ( kind = 8 ) r8_sigmoid, the value. ! implicit none real ( kind = 8 ) b real ( kind = 8 ) l real ( kind = 8 ) m real ( kind = 8 ) r8_sigmoid real ( kind = 8 ) x r8_sigmoid = l / ( 1.0D+00 + exp ( - m * ( x - b ) ) ) return end function r8_sign ( x ) !*****************************************************************************80 ! !! R8_SIGN returns the sign of an R8. ! ! Discussion: ! ! value = -1 if X < 0; ! value = +1 if X => 0. ! ! Note that the standard FORTRAN90 "sign" function is more complicated. ! In particular, ! ! Z = sign ( X, Y ) ! ! means that ! ! Z = |X| if 0 <= Y; ! - |X| if Y < 0; ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 March 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose sign is desired. ! ! Output, real ( kind = 8 ) R8_SIGN, the sign of X: ! implicit none real ( kind = 8 ) r8_sign real ( kind = 8 ) value real ( kind = 8 ) x if ( x < 0.0D+00 ) then value = -1.0D+00 else value = +1.0D+00 end if r8_sign = value return end function r8_sign3 ( x ) !*****************************************************************************80 ! !! R8_SIGN3 returns the three-way sign of an R8. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 28 September 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose sign is desired. ! ! Output, real ( kind = 8 ) R8_SIGN3, the sign of X: ! implicit none real ( kind = 8 ) r8_sign3 real ( kind = 8 ) value real ( kind = 8 ) x if ( x < 0.0D+00 ) then value = -1.0D+00 else if ( x == 0.0D+00 ) then value = 0.0D+00 else value = +1.0D+00 end if r8_sign3 = value return end function r8_sign_char ( x ) !*****************************************************************************80 ! !! R8_SIGN_CHAR returns a character indicating the sign of an R8. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 28 April 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the number whose sign is desired. ! ! Output, character R8_SIGN_CHAR, the sign of X, '-', '0' or '+'. ! implicit none character r8_sign_char character value real ( kind = 8 ) x if ( x < 0.0D+00 ) then value = '-' else if ( x == 0.0D+00 ) then value = '0' else value = '+' end if r8_sign_char = value return end function r8_sign_match ( r1, r2 ) !*****************************************************************************80 ! !! R8_SIGN_MATCH is TRUE if two R8's are of the same sign. ! ! Discussion: ! ! This test could be coded numerically as ! ! if ( 0 <= r1 * r2 ) then ... ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 April 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R1, R2, the values to check. ! ! Output, logical ( kind = 4 ) R8_SIGN_MATCH, is TRUE if ! ( R1 <= 0 and R2 <= 0 ) or ( 0 <= R1 and 0 <= R2 ). ! implicit none real ( kind = 8 ) r1 real ( kind = 8 ) r2 logical ( kind = 4 ) r8_sign_match r8_sign_match = ( r1 <= 0.0D+00 .and. r2 <= 0.0D+00 ) .or. & ( 0.0D+00 <= r1 .and. 0.0D+00 <= r2 ) return end function r8_sign_match_strict ( r1, r2 ) !*****************************************************************************80 ! !! R8_SIGN_MATCH_STRICT is TRUE if two R8's are of the same strict sign. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 28 April 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R1, R2, the values to check. ! ! Output, logical ( kind = 4 ) R8_SIGN_MATCH_STRICT, is TRUE if the ! signs match. ! implicit none real ( kind = 8 ) r1 real ( kind = 8 ) r2 logical ( kind = 4 ) r8_sign_match_strict r8_sign_match_strict = & ( r1 < 0.0D+00 .and. r2 < 0.0D+00 ) .or. & ( r1 == 0.0D+00 .and. r2 == 0.0D+00 ) .or. & ( 0.0D+00 < r1 .and. 0.0D+00 < r2 ) return end function r8_sign_opposite ( r1, r2 ) !*****************************************************************************80 ! !! R8_SIGN_OPPOSITE is TRUE if two R8's are not of the same sign. ! ! Discussion: ! ! This test could be coded numerically as ! ! if ( r1 * r2 <= 0.0 ) then ... ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 June 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R1, R2, the values to check. ! ! Output, logical ( kind = 4 ) R8_SIGN_OPPOSITE, is TRUE if ! ( R1 <= 0 and 0 <= R2 ) or ( R2 <= 0 and 0 <= R1 ). ! implicit none real ( kind = 8 ) r1 real ( kind = 8 ) r2 logical ( kind = 4 ) r8_sign_opposite r8_sign_opposite = ( r1 <= 0.0D+00 .and. 0.0D+00 <= r2 ) .or. & ( r2 <= 0.0D+00 .and. 0.0D+00 <= r1 ) return end function r8_sign_opposite_strict ( r1, r2 ) !*****************************************************************************80 ! !! R8_SIGN_OPPOSITE_STRICT is TRUE if two R8's are strictly of opposite sign. ! ! Discussion: ! ! This test could be coded numerically as ! ! if ( r1 * r2 < 0.0 ) then ... ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 June 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R1, R2, the values to check. ! ! Output, logical ( kind = 4 ) R8_SIGN_OPPOSITE_STRICT, is TRUE if ! ( R1 < 0 and 0 < R2 ) or ( R2 < 0 and 0 < R1 ). ! implicit none real ( kind = 8 ) r1 real ( kind = 8 ) r2 logical ( kind = 4 ) r8_sign_opposite_strict r8_sign_opposite_strict = ( r1 < 0.0D+00 .and. 0.0D+00 < r2 ) .or. & ( r2 < 0.0D+00 .and. 0.0D+00 < r1 ) return end subroutine r8_sincos_sum ( a, b, d, e, f ) !*****************************************************************************80 ! !! R8_SINCOS_SUM simplifies a*sin(cx)+b*cos(cx). ! ! Discussion: ! ! The expression ! a * sin ( c * x ) + b * cos ( c * x ) ! can be rewritten as ! d * sin ( c * x + e ) ! or ! d * cos ( c * x + f ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 January 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A, B, the coefficients in the linear combination. ! ! Output, real ( kind = 8 ) D, E, F, the new coefficient, and the shift for ! sine or for cosine. ! implicit none real ( kind = 8 ) a real ( kind = 8 ) b real ( kind = 8 ) d real ( kind = 8 ) e real ( kind = 8 ) f real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 d = sqrt ( a * a + b * b ) e = atan2 ( b, a ) f = atan2 ( b, a ) - r8_pi / 2.0D+00 if ( f < - r8_pi ) then f = f + 2.0D+00 * r8_pi end if return end function r8_sind ( degrees ) !*****************************************************************************80 ! !! R8_SIND returns the sine of an angle given in degrees. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 July 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) DEGREES, the angle in degrees. ! ! Output, real ( kind = 8 ) R8_SIND, the sine of the angle. ! implicit none real ( kind = 8 ) degrees real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) r8_sind real ( kind = 8 ) radians radians = r8_pi * ( degrees / 180.0D+00 ) r8_sind = sin ( radians ) return end function r8_softplus ( x ) !*****************************************************************************80 ! !! R8_SOFTPLUS evaluates the softplus function of an R8. ! ! Discussion: ! ! An R8 is a double precision real value. ! ! The softplus function is a smoothed (differentiable) version of max(x,0.0). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 September 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument. ! ! Output, real ( kind = 8 ) VALUE, the function value. ! implicit none real ( kind = 8 ) r8_softplus real ( kind = 8 ) value real ( kind = 8 ) x if ( x <= -36.841D+00 ) then value = 0.0D+00 else if ( +36.841D+00 <= x ) then value = x else value = log ( 1.0D+00 + exp ( x ) ) end if r8_softplus = value return end function r8_sqrt_i4 ( i ) !*****************************************************************************80 ! !! R8_SQRT_I4 returns the square root of an I4 as an R8. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 June 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) I, the number whose square root is desired. ! ! Output, real ( kind = 8 ) R8_SQRT_I4, the value of sqrt(I). ! implicit none integer ( kind = 4 ) i real ( kind = 8 ) r8_sqrt_i4 r8_sqrt_i4 = sqrt ( real ( i, kind = 8 ) ) return end subroutine r8_swap ( x, y ) !*****************************************************************************80 ! !! R8_SWAP swaps two R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 22 December 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, real ( kind = 8 ) X, Y. On output, the values of X and ! Y have been interchanged. ! implicit none real ( kind = 8 ) x real ( kind = 8 ) y real ( kind = 8 ) z z = x x = y y = z return end subroutine r8_swap3 ( x, y, z ) !*****************************************************************************80 ! !! R8_SWAP3 swaps three R8's. ! ! Example: ! ! Input: ! ! X = 1, Y = 2, Z = 3 ! ! Output: ! ! X = 2, Y = 3, Z = 1 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 June 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, real ( kind = 8 ) X, Y, Z, three values to be swapped. ! implicit none real ( kind = 8 ) w real ( kind = 8 ) x real ( kind = 8 ) y real ( kind = 8 ) z w = x x = y y = z z = w return end function r8_tand ( degrees ) !*****************************************************************************80 ! !! R8_TAND returns the tangent of an angle given in degrees. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 July 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) DEGREES, the angle in degrees. ! ! Output, real ( kind = 8 ) R8_TAND, the tangent of the angle. ! implicit none real ( kind = 8 ) degrees real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) r8_tand real ( kind = 8 ) radians radians = r8_pi * ( degrees / 180.0D+00 ) r8_tand = tan ( radians ) return end function r8_tiny ( ) !*****************************************************************************80 ! !! R8_TINY returns a very small but positive R8. ! ! Discussion: ! ! FORTRAN90 provides a built-in routine TINY ( X ) that ! is more suitable for this purpose, returning the smallest positive ! but normalized real number. ! ! This routine does NOT try to provide an accurate value for TINY. ! Instead, it simply returns a "reasonable" value, that is, a rather ! small, but representable, real number. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 March 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Output, real ( kind = 8 ) R8_TINY, a "tiny" value. ! implicit none real ( kind = 8 ) r8_tiny r8_tiny = 1.0D-30 return end subroutine r8_to_r8_discrete ( r, rmin, rmax, nr, rd ) !*****************************************************************************80 ! !! R8_TO_R8_DISCRETE maps R to RD in [RMIN, RMAX] with NR possible values. ! ! Formula: ! ! if ( R < RMIN ) then ! RD = RMIN ! else if ( RMAX < R ) then ! RD = RMAX ! else ! T = nint ( ( NR - 1 ) * ( R - RMIN ) / ( RMAX - RMIN ) ) ! RD = RMIN + T * ( RMAX - RMIN ) / real ( NR - 1 ) ! ! In the special case where NR = 1, when ! ! XD = 0.5 * ( RMAX + RMIN ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 April 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, the number to be converted. ! ! Input, real ( kind = 8 ) RMAX, RMIN, the maximum and minimum ! values for RD. ! ! Input, integer ( kind = 4 ) NR, the number of allowed values for XD. ! NR should be at least 1. ! ! Output, real ( kind = 8 ) RD, the corresponding discrete value. ! implicit none integer ( kind = 4 ) f integer ( kind = 4 ) nr real ( kind = 8 ) r real ( kind = 8 ) rd real ( kind = 8 ) rmax real ( kind = 8 ) rmin ! ! Check for errors. ! if ( nr < 1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_TO_R8_DISCRETE - Fatal error!' write ( *, '(a,i8)' ) ' NR = ', nr write ( *, '(a)' ) ' but NR must be at least 1.' stop 1 end if if ( nr == 1 ) then rd = 0.5D+00 * ( rmin + rmax ) return end if if ( rmax == rmin ) then rd = rmax return end if f = nint ( real ( nr, kind = 8 ) * ( rmax - r ) / ( rmax - rmin ) ) f = max ( f, 0 ) f = min ( f, nr ) rd = ( real ( f, kind = 8 ) * rmin & + real ( nr - f, kind = 8 ) * rmax ) & / real ( nr, kind = 8 ) return end subroutine r8_to_dhms ( r, d, h, m, s ) !*****************************************************************************80 ! !! R8_TO_DHMS converts decimal days into days, hours, minutes, seconds. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 April 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, a decimal number representing a time ! period measured in days. ! ! Output, integer ( kind = 4 ) D, H, M, S, the equivalent number of days, ! hours, minutes and seconds. ! implicit none integer ( kind = 4 ) d integer ( kind = 4 ) h integer ( kind = 4 ) m real ( kind = 8 ) r real ( kind = 8 ) r_copy integer ( kind = 4 ) s r_copy = abs ( r ) d = int ( r_copy ) r_copy = r_copy - d r_copy = 24.0D+00 * r_copy h = int ( r_copy ) r_copy = r_copy - h r_copy = 60.0D+00 * r_copy m = int ( r_copy ) r_copy = r_copy - m r_copy = 60.0D+00 * r_copy s = int ( r_copy ) if ( r < 0.0D+00 ) then d = -d h = -h m = -m s = -s end if return end subroutine r8_to_i4 ( xmin, xmax, x, ixmin, ixmax, ix ) !*****************************************************************************80 ! !! R8_TO_I4 maps X in [XMIN, XMAX] to integer IX in [IXMIN, IXMAX]. ! ! Formula: ! ! IX := IXMIN + ( IXMAX - IXMIN ) * ( X - XMIN ) / ( XMAX - XMIN ) ! IX := min ( IX, max ( IXMIN, IXMAX ) ) ! IX := max ( IX, min ( IXMIN, IXMAX ) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 April 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) XMIN, XMAX, the range. XMAX and ! XMIN must not be equal. It is not necessary that XMIN be less than XMAX. ! ! Input, real ( kind = 8 ) X, the number to be converted. ! ! Input, integer ( kind = 4 ) IXMIN, IXMAX, the allowed range of the output ! variable. IXMAX corresponds to XMAX, and IXMIN to XMIN. ! It is not necessary that IXMIN be less than IXMAX. ! ! Output, integer ( kind = 4 ) IX, the value in the range [IXMIN,IXMAX] that ! corresponds to X. ! implicit none integer ( kind = 4 ) ix integer ( kind = 4 ) ixmax integer ( kind = 4 ) ixmin real ( kind = 8 ) temp real ( kind = 8 ) x real ( kind = 8 ) xmax real ( kind = 8 ) xmin if ( xmax == xmin ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_TO_I4 - Fatal error!' write ( *, '(a)' ) ' XMAX = XMIN, making a zero divisor.' write ( *, '(a,g14.6)' ) ' XMAX = ', xmax write ( *, '(a,g14.6)' ) ' XMIN = ', xmin stop 1 end if temp = & ( ( xmax - x ) * real ( ixmin, kind = 8 ) & + ( x - xmin ) * real ( ixmax, kind = 8 ) ) & / ( xmax - xmin ) if ( 0.0D+00 <= temp ) then temp = temp + 0.5D+00 else temp = temp - 0.5D+00 end if ix = int ( temp ) return end function r8_uniform_01 ( seed ) !*****************************************************************************80 ! !! R8_UNIFORM_01 returns a unit pseudorandom R8. ! ! Discussion: ! ! An R8 is a real ( kind = 8 ) value. ! ! For now, the input quantity SEED is an integer variable. ! ! This routine implements the recursion ! ! seed = 16807 * seed mod ( 2^31 - 1 ) ! r8_uniform_01 = seed / ( 2^31 - 1 ) ! ! The integer arithmetic never requires more than 32 bits, ! including a sign bit. ! ! If the initial seed is 12345, then the first three computations are ! ! Input Output R8_UNIFORM_01 ! SEED SEED ! ! 12345 207482415 0.096616 ! 207482415 1790989824 0.833995 ! 1790989824 2035175616 0.947702 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 July 2006 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Paul Bratley, Bennett Fox, Linus Schrage, ! A Guide to Simulation, ! Springer Verlag, pages 201-202, 1983. ! ! Pierre L'Ecuyer, ! Random Number Generation, ! in Handbook of Simulation, ! edited by Jerry Banks, ! Wiley Interscience, page 95, 1998. ! ! Bennett Fox, ! Algorithm 647: ! Implementation and Relative Efficiency of Quasirandom ! Sequence Generators, ! ACM Transactions on Mathematical Software, ! Volume 12, Number 4, pages 362-376, 1986. ! ! Peter Lewis, Allen Goodman, James Miller ! A Pseudo-Random Number Generator for the System/360, ! IBM Systems Journal, ! Volume 8, pages 136-143, 1969. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) SEED, the "seed" value, which should ! NOT be 0. On output, SEED has been updated. ! ! Output, real ( kind = 8 ) R8_UNIFORM_01, a new pseudorandom variate, ! strictly between 0 and 1. ! implicit none integer ( kind = 4 ), parameter :: i4_huge = 2147483647 integer ( kind = 4 ) k real ( kind = 8 ) r8_uniform_01 integer ( kind = 4 ) seed if ( seed == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_UNIFORM_01 - Fatal error!' write ( *, '(a)' ) ' Input value of SEED = 0.' stop 1 end if k = seed / 127773 seed = 16807 * ( seed - k * 127773 ) - k * 2836 if ( seed < 0 ) then seed = seed + i4_huge end if r8_uniform_01 = real ( seed, kind = 8 ) * 4.656612875D-10 return end function r8_uniform_ab ( a, b, seed ) !*****************************************************************************80 ! !! R8_UNIFORM_AB returns a scaled pseudorandom R8. ! ! Discussion: ! ! An R8 is a real ( kind = 8 ) value. ! ! For now, the input quantity SEED is an integer variable. ! ! The pseudorandom number should be uniformly distributed ! between A and B. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 July 2006 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A, B, the limits of the interval. ! ! Input/output, integer ( kind = 4 ) SEED, the "seed" value, which should ! NOT be 0. On output, SEED has been updated. ! ! Output, real ( kind = 8 ) R8_UNIFORM_AB, a number strictly between A and B. ! implicit none real ( kind = 8 ) a real ( kind = 8 ) b integer ( kind = 4 ), parameter :: i4_huge = 2147483647 integer ( kind = 4 ) k real ( kind = 8 ) r8_uniform_ab integer ( kind = 4 ) seed if ( seed == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8_UNIFORM_AB - Fatal error!' write ( *, '(a)' ) ' Input value of SEED = 0.' stop 1 end if k = seed / 127773 seed = 16807 * ( seed - k * 127773 ) - k * 2836 if ( seed < 0 ) then seed = seed + i4_huge end if r8_uniform_ab = a + ( b - a ) * real ( seed, kind = 8 ) * 4.656612875D-10 return end subroutine r8_unswap3 ( x, y, z ) !*****************************************************************************80 ! !! R8_UNSWAP3 unswaps three R8's. ! ! Example: ! ! Input: ! ! X = 2, Y = 3, Z = 1 ! ! Output: ! ! X = 1, Y = 2, Z = 3 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 July 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, real ( kind = 8 ) X, Y, Z, three values to be swapped. ! implicit none real ( kind = 8 ) w real ( kind = 8 ) x real ( kind = 8 ) y real ( kind = 8 ) z w = z z = y y = x x = w return end function r8_walsh_1d ( x, digit ) !*****************************************************************************80 ! !! R8_WALSH_1D evaluates the Walsh function. ! ! Discussion: ! ! Consider the binary representation of X, and number the digits ! in descending order, from leading to lowest, with the units digit ! being numbered 0. ! ! The Walsh function W(J)(X) is equal to the J-th binary digit of X. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 May 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, the argument of the Walsh function. ! ! Input, integer ( kind = 4 ) DIGIT, the index of the Walsh function. ! ! Output, real ( kind = 8 ) R8_WALSH_1D, the value of the Walsh function. ! implicit none integer ( kind = 4 ) digit integer ( kind = 4 ) n real ( kind = 8 ) r8_walsh_1d real ( kind = 8 ) x real ( kind = 8 ) x_copy ! ! Hide the effect of the sign of X. ! x_copy = abs ( x ) ! ! If DIGIT is positive, divide by 2 DIGIT times. ! If DIGIT is negative, multiply by 2 (-DIGIT) times. ! x_copy = x_copy / 2.0D+00 ** digit ! ! Make it an integer. ! Because it's positive, and we're using INT, we don't change the ! units digit. ! n = int ( x_copy ) ! ! Is the units digit odd or even? ! if ( mod ( n, 2 ) == 0 ) then r8_walsh_1d = 0.0D+00 else r8_walsh_1d = 1.0D+00 end if return end function r8_wrap ( r, rlo, rhi ) !*****************************************************************************80 ! !! R8_WRAP forces an R8 to lie between given limits by wrapping. ! ! Discussion: ! ! An R8 is a real ( kind = 8 ) value. ! ! Example: ! ! RLO = 4.0, RHI = 8.0 ! ! R Value ! ! -2 8 ! -1 4 ! 0 5 ! 1 6 ! 2 7 ! 3 8 ! 4 4 ! 5 5 ! 6 6 ! 7 7 ! 8 8 ! 9 4 ! 10 5 ! 11 6 ! 12 7 ! 13 8 ! 14 4 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 July 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, a value. ! ! Input, real ( kind = 8 ) RLO, RHI, the desired bounds. ! ! Output, real ( kind = 8 ) R8_WRAP, a "wrapped" version of the value. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) r real ( kind = 8 ) r8_wrap real ( kind = 8 ) rhi real ( kind = 8 ) rhi2 real ( kind = 8 ) rlo real ( kind = 8 ) rlo2 real ( kind = 8 ) rwide real ( kind = 8 ) value ! ! Guarantee RLO2 < RHI2. ! rlo2 = min ( rlo, rhi ) rhi2 = max ( rlo, rhi ) ! ! Find the width. ! rwide = rhi2 - rlo2 ! ! Add enough copies of (RHI2-RLO2) to R so that the ! result ends up in the interval RLO2 - RHI2. ! if ( rwide == 0.0D+00 ) then value = rlo else if ( r < rlo2 ) then n = int ( ( rlo2 - r ) / rwide ) + 1 value = r + n * rwide if ( value == rhi ) then value = rlo end if else n = int ( ( r - rlo2 ) / rwide ) value = r - n * rwide if ( value == rlo ) then value = rhi end if end if r8_wrap = value return end function r82_dist_l2 ( a1, a2 ) !*****************************************************************************80 ! !! R82_DIST_L2 returns the L2 distance between a pair of R82's. ! ! Discussion: ! ! An R82 is a vector of type R8, with two entries. ! ! The vector L2 norm is defined as: ! ! sqrt ( sum ( 1 <= I <= N ) A(I) * A(I) ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A1(2), A2(2), the vectors. ! ! Output, real ( kind = 8 ) R82_DIST_L2, the L2 norm of the distance ! between A1 and A2. ! implicit none real ( kind = 8 ) a1(2) real ( kind = 8 ) a2(2) real ( kind = 8 ) r82_dist_l2 r82_dist_l2 = sqrt ( sum ( ( a1(1:2) - a2(1:2) ) ** 2 ) ) return end function r82_eq ( a1, a2 ) !*****************************************************************************80 ! !! R82_EQ == ( A1 == A2 ) for two R82's. ! ! Discussion: ! ! An R82 is a vector of type R8, with two entries. ! ! The comparison is lexicographic. ! ! A1 == A2 <=> A1(1) == A2(1) and A1(2) == A2(2). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A1(2), A2(2), two R82 vectors to be compared. ! ! Output, logical ( kind = 4 ) R82_EQ, is TRUE if and only if A1 == A2. ! implicit none real ( kind = 8 ) a1(2) real ( kind = 8 ) a2(2) logical ( kind = 4 ) r82_eq if ( all ( a1(1:2) == a2(1:2) ) ) then r82_eq = .true. else r82_eq = .false. end if return end function r82_ge ( a1, a2 ) !*****************************************************************************80 ! !! R82_GE == ( A1 >= A2 ) for two R82's. ! ! Discussion: ! ! An R82 is a vector of type R8, with two entries. ! ! The comparison is lexicographic. ! ! A1 >= A2 <=> A1(1) > A2(1) or ( A1(1) == A2(1) and A1(2) >= A2(2) ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A1(2), A2(2), R82 vectors to be compared. ! ! Output, logical ( kind = 4 ) R82_GE, is TRUE if and only if A1 >= A2. ! implicit none real ( kind = 8 ) a1(2) real ( kind = 8 ) a2(2) integer ( kind = 4 ) i logical ( kind = 4 ) r82_ge r82_ge = .true. do i = 1, 2 if ( a2(i) < a1(i) ) then r82_ge = .true. exit else if ( a1(i) < a2(i) ) then r82_ge = .false. exit end if end do return end function r82_gt ( a1, a2 ) !*****************************************************************************80 ! !! R82_GT == ( A1 > A2 ) for two R82's. ! ! Discussion: ! ! An R82 is a vector of type R2, with two entries. ! ! The comparison is lexicographic. ! ! A1 > A2 <=> A1(1) > A2(1) or ( A1(1) == A2(1) and A1(2) > A2(2) ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A1(2), A2(2), R82 vectors to be compared. ! ! Output, logical ( kind = 4 ) R82_GT, is TRUE if and only if A1 > A2. ! implicit none real ( kind = 8 ) a1(2) real ( kind = 8 ) a2(2) integer ( kind = 4 ) i logical ( kind = 4 ) r82_gt r82_gt = .false. do i = 1, 2 if ( a2(i) < a1(i) ) then r82_gt = .true. exit else if ( a1(i) < a2(i) ) then r82_gt = .false. exit end if end do return end function r82_le ( a1, a2 ) !*****************************************************************************80 ! !! R82_LE == ( A1 <= A2 ) for two R82's. ! ! Discussion: ! ! An R82 is a vector of type R8, with two entries. ! ! The comparison is lexicographic. ! ! A1 <= A2 <=> A1(1) < A2(1) or ( A1(1) == A2(1) and A1(2) <= A2(2) ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A1(2), A2(2), R82 vectors to be compared. ! ! Output, logical ( kind = 4 ) R82_LE, is TRUE if and only if A1 <= A2. ! implicit none real ( kind = 8 ) a1(2) real ( kind = 8 ) a2(2) integer ( kind = 4 ) i logical ( kind = 4 ) r82_le r82_le = .true. do i = 1, 2 if ( a1(i) < a2(i) ) then r82_le = .true. exit else if ( a2(i) < a1(i) ) then r82_le = .false. exit end if end do return end function r82_lt ( a1, a2 ) !*****************************************************************************80 ! !! R82_LT == ( A1 < A2 ) for two R82's. ! ! Discussion: ! ! An R82 is a vector of type R8, with two entries. ! ! The comparison is lexicographic. ! ! A1 < A2 <=> A1(1) < A2(1) or ( A1(1) == A2(1) and A1(2) < A2(2) ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A1(2), A2(2), R82 vectors to be compared. ! ! Output, logical ( kind = 4 ) R82_LT, is TRUE if and only if A1 < A2. ! implicit none integer ( kind = 4 ), parameter :: dim_num = 2 real ( kind = 8 ) a1(dim_num) real ( kind = 8 ) a2(dim_num) integer ( kind = 4 ) i logical ( kind = 4 ) r82_lt r82_lt = .false. do i = 1, dim_num if ( a1(i) < a2(i) ) then r82_lt = .true. exit else if ( a2(i) < a1(i) ) then r82_lt = .false. exit end if end do return end function r82_ne ( a1, a2 ) !*****************************************************************************80 ! !! R82_NE == ( A1 /= A2 ) for two R82's. ! ! Discussion: ! ! An R82 is a vector of type R8, with two entries. ! ! The comparison is lexicographic. ! ! A1 /= A2 <=> A1(1) /= A2(1) or A1(2) /= A2(2). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A1(2), A2(2), R82 vectors to be compared. ! ! Output, logical ( kind = 4 ) R82_NE, is TRUE if and only if A1 /= A2. ! implicit none integer ( kind = 4 ), parameter :: dim_num = 2 real ( kind = 8 ) a1(dim_num) real ( kind = 8 ) a2(dim_num) logical ( kind = 4 ) r82_ne if ( any ( a1(1:dim_num) /= a2(1:dim_num) ) ) then r82_ne = .true. else r82_ne = .false. end if return end function r82_norm ( a ) !*****************************************************************************80 ! !! R82_NORM returns the Euclidean norm of an R82. ! ! Discussion: ! ! An R82 is a vector of type R8, with two entries. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A(2), the vector. ! ! Output, real ( kind = 8 ) R82_NORM, the norm. ! implicit none real ( kind = 8 ) a(2) real ( kind = 8 ) r82_norm r82_norm = sqrt ( a(1) * a(1) + a(2) * a(2) ) return end subroutine r82_normalize ( a ) !*****************************************************************************80 ! !! R82_NORMALIZE Euclidean normalizes an R82. ! ! Discussion: ! ! An R82 is a vector of type R8, with two entries. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 October 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, real ( kind = 8 ) A(2), the components of the vector. ! implicit none real ( kind = 8 ) a(2) real ( kind = 8 ) norm norm = sqrt ( a(1) * a(1) + a(2) * a(2) ) if ( norm /= 0.0D+00 ) then a(1:2) = a(1:2) / norm end if return end subroutine r82_print ( a, title ) !*****************************************************************************80 ! !! R82_PRINT prints an R82. ! ! Discussion: ! ! An R82 is a vector of type R8, with two entries. ! ! A format is used which suggests a coordinate pair: ! ! Example: ! ! Center : ( 1.23, 7.45 ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A(2), the coordinates of the vector. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none real ( kind = 8 ) a(2) character ( len = * ) title if ( 0 < len_trim ( title ) ) then write ( *, '( 2x, a, a4, g14.6, a1, g14.6, a1 )' ) & trim ( title ), ' : (', a(1), ',', a(2), ')' else write ( *, '( 2x, a1, g14.6, a1, g14.6, a1 )' ) '(', a(1), ',', a(2), ')' end if return end subroutine r82_swap ( x, y ) !*****************************************************************************80 ! !! R82_SWAP swaps two R82 values. ! ! Discussion: ! ! An R82 is a vector of type R8, with two entries. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, real ( kind = 8 ) X(2), Y(2). On output, the values of X and ! Y have been interchanged. ! implicit none integer ( kind = 4 ), parameter :: dim_num = 2 real ( kind = 8 ) x(dim_num) real ( kind = 8 ) y(dim_num) real ( kind = 8 ) z(dim_num) z(1:dim_num) = x(1:dim_num) x(1:dim_num) = y(1:dim_num) y(1:dim_num) = z(1:dim_num) return end subroutine r82_uniform_ab ( b, c, seed, a ) !*****************************************************************************80 ! !! R82_UNIFORM_AB returns a random R82 value in a given range. ! ! Discussion: ! ! An R82 is a vector of type R8, with two entries. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) B, C, the minimum and maximum values. ! ! Input/output, integer ( kind = 4 ) SEED, a seed for the random number ! generator. ! ! Output, real ( kind = 8 ) A(2), the randomly chosen value. ! implicit none integer ( kind = 4 ), parameter :: dim_num = 2 real ( kind = 8 ) a(dim_num) real ( kind = 8 ) b real ( kind = 8 ) c real ( kind = 8 ) r8_uniform_ab integer ( kind = 4 ) i integer ( kind = 4 ) seed do i = 1, dim_num a(i) = r8_uniform_ab ( b, c, seed ) end do return end subroutine r82col_print_part ( n, a, max_print, title ) !*****************************************************************************80 ! !! R82COL_PRINT_PART prints "part" of an R82COL. ! ! Discussion: ! ! An R82COL is an (N,2) array of R8's. ! ! The user specifies MAX_PRINT, the maximum number of lines to print. ! ! If N, the size of the vector, is no more than MAX_PRINT, then ! the entire vector is printed, one entry per line. ! ! Otherwise, if possible, the first MAX_PRINT-2 entries are printed, ! followed by a line of periods suggesting an omission, ! and the last entry. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of the vector. ! ! Input, real ( kind = 8 ) A(N,2), the vector to be printed. ! ! Input, integer ( kind = 4 ) MAX_PRINT, the maximum number of lines ! to print. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,2) integer ( kind = 4 ) i integer ( kind = 4 ) max_print character ( len = * ) title if ( max_print <= 0 ) then return end if if ( n <= 0 ) then return end if write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' if ( n <= max_print ) then do i = 1, n write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6)' ) i, ':', a(i,1:2) end do else if ( 3 <= max_print ) then do i = 1, max_print - 2 write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6)' ) i, ':', a(i,1:2) end do write ( *, '(a)' ) ' ........ .............. ..............' i = n write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6)' ) i, ':', a(i,1:2) else do i = 1, max_print - 1 write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6)' ) i, ':', a(i,1:2) end do i = max_print write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,a)' ) i, ':', a(i,1:2), & '...more entries...' end if return end subroutine r82row_max ( n, a, amax ) !*****************************************************************************80 ! !! R82ROW_MAX returns the maximum value in an R82ROW. ! ! Discussion: ! ! An R82ROW is a (2,N) array of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 June 2006 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(2,N), the array. ! ! Output, real ( kind = 8 ) AMAX(2); the largest entries in each row. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(2,n) real ( kind = 8 ) amax(2) amax(1) = maxval ( a(1,1:n) ) amax(2) = maxval ( a(2,1:n) ) return end subroutine r82row_min ( n, a, amin ) !*****************************************************************************80 ! !! R82ROW_MIN returns the minimum value in an R82ROW. ! ! Discussion: ! ! An R82ROW is a (2,N) array of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 June 2006 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(2,N), the array. ! ! Output, real ( kind = 8 ) AMIN(2); the smallest entries in each row. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(2,n) real ( kind = 8 ) amin(2) amin(1) = minval ( a(1,1:n) ) amin(2) = minval ( a(2,1:n) ) return end subroutine r82row_order_type ( n, a, order ) !*****************************************************************************80 ! !! R82ROW_ORDER_TYPE finds the order type of an R82ROW. ! ! Discussion: ! ! An R82ROW is a (2,N) array of R8's. ! ! The dictionary or lexicographic ordering is used. ! ! (X1,Y1) < (X2,Y2) <=> X1 < X2 or ( X1 = X2 and Y1 < Y2). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of the array. ! ! Input, real ( kind = 8 ) A(2,N), the array to be checked. ! ! Output, integer ( kind = 4 ) ORDER, order indicator: ! -1, no discernable order; ! 0, all entries are equal; ! 1, ascending order; ! 2, strictly ascending order; ! 3, descending order; ! 4, strictly descending order. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ), parameter :: dim_num = 2 real ( kind = 8 ) a(dim_num,n) integer ( kind = 4 ) i integer ( kind = 4 ) order ! ! Search for the first value not equal to A(1,1). ! i = 1 do i = i + 1 if ( n < i ) then order = 0 return end if if ( & a(1,1) < a(1,i) .or. & ( a(1,1) == a(1,i) .and. a(2,1) < a(2,i) ) & ) then if ( i == 2 ) then order = 2 else order = 1 end if exit else if ( & a(1,i) < a(1,1) .or. & ( a(1,i) == a(1,1) .and. a(2,i) < a(2,1) ) & ) then if ( i == 2 ) then order = 4 else order = 3 end if exit end if end do ! ! Now we have a "direction". Examine subsequent entries. ! do i = i + 1 if ( n < i ) then exit end if if ( order == 1 ) then if ( & a(1,i) < a(1,i-1) .or. & ( a(1,i) == a(1,i-1) .and. a(2,i) < a(2,i-1) ) & ) then order = -1 exit end if else if ( order == 2 ) then if ( & a(1,i) < a(1,i-1) .or. & ( a(1,i) == a(1,i-1) .and. a(2,i) < a(2,i-1) ) & ) then order = -1 exit else if ( & a(1,i) == a(1,i-1) .and. a(2,i) == a(2,i-1) ) then order = 1 end if else if ( order == 3 ) then if ( & a(1,i-1) < a(1,i) .or. & ( a(1,i-1) == a(1,i) .and. a(2,i-1) < a(2,i) ) & ) then order = -1 exit end if else if ( order == 4 ) then if ( & a(1,i-1) < a(1,i) .or. & ( a(1,i-1) == a(1,i) .and. a(2,i-1) < a(2,i) ) & ) then order = -1 exit else if ( a(1,i) == a(1,i-1) .and. a(2,i) == a(2,i-1) ) then order = 3 end if end if end do return end subroutine r82row_part_quick_a ( n, a, l, r ) !*****************************************************************************80 ! !! R82ROW_PART_QUICK_A reorders an R82ROW as part of a quick sort. ! ! Discussion: ! ! An R82ROW is a (2,N) array of R8's. ! ! The routine reorders the entries of A. Using A(1:2,1) as a ! key, all entries of A that are less than or equal to the key will ! precede the key, which precedes all entries that are greater than the key. ! ! Example: ! ! Input: ! ! N = 8 ! ! A = ( (2,4), (8,8), (6,2), (0,2), (10,6), (10,0), (0,6), (4,8) ) ! ! Output: ! ! L = 2, R = 4 ! ! A = ( (0,2), (0,6), (2,4), (8,8), (6,2), (10,6), (10,0), (4,8) ) ! ----------- ---------------------------------- ! LEFT KEY RIGHT ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of A. ! ! Input/output, real ( kind = 8 ) A(2,N). On input, the array to be checked. ! On output, A has been reordered as described above. ! ! Output, integer ( kind = 4 ) L, R, the indices of A that define the three ! segments. Let KEY = the input value of A(1:2,1). Then ! I <= L A(1:2,I) < KEY; ! L < I < R A(1:2,I) = KEY; ! R <= I KEY < A(1:2,I). ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ), parameter :: dim_num = 2 real ( kind = 8 ) a(dim_num,n) integer ( kind = 4 ) i real ( kind = 8 ) key(dim_num) integer ( kind = 4 ) l integer ( kind = 4 ) m integer ( kind = 4 ) r logical ( kind = 4 ) r8vec_eq logical ( kind = 4 ) r8vec_gt logical ( kind = 4 ) r8vec_lt if ( n < 1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R82ROW_PART_QUICK_A - Fatal error!' write ( *, '(a)' ) ' N < 1.' write ( *, '(a,i8)' ) ' N = ', n stop 1 else if ( n == 1 ) then l = 0 r = 2 return end if key(1:dim_num) = a(1:dim_num,1) m = 1 ! ! The elements of unknown size have indices between L+1 and R-1. ! l = 1 r = n + 1 do i = 2, n if ( r8vec_gt ( dim_num, a(1:dim_num,l+1), key(1:dim_num) ) ) then r = r - 1 call r8vec_swap ( dim_num, a(1:dim_num,r), a(1:dim_num,l+1) ) else if ( r8vec_eq ( dim_num, a(1:dim_num,l+1), key(1:dim_num) ) ) then m = m + 1 call r8vec_swap ( dim_num, a(1:dim_num,m), a(1:dim_num,l+1) ) l = l + 1 else if ( r8vec_lt ( dim_num, a(1:dim_num,l+1), key(1:dim_num) ) ) then l = l + 1 end if end do ! ! Now shift small elements to the left, and KEY elements to center. ! do i = 1, l - m a(1:dim_num,i) = a(1:dim_num,i+m) end do l = l - m do i = 1, dim_num a(i,l+1:l+m) = key(i) end do return end subroutine r82row_permute ( n, p, a ) !*****************************************************************************80 ! !! R82ROW_PERMUTE permutes an R82ROW in place. ! ! Discussion: ! ! An R82ROW is a (2,N) array of R8's. ! ! The same logic can be used to permute an array of objects of any ! arithmetic type, or an array of objects of any complexity. The only ! temporary storage required is enough to store a single object. The number ! of data movements made is N + the number of cycles of order 2 or more, ! which is never more than N + N/2. ! ! Example: ! ! Input: ! ! N = 5 ! P = ( 2, 4, 5, 1, 3 ) ! A = ( 1.0, 2.0, 3.0, 4.0, 5.0 ) ! (11.0, 22.0, 33.0, 44.0, 55.0 ) ! ! Output: ! ! A = ( 2.0, 4.0, 5.0, 1.0, 3.0 ) ! ( 22.0, 44.0, 55.0, 11.0, 33.0 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 March 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of objects. ! ! Input, integer ( kind = 4 ) P(N), the permutation. P(I) = J means ! that the I-th element of the output array should be the J-th ! element of the input array. ! ! Input/output, real ( kind = 8 ) A(2,N), the array to be permuted. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ), parameter :: dim_num = 2 real ( kind = 8 ) a(2,n) real ( kind = 8 ) a_temp(2) integer ( kind = 4 ) ierror integer ( kind = 4 ) iget integer ( kind = 4 ) iput integer ( kind = 4 ) istart integer ( kind = 4 ) p(n) integer ( kind = 4 ) perm1_check ierror = perm1_check ( n, p ) if ( ierror .ne. 0 ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'R82ROW_PERMUTE - Fatal error!' write ( *, '(a)' ) ' PERM1_CHECK returned error.' stop 1 end if ! ! Search for the next element of the permutation that has not been used. ! do istart = 1, n if ( p(istart) < 0 ) then else if ( p(istart) == istart ) then p(istart) = - p(istart) else a_temp(1:2) = a(1:2,istart) iget = istart ! ! Copy the new value into the vacated entry. ! do iput = iget iget = p(iget) p(iput) = - p(iput) if ( iget < 1 .or. n < iget ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R82ROW_PERMUTE - Fatal error!' write ( *, '(a)' ) ' A permutation index is out of range.' write ( *, '(a,i8,a,i8)' ) ' P(', iput, ') = ', iget stop 1 end if if ( iget == istart ) then a(1:2,iput) = a_temp(1:2) exit end if a(1:2,iput) = a(1:2,iget) end do end if end do ! ! Restore the signs of the entries. ! p(1:n) = - p(1:n) return end subroutine r82row_print ( n, a, title ) !*****************************************************************************80 ! !! R82ROW_PRINT prints an R82ROW. ! ! Discussion: ! ! An R82ROW is a (2,N) array of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of components of the vector. ! ! Input, real ( kind = 8 ) A(2,N), the R82 vector to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ), parameter :: dim_num = 2 real ( kind = 8 ) a(dim_num,n) integer ( kind = 4 ) i character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' do i = 1, n write ( *, '(2x,i8,(5g14.6))' ) i, a(1:dim_num,i) end do return end subroutine r82row_print_part ( n, a, max_print, title ) !*****************************************************************************80 ! !! R82ROW_PRINT_PART prints "part" of an R82ROW. ! ! Discussion: ! ! An R82ROW is a (2,N) array of R8's. ! ! The user specifies MAX_PRINT, the maximum number of lines to print. ! ! If N, the size of the vector, is no more than MAX_PRINT, then ! the entire vector is printed, one entry per line. ! ! Otherwise, if possible, the first MAX_PRINT-2 entries are printed, ! followed by a line of periods suggesting an omission, ! and the last entry. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of the vector. ! ! Input, real ( kind = 8 ) A(2,N), the vector to be printed. ! ! Input, integer ( kind = 4 ) MAX_PRINT, the maximum number of lines ! to print. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(2,n) integer ( kind = 4 ) i integer ( kind = 4 ) max_print character ( len = * ) title if ( max_print <= 0 ) then return end if if ( n <= 0 ) then return end if write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' if ( n <= max_print ) then do i = 1, n write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6)' ) i, ':', a(1:2,i) end do else if ( 3 <= max_print ) then do i = 1, max_print - 2 write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6)' ) i, ':', a(1:2,i) end do write ( *, '(a)' ) ' ........ .............. ..............' i = n write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6)' ) i, ':', a(1:2,i) else do i = 1, max_print - 1 write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6)' ) i, ':', a(1:2,i) end do i = max_print write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,a)' ) i, ':', a(1:2,i), & '...more entries...' end if return end subroutine r82row_sort_heap_index_a ( n, a, indx ) !*****************************************************************************80 ! !! R82ROW_SORT_HEAP_INDEX_A ascending index heaps an R82ROW. ! ! Discussion: ! ! An R82ROW is a (2,N) array of R8's. ! ! The sorting is not actually carried out. Rather an index array is ! created which defines the sorting. This array may be used to sort ! or index the array, or to sort or index related arrays keyed on the ! original array. ! ! Once the index array is computed, the sorting can be carried out ! "implicitly: ! ! A(1:2,INDX(1:N)) is sorted, ! ! or explicitly, by the call ! ! call r82row_permute ( n, indx, a ) ! ! after which A(1:2,I), I = 1 to N is sorted. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(2,N), an array to be index-sorted. ! ! Output, integer ( kind = 4 ) INDX(N), the sort index. The ! I-th element of the sorted array is A(1:2,INDX(I)). ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ), parameter :: dim_num = 2 real ( kind = 8 ) a(dim_num,n) real ( kind = 8 ) aval(dim_num) integer ( kind = 4 ) i integer ( kind = 4 ) indx(n) integer ( kind = 4 ) indxt integer ( kind = 4 ) ir integer ( kind = 4 ) j integer ( kind = 4 ) l if ( n < 1 ) then return end if do i = 1, n indx(i) = i end do if ( n == 1 ) then return end if l = n / 2 + 1 ir = n do if ( 1 < l ) then l = l - 1 indxt = indx(l) aval(1:dim_num) = a(1:dim_num,indxt) else indxt = indx(ir) aval(1:dim_num) = a(1:dim_num,indxt) indx(ir) = indx(1) ir = ir - 1 if ( ir == 1 ) then indx(1) = indxt exit end if end if i = l j = l + l do while ( j <= ir ) if ( j < ir ) then if ( a(1,indx(j)) < a(1,indx(j+1)) .or. & ( a(1,indx(j)) == a(1,indx(j+1)) .and. & a(2,indx(j)) < a(2,indx(j+1)) ) ) then j = j + 1 end if end if if ( aval(1) < a(1,indx(j)) .or. & ( aval(1) == a(1,indx(j)) .and. & aval(2) < a(2,indx(j)) ) ) then indx(i) = indx(j) i = j j = j + j else j = ir + 1 end if end do indx(i) = indxt end do return end subroutine r82row_sort_quick_a ( n, a ) !*****************************************************************************80 ! !! R82ROW_SORT_QUICK_A ascending sorts an R82ROW using quick sort. ! ! Discussion: ! ! An R82ROW is a (2,N) array of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input/output, real ( kind = 8 ) A(2,N). ! On input, the array to be sorted. ! On output, the array has been sorted. ! implicit none integer ( kind = 4 ), parameter :: level_max = 30 integer ( kind = 4 ) n integer ( kind = 4 ), parameter :: dim_num = 2 real ( kind = 8 ) a(dim_num,n) integer ( kind = 4 ) base integer ( kind = 4 ) l_segment integer ( kind = 4 ) level integer ( kind = 4 ) n_segment integer ( kind = 4 ) rsave(level_max) integer ( kind = 4 ) r_segment if ( n < 1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R82ROW_SORT_QUICK_A - Fatal error!' write ( *, '(a)' ) ' N < 1.' write ( *, '(a,i8)' ) ' N = ', n stop 1 else if ( n == 1 ) then return end if level = 1 rsave(level) = n + 1 base = 1 n_segment = n do ! ! Partition the segment. ! call r82row_part_quick_a ( n_segment, a(1,base), l_segment, r_segment ) ! ! If the left segment has more than one element, we need to partition it. ! if ( 1 < l_segment ) then if ( level_max < level ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R82ROW_SORT_QUICK_A - Fatal error!' write ( *, '(a,i8)' ) ' Exceeding recursion maximum of ', level_max stop 1 end if level = level + 1 n_segment = l_segment rsave(level) = r_segment + base - 1 ! ! The left segment and the middle segment are sorted. ! Must the right segment be partitioned? ! else if ( r_segment < n_segment ) then n_segment = n_segment + 1 - r_segment base = base + r_segment - 1 ! ! Otherwise, we back up a level if there is an earlier one. ! else do if ( level <= 1 ) then return end if base = rsave(level) n_segment = rsave(level-1) - rsave(level) level = level - 1 if ( 0 < n_segment ) then exit end if end do end if end do return end function r83_norm ( x, y, z ) !*****************************************************************************80 ! !! R83_NORM returns the Euclidean norm of an R83. ! ! Discussion: ! ! An R83 is a vector of 3 R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, Y, Z, the vector. ! ! Output, real ( kind = 8 ) R83_NORM, the norm of the vector. ! implicit none real ( kind = 8 ) r83_norm real ( kind = 8 ) x real ( kind = 8 ) y real ( kind = 8 ) z r83_norm = sqrt ( x * x + y * y + z * z ) return end subroutine r83_normalize ( x, y, z ) !*****************************************************************************80 ! !! R83_NORMALIZE normalizes an R83. ! ! Discussion: ! ! An R83 is a vector of 3 R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 January 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, real ( kind = 8 ) X, Y, Z, the components of the vector. ! implicit none real ( kind = 8 ) norm real ( kind = 8 ) x real ( kind = 8 ) y real ( kind = 8 ) z norm = sqrt ( x * x + y * y + z * z ) if ( norm /= 0.0D+00 ) then x = x / norm y = y / norm z = z / norm end if return end subroutine r83_print ( x, y, z, title ) !*****************************************************************************80 ! !! R83_PRINT prints an R83. ! ! Discussion: ! ! An R83 is a vector of 3 R8's. ! ! A format is used which suggests a coordinate triple: ! ! Example: ! ! Center : ( 1.23, 7.45, -1.45 ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 03 July 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X, Y, Z, the coordinates of the vector. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none character ( len = * ) title real ( kind = 8 ) x real ( kind = 8 ) y real ( kind = 8 ) z if ( 0 < len_trim ( title ) ) then write ( *, '( 2x, a, a4, g14.6, a1, g14.6, a1, g14.6, a1 )' ) & trim ( title ), ' : (', x, ',', y, ',', z, ')' else write ( *, '( 2x, a1, g14.6, a1, g14.6, a1, g14.6, a1 )' ) & '(', x, ',', y, ',', z, ')' end if return end subroutine r83_swap ( x, y ) !*****************************************************************************80 ! !! R83_SWAP swaps two R83's. ! ! Discussion: ! ! An R83 is a vector of 3 R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, real ( kind = 8 ) X(3), Y(3). On output, the values ! of X and Y have been interchanged. ! implicit none integer ( kind = 4 ), parameter :: dim_num = 3 real ( kind = 8 ) x(dim_num) real ( kind = 8 ) y(dim_num) real ( kind = 8 ) z(dim_num) z(1:dim_num) = x(1:dim_num) x(1:dim_num) = y(1:dim_num) y(1:dim_num) = z(1:dim_num) return end subroutine r83col_print_part ( n, a, max_print, title ) !*****************************************************************************80 ! !! R83COL_PRINT_PART prints "part" of an R83COL. ! ! Discussion: ! ! An R83COL is an (N,3) array of R8's. ! ! The user specifies MAX_PRINT, the maximum number of lines to print. ! ! If N, the size of the vector, is no more than MAX_PRINT, then ! the entire vector is printed, one entry per line. ! ! Otherwise, if possible, the first MAX_PRINT-2 entries are printed, ! followed by a line of periods suggesting an omission, ! and the last entry. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of the vector. ! ! Input, real ( kind = 8 ) A(N,3), the vector to be printed. ! ! Input, integer ( kind = 4 ) MAX_PRINT, the maximum number of lines ! to print. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,3) integer ( kind = 4 ) i integer ( kind = 4 ) max_print character ( len = * ) title if ( max_print <= 0 ) then return end if if ( n <= 0 ) then return end if write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' if ( n <= max_print ) then do i = 1, n write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,g14.6)' ) i, ':', a(i,1:3) end do else if ( 3 <= max_print ) then do i = 1, max_print - 2 write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,g14.6)' ) i, ':', a(i,1:3) end do write ( *, '(a)' ) & ' ........ .............. .............. ..............' i = n write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,g14.6)' ) i, ':', a(i,1:3) else do i = 1, max_print - 1 write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,g14.6)' ) i, ':', a(i,1:3) end do i = max_print write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,g14.6,2x,a)' ) i, ':', a(i,1:3), & '...more entries...' end if return end subroutine r83row_max ( n, a, amax ) !*****************************************************************************80 ! !! R83ROW_MAX returns the maximum value in an R83ROW. ! ! Discussion: ! ! An R83ROW is a (3,N) array of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(3,N), the array. ! ! Output, real ( kind = 8 ) AMAX(3); the largest entries in each row. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(3,n) real ( kind = 8 ) amax(3) integer ( kind = 4 ) i do i = 1, 3 amax(i) = maxval ( a(i,1:n) ) end do return end subroutine r83row_min ( n, a, amin ) !*****************************************************************************80 ! !! R83ROW_MIN returns the minimum value in an R83ROW. ! ! Discussion: ! ! An R83ROW is a (3,N) array of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(3,N), the array. ! ! Output, real ( kind = 8 ) AMIN(3); the smallest entries in each row. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(3,n) real ( kind = 8 ) amin(3) integer ( kind = 4 ) i do i = 1, 3 amin(i) = minval ( a(i,1:n) ) end do return end subroutine r83row_normalize ( n, x ) !*****************************************************************************80 ! !! R83ROW_NORMALIZE normalizes each R83 in an R83ROW. ! ! Discussion: ! ! An R83ROW is a (3,N) array of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of R83 vectors. ! ! Input/output, real ( kind = 8 ) X(3,N), the coordinates of N R83 vectors. ! On output, the nonzero vectors have been scaled to have unit L2 norm. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i real ( kind = 8 ) norm real ( kind = 8 ) x(3,n) do i = 1, n norm = sqrt ( sum ( x(1:3,i) ** 2 ) ) if ( norm /= 0.0D+00 ) then x(1:3,i) = x(1:3,i) / norm end if end do return end subroutine r83row_print_part ( n, a, max_print, title ) !*****************************************************************************80 ! !! R83ROW_PRINT_PART prints "part" of an R83ROW. ! ! Discussion: ! ! An R83ROW is a (3,N) array of R8's. ! ! The user specifies MAX_PRINT, the maximum number of lines to print. ! ! If N, the size of the vector, is no more than MAX_PRINT, then ! the entire vector is printed, one entry per line. ! ! Otherwise, if possible, the first MAX_PRINT-2 entries are printed, ! followed by a line of periods suggesting an omission, ! and the last entry. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of the vector. ! ! Input, real ( kind = 8 ) A(3,N), the vector to be printed. ! ! Input, integer ( kind = 4 ) MAX_PRINT, the maximum number of lines ! to print. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(3,n) integer ( kind = 4 ) i integer ( kind = 4 ) max_print character ( len = * ) title if ( max_print <= 0 ) then return end if if ( n <= 0 ) then return end if write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' if ( n <= max_print ) then do i = 1, n write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,g14.6)' ) i, ':', a(1:3,i) end do else if ( 3 <= max_print ) then do i = 1, max_print - 2 write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,g14.6)' ) i, ':', a(1:3,i) end do write ( *, '(a)' ) & ' ........ .............. .............. ..............' i = n write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,g14.6)' ) i, ':', a(1:3,i) else do i = 1, max_print - 1 write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,g14.6)' ) i, ':', a(1:3,i) end do i = max_print write ( *, '(2x,i8,a,1x,g14.6,2x,g14.6,2x,g14.6,2x,a)' ) i, ':', a(1:3,i), & '...more entries...' end if return end subroutine r84_normalize ( v ) !*****************************************************************************80 ! !! R84_NORMALIZE normalizes an R84. ! ! Discussion: ! ! An R84 is a vector of four R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, real ( kind = 8 ) V(4), the components of the vector. ! implicit none real ( kind = 8 ) norm real ( kind = 8 ) v(4) norm = sqrt ( sum ( v(1:4) ** 2 ) ) if ( norm /= 0.0D+00 ) then v(1:4) = v(1:4) / norm end if return end subroutine r8block_expand_linear ( l, m, n, x, lfat, mfat, nfat, xfat ) !*****************************************************************************80 ! !! R8BLOCK_EXPAND_LINEAR linearly interpolates new data into an R8BLOCK. ! ! Discussion: ! ! An R8BLOCK is a 3D array of R8 values. ! ! In this routine, the expansion is specified by giving the number ! of intermediate values to generate between each pair of original ! data rows and columns. ! ! The interpolation is not actually linear. It uses the functions ! ! 1, x, y, z, xy, xz, yz, xyz. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 October 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) L, M, N, the dimensions of the input data. ! ! Input, real ( kind = 8 ) X(L,M,N), the original data. ! ! Input, integer ( kind = 4 ) LFAT, MFAT, NFAT, the number of data values ! to interpolate original data values in the first, second and third ! dimensions. ! ! Output, real ( kind = 8 ) XFAT(L2,M2,N2), the fattened data, where ! L2 = (L-1)*(LFAT+1)+1, ! M2 = (M-1)*(MFAT+1)+1, ! N2 = (N-1)*(NFAT+1)+1. ! implicit none integer ( kind = 4 ) l integer ( kind = 4 ) lfat integer ( kind = 4 ) m integer ( kind = 4 ) mfat integer ( kind = 4 ) n integer ( kind = 4 ) nfat integer ( kind = 4 ) i integer ( kind = 4 ) ihi integer ( kind = 4 ) ii integer ( kind = 4 ) iii integer ( kind = 4 ) ip1 integer ( kind = 4 ) j integer ( kind = 4 ) jhi integer ( kind = 4 ) jj integer ( kind = 4 ) jjj integer ( kind = 4 ) jp1 integer ( kind = 4 ) k integer ( kind = 4 ) khi integer ( kind = 4 ) kk integer ( kind = 4 ) kkk integer ( kind = 4 ) kp1 real ( kind = 8 ) r real ( kind = 8 ) s real ( kind = 8 ) t real ( kind = 8 ) x(l,m,n) real ( kind = 8 ) x000 real ( kind = 8 ) x001 real ( kind = 8 ) x010 real ( kind = 8 ) x011 real ( kind = 8 ) x100 real ( kind = 8 ) x101 real ( kind = 8 ) x110 real ( kind = 8 ) x111 real ( kind = 8 ) xfat((l-1)*(lfat+1)+1,(m-1)*(mfat+1)+1,(n-1)*(nfat+1)+1) do i = 1, l if ( i < l ) then ihi = lfat else ihi = 0 end if do j = 1, m if ( j < m ) then jhi = mfat else jhi = 0 end if do k = 1, n if ( k < n ) then khi = nfat else khi = 0 end if if ( i < l ) then ip1 = i + 1 else ip1 = i end if if ( j < m ) then jp1 = j + 1 else jp1 = j end if if ( k < n ) then kp1 = k + 1 else kp1 = k end if x000 = x(i,j,k) x001 = x(i,j,kp1) x100 = x(ip1,j,k) x101 = x(ip1,j,kp1) x010 = x(i,jp1,k) x011 = x(i,jp1,kp1) x110 = x(ip1,jp1,k) x111 = x(ip1,jp1,kp1) do ii = 0, ihi r = real ( ii, kind = 8 ) & / real ( ihi + 1, kind = 8 ) do jj = 0, jhi s = real ( jj, kind = 8 ) & / real ( jhi + 1, kind = 8 ) do kk = 0, khi t = real ( kk, kind = 8 ) & / real ( khi + 1, kind = 8 ) iii = 1 + ( i - 1 ) * ( lfat + 1 ) + ii jjj = 1 + ( j - 1 ) * ( mfat + 1 ) + jj kkk = 1 + ( k - 1 ) * ( nfat + 1 ) + kk xfat(iii,jjj,kkk) = & x000 * ( 1.0D+00 - r ) * ( 1.0D+00 - s ) * ( 1.0D+00 - t ) & + x001 * ( 1.0D+00 - r ) * ( 1.0D+00 - s ) * ( t ) & + x010 * ( 1.0D+00 - r ) * ( s ) * ( 1.0D+00 - t ) & + x011 * ( 1.0D+00 - r ) * ( s ) * ( t ) & + x100 * ( r ) * ( 1.0D+00 - s ) * ( 1.0D+00 - t ) & + x101 * ( r ) * ( 1.0D+00 - s ) * ( t ) & + x110 * ( r ) * ( s ) * ( 1.0D+00 - t ) & + x111 * ( r ) * ( s ) * ( t ) end do end do end do end do end do end do return end subroutine r8block_print ( l, m, n, a, title ) !*****************************************************************************80 ! !! R8BLOCK_PRINT prints an R8BLOCK. ! ! Discussion: ! ! An R8BLOCK is a 3D array of R8 values. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 October 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) L, M, N, the dimensions of the block. ! ! Input, real ( kind = 8 ) A(L,M,N), the matrix to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) l integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(l,m,n) integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo integer ( kind = 4 ) k character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) do k = 1, n write ( *, '(a)' ) ' ' write ( *, '(a,i8)' ) ' K = ', k do jlo = 1, m, 5 jhi = min ( jlo + 4, m ) write ( *, '(a)' ) ' ' write ( *, '(10x,5(i8,6x))' ) (j, j = jlo, jhi ) write ( *, '(a)' ) ' ' do i = 1, l write ( *, '(2x,i8,5g14.6)' ) i, a(i,jlo:jhi,k) end do end do end do return end subroutine r8cmat_print ( lda, m, n, a, title ) !*****************************************************************************80 ! !! R8CMAT_PRINT prints an R8CMAT. ! ! Discussion: ! ! An R8CMAT is an MxN array of R8's, stored with a leading dimension ! of LD, and hence accessed either as a double indexed array: ! (I,J) -> (I,J) ! or as a vector: ! (I,J) -> (I+J*LD). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 March 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) LDA, the leading dimension of A. ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(LDA,N), the M by N matrix. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) lda integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(lda,n) character ( len = * ) title call r8cmat_print_some ( lda, m, n, a, 1, 1, m, n, title ) return end subroutine r8cmat_print_some ( lda, m, n, a, ilo, jlo, ihi, jhi, title ) !*****************************************************************************80 ! !! R8CMAT_PRINT_SOME prints some of an R8CMAT. ! ! Discussion: ! ! An R8CMAT is an MxN array of R8's, stored with a leading dimension ! of LD, and hence accessed either as a double indexed array: ! (I,J) -> (I,J) ! or as a vector: ! (I,J) -> (I+J*LD). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 March 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) LDA, the leading dimension of A. ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) A(LDA,N), the M by N matrix. ! ! Input, integer ( kind = 4 ) ILO, JLO, the first row and column to print. ! ! Input, integer ( kind = 4 ) IHI, JHI, the last row and column to print. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ), parameter :: incx = 5 integer ( kind = 4 ) lda integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(lda,n) character ( len = 14 ) ctemp(incx) integer ( kind = 4 ) i integer ( kind = 4 ) i2hi integer ( kind = 4 ) i2lo integer ( kind = 4 ) ihi integer ( kind = 4 ) ilo integer ( kind = 4 ) inc integer ( kind = 4 ) j integer ( kind = 4 ) j2 integer ( kind = 4 ) j2hi integer ( kind = 4 ) j2lo integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) if ( m <= 0 .or. n <= 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) ' (None)' return end if do j2lo = max ( jlo, 1 ), min ( jhi, n ), incx j2hi = j2lo + incx - 1 j2hi = min ( j2hi, n ) j2hi = min ( j2hi, jhi ) inc = j2hi + 1 - j2lo write ( *, '(a)' ) ' ' do j = j2lo, j2hi j2 = j + 1 - j2lo write ( ctemp(j2), '(i8,6x)' ) j end do write ( *, '('' Col '',5a14)' ) ctemp(1:inc) write ( *, '(a)' ) ' Row' write ( *, '(a)' ) ' ' i2lo = max ( ilo, 1 ) i2hi = min ( ihi, m ) do i = i2lo, i2hi do j2 = 1, inc j = j2lo - 1 + j2 if ( a(i,j) == real ( int ( a(i,j) ), kind = 8 ) ) then write ( ctemp(j2), '(f8.0,6x)' ) a(i,j) else write ( ctemp(j2), '(g14.6)' ) a(i,j) end if end do write ( *, '(i5,a,5a14)' ) i, ':', ( ctemp(j), j = 1, inc ) end do end do return end subroutine r8cmat_to_r8mat ( lda, m, n, a1, a2 ) !*****************************************************************************80 ! !! R8CMAT_TO_R8MAT transfers data from an R8CMAT to an R8MAT. ! ! Discussion: ! ! An R8CMAT is an MxN array of R8's, stored with a leading dimension LD, ! accessible as a vector: ! (I,J) -> (I+J*LD). ! or as a doubly-dimensioned array, if declared A(LD,N): ! (I,J) -> A(I,J) ! ! An R8MAT is an MxN array of R8's, ! accessible as a vector: ! (I,J) -> (I+J*M). ! or as a doubly-dimensioned array, if declared A(M,N): ! (I,J) -> A(I,J) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 March 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) LDA, the leading dimension of A1. ! ! Input, integer ( kind = 4 ) M, the number of rows of data. ! M <= LDA. ! ! Input, integer ( kind = 4 ) N, the number of columns of data. ! ! Input, real ( kind = 8 ) A1(LDA,N), the matrix to be copied. ! ! Output, real ( kind = 8 ) A2(M,N), a copy of the ! information in the MxN submatrix of A1. ! implicit none integer ( kind = 4 ) lda integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a1(lda,n) real ( kind = 8 ) a2(m,n) a2(1:m,1:n) = a1(1:m,1:n) return end subroutine r8int_to_r8int ( rmin, rmax, r, r2min, r2max, r2 ) !*****************************************************************************80 ! !! R8INT_TO_R8INT maps one R8INT to another. ! ! Discussion: ! ! The formula used is ! ! R2 := R2MIN + ( R2MAX - R2MIN ) * ( R - RMIN ) / ( RMAX - RMIN ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 January 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) RMIN, RMAX, the first range. ! ! Input, real ( kind = 8 ) R, the number to be converted. ! ! Input, real ( kind = 8 ) R2MAX, R2MIN, the second range. ! ! Output, real ( kind = 8 ) R2, the corresponding value in ! the range [R2MIN,R2MAX]. ! implicit none real ( kind = 8 ) r real ( kind = 8 ) rmax real ( kind = 8 ) rmin real ( kind = 8 ) r2 real ( kind = 8 ) r2max real ( kind = 8 ) r2min if ( rmax == rmin ) then r2 = ( r2max + r2min ) / 2.0D+00 else r2 = ( ( ( rmax - r ) * r2min & + ( r - rmin ) * r2max ) & / ( rmax - rmin ) ) end if return end subroutine r8int_to_i4int ( rmin, rmax, r, imin, imax, i ) !*****************************************************************************80 ! !! R8INT_TO_I4INT maps an R8INT to an integer interval. ! ! Discussion: ! ! The formula used is ! ! I := IMIN + ( IMAX - IMIN ) * ( R - RMIN ) / ( RMAX - RMIN ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 January 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) RMIN, RMAX, the range. ! ! Input, real ( kind = 8 ) R, the number to be converted. ! ! Input, integer ( kind = 4 ) IMAX, IMIN, the integer range. ! ! Output, integer ( kind = 4 ) I, the corresponding value in the ! range [IMIN,IMAX]. ! implicit none integer ( kind = 4 ) i integer ( kind = 4 ) imax integer ( kind = 4 ) imin real ( kind = 8 ) r real ( kind = 8 ) rmax real ( kind = 8 ) rmin if ( rmax == rmin ) then i = ( imax + imin ) / 2 else i = nint ( & ( ( rmax - r ) * real ( imin, kind = 8 ) & + ( r - rmin ) * real ( imax, kind = 8 ) ) & / ( rmax - rmin ) ) end if return end subroutine r8mat_add ( m, n, alpha, a, beta, b, c ) !*****************************************************************************80 ! !! R8MAT_ADD computes C = alpha * A + beta * B for R8MAT's. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 December 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) ALPHA, the multiplier for A. ! ! Input, real ( kind = 8 ) A(M,N), the first matrix. ! ! Input, real ( kind = 8 ) BETA, the multiplier for A. ! ! Input, real ( kind = 8 ) B(M,N), the second matrix. ! ! Output, real ( kind = 8 ) C(M,N), the sum of alpha*A+beta*B. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) alpha real ( kind = 8 ) b(m,n) real ( kind = 8 ) beta real ( kind = 8 ) c(m,n) c(1:m,1:n) = alpha * a(1:m,1:n) + beta * b(1:m,1:n) return end function r8mat_amax ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_AMAX returns the maximum absolute value entry of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 April 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix. ! ! Output, real ( kind = 8 ) R8MAT_AMAX, the maximum absolute value ! entry of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) r8mat_amax r8mat_amax = maxval ( abs ( a(1:m,1:n) ) ) return end subroutine r8mat_border_add ( m, n, table, table2 ) !*****************************************************************************80 ! !! R8MAT_BORDER_ADD adds a "border" to an R8MAT. ! ! Discussion: ! ! We suppose the input data gives values of a quantity on nodes ! in the interior of a 2D grid, and we wish to create a new table ! with additional positions for the nodes that would be on the ! border of the 2D grid. ! ! 0 0 0 0 0 0 ! * * * * 0 * * * * 0 ! * * * * --> 0 * * * * 0 ! * * * * 0 * * * * 0 ! 0 0 0 0 0 0 ! ! The illustration suggests the situation in which a 3 by 4 array ! is input, and a 5 by 6 array is to be output. ! ! The old data is shifted to its correct positions in the new array. ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 January 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the spatial dimension. ! ! Input, integer ( kind = 4 ) N, the number of points. ! ! Input, real ( kind = 8 ) TABLE(M,N), the table data. ! ! Output, real ( kind = 8 ) TABLE2(M+2,N+2), the augmented table data. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) table(m,n) real ( kind = 8 ) table2(m+2,n+2) table2(1,1:n+2) = 0.0D+00 table2(m+2,1:n+2) = 0.0D+00 table2(2:m+1,1) = 0.0D+00 table2(2:m+1,n+2) = 0.0D+00 table2(2:m+1,2:n+1) = table(1:m,1:n) return end subroutine r8mat_border_cut ( m, n, table, table2 ) !*****************************************************************************80 ! !! R8MAT_BORDER_CUT cuts the "border" of an R8MAT. ! ! Discussion: ! ! We suppose the input data gives values of a quantity on nodes ! on a 2D grid, and we wish to create a new table corresponding only ! to those nodes in the interior of the 2D grid. ! ! 0 0 0 0 0 0 ! 0 * * * * 0 * * * * ! 0 * * * * 0 -> * * * * ! 0 * * * * 0 * * * * ! 0 0 0 0 0 0 ! ! The illustration suggests the situation in which a 5 by 6 array ! is input, and a 3 by 4 array is to be output. ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 January 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the spatial dimension. ! ! Input, integer ( kind = 4 ) N, the number of points. ! ! Input, real ( kind = 8 ) TABLE(M,N), the table data. ! ! Output, real ( kind = 8 ) TABLE2(M-2,N-2), the new table data. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) table(m,n) real ( kind = 8 ) table2(m-2,n-2) if ( m <= 2 .or. n <= 2 ) then return end if table2(1:m-2,1:n-2) = table(2:m-1,2:n-1) return end subroutine r8mat_cholesky_factor ( n, a, c, flag ) !*****************************************************************************80 ! !! R8MAT_CHOLESKY_FACTOR computes the Cholesky factor of a symmetric matrix. ! ! Discussion: ! ! The matrix must be symmetric and positive semidefinite. ! ! For a positive semidefinite symmetric matrix A, the Cholesky factorization ! is a lower triangular matrix L such that: ! ! A = L * L' ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 April 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns of ! the matrix A. ! ! Input, real ( kind = 8 ) A(N,N), the N by N matrix. ! ! Output, real ( kind = 8 ) C(N,N), the N by N lower triangular ! Cholesky factor. ! ! Output, integer ( kind = 4 ) FLAG: ! 0, no error occurred. ! 1, the matrix is not positive definite. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) c(n,n) integer ( kind = 4 ) flag integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) sum2 flag = 0 c(1:n,1:n) = a(1:n,1:n) do j = 1, n c(1:j-1,j) = 0.0D+00 do i = j, n sum2 = c(j,i) - dot_product ( c(j,1:j-1), c(i,1:j-1) ) if ( i == j ) then if ( sum2 <= 0.0D+00 ) then flag = 1 return else c(i,j) = sqrt ( sum2 ) end if else if ( c(j,j) /= 0.0D+00 ) then c(i,j) = sum2 / c(j,j) else c(i,j) = 0.0D+00 end if end if end do end do return end subroutine r8mat_cholesky_factor_upper ( n, a, c, flag ) !*****************************************************************************80 ! !! R8MAT_CHOLESKY_FACTOR_UPPER: upper Cholesky factor of a symmetric matrix. ! ! Discussion: ! ! The matrix must be symmetric and positive semidefinite. ! ! For a positive semidefinite symmetric matrix A, the Cholesky factorization ! is an upper triangular matrix R such that: ! ! A = R * R' ! ! The lower Cholesky factor is a lower triangular matrix L such that ! ! A = L * L' ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 03 August 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns of ! the matrix A. ! ! Input, real ( kind = 8 ) A(N,N), the N by N matrix. ! ! Output, real ( kind = 8 ) C(N,N), the N by N upper triangular ! Cholesky factor. ! ! Output, integer ( kind = 4 ) FLAG: ! 0, no error occurred. ! 1, the matrix is not positive definite. ! 2, the matrix is not nonnegative definite. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) c(n,n) integer ( kind = 4 ) flag integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) sum2 flag = 0 c(1:n,1:n) = a(1:n,1:n) do j = 1, n c(j,1:j-1) = 0.0D+00 do i = j, n sum2 = c(i,j) - dot_product ( c(1:j-1,j), c(1:j-1,i) ) if ( i == j ) then if ( sum2 <= 0.0D+00 ) then flag = 1 return else c(j,i) = sqrt ( sum2 ) end if else if ( c(j,j) /= 0.0D+00 ) then c(j,i) = sum2 / c(j,j) else c(j,i) = 0.0D+00 end if end if end do end do return end subroutine r8mat_cholesky_inverse ( n, a ) !*****************************************************************************80 ! !! R8MAT_CHOLESKY_INVERSE computes the inverse of a symmetric matrix. ! ! Discussion: ! ! The matrix must be symmetric and positive semidefinite. ! ! The upper triangular Cholesky factorization R is computed, so that: ! ! A = R' * R ! ! Then the inverse B is computed by ! ! B = inv ( A ) = inv ( R ) * inv ( R' ) ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 October 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns of ! the matrix A. ! ! Input/output, real ( kind = 8 ) A(N,N). On input, the matrix. ! On output, the inverse of the matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) s real ( kind = 8 ) t do j = 1, n s = 0.0D+00 do k = 1, j - 1 t = a(k,j) - dot_product ( a(1:k-1,k), a(1:k-1,j) ) t = t / a(k,k) a(k,j) = t s = s + t * t end do s = a(j,j) - s if ( s <= 0.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_CHOLESKY_INVERSE - Fatal error!' write ( *, '(a)' ) ' The matrix is singular.' stop 1 end if a(j,j) = sqrt ( s ) a(j+1:n,j) = 0.0D+00 end do ! ! Compute inverse(R). ! do k = 1, n a(k,k) = 1.0D+00 / a(k,k) a(1:k-1,k) = - a(1:k-1,k) * a(k,k) do j = k + 1, n t = a(k,j) a(k,j) = 0.0D+00 a(1:k,j) = a(1:k,j) + t * a(1:k,k) end do end do ! ! Form inverse(R) * (inverse(R))'. ! do j = 1, n do k = 1, j - 1 t = a(k,j) a(1:k,k) = a(1:k,k) + t * a(1:k,j) end do t = a(j,j) a(1:j,j) = a(1:j,j) * t end do ! ! Use reflection. ! do i = 1, n do j = 1, i - 1 a(i,j) = a(j,i) end do end do return end subroutine r8mat_cholesky_solve ( n, l, b, x ) !*****************************************************************************80 ! !! R8MAT_CHOLESKY_SOLVE solves a Cholesky factored linear system A * x = b. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! This routine works with the lower triangular Cholesky factor A = L * L'. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns of ! the matrix A. ! ! Input, real ( kind = 8 ) L(N,N), the N by N lower Cholesky factor of the ! system matrix A. ! ! Input, real ( kind = 8 ) B(N), the right hand side of the linear system. ! ! Output, real ( kind = 8 ) X(N), the solution of the linear system. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) b(n) real ( kind = 8 ) l(n,n) real ( kind = 8 ) x(n) ! ! Solve L * y = b. ! call r8mat_l_solve ( n, l, b, x ) ! ! Solve L' * x = y. ! call r8mat_lt_solve ( n, l, x, x ) return end subroutine r8mat_cholesky_solve_upper ( n, r, b, x ) !*****************************************************************************80 ! !! R8MAT_CHOLESKY_SOLVE_UPPER solves a Cholesky factored system A * x = b. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! This routine works with the upper triangular Cholesky factor A = R' * R. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 October 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns of ! the matrix A. ! ! Input, real ( kind = 8 ) R(N,N), the N by N upper Cholesky factor of the ! system matrix. ! ! Input, real ( kind = 8 ) B(N), the right hand side of the linear system. ! ! Output, real ( kind = 8 ) X(N), the solution of the linear system. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) b(n) real ( kind = 8 ) r(n,n) real ( kind = 8 ) x(n) ! ! Solve R' * y = b. ! call r8mat_ut_solve ( n, r, b, x ) ! ! Solve R * x = y. ! call r8mat_u_solve ( n, r, x, x ) return end subroutine r8mat_copy ( m, n, a, b ) !*****************************************************************************80 ! !! R8MAT_COPY copies an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 July 2008 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the order of the matrix. ! ! Input, real ( kind = 8 ) A(M,N), the matrix to be copied. ! ! Output, real ( kind = 8 ) B(M,N), a copy of the matrix. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) b(m,n) b(1:m,1:n) = a(1:m,1:n) return end subroutine r8mat_covariance ( m, n, x, c ) !*****************************************************************************80 ! !! R8MAT_COVARIANCE computes the sample covariance of a set of vector data. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 June 2013 ! ! Author: ! ! John Burkardt. ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the size of a single data vectors. ! ! Input, integer ( kind = 4 ) N, the number of data vectors. ! N should be greater than 1. ! ! Input, real ( kind = 8 ) X(M,N), an array of N data vectors, each ! of length M. ! ! Output, real ( kind = 8 ) C(M,M), the covariance matrix for the data. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) c(m,m) integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) x(m,n) real ( kind = 8 ) x_mean(m) c(1:m,1:m) = 0.0D+00 ! ! Special case of N = 1. ! if ( n == 1 ) then do i = 1, m c(i,i) = 1.0D+00 end do return end if ! ! Determine the sample means. ! do i = 1, m x_mean(i) = sum ( x(i,1:n) ) / real ( n, kind = 8 ) end do ! ! Determine the sample covariance. ! do j = 1, m do i = 1, m do k = 1, n c(i,j) = c(i,j) + ( x(i,k) - x_mean(i) ) * ( x(j,k) - x_mean(j) ) end do end do end do c(1:m,1:m) = c(1:m,1:m) / real ( n - 1, kind = 8 ) return end subroutine r8mat_det ( n, a, det ) !*****************************************************************************80 ! !! R8MAT_DET computes the determinant of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! Original FORTRAN77 version by Helmut Spaeth. ! FORTRAN90 version by John Burkardt. ! ! Reference: ! ! Helmut Spaeth, ! Cluster Analysis Algorithms ! for Data Reduction and Classification of Objects, ! Ellis Horwood, 1980, page 125-127. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! ! Input, real ( kind = 8 ) A(N,N), the matrix whose determinant is desired. ! ! Output, real ( kind = 8 ) DET, the determinant of the matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) b(n,n) real ( kind = 8 ) det integer ( kind = 4 ) j integer ( kind = 4 ) k integer ( kind = 4 ) m integer ( kind = 4 ) piv(1) real ( kind = 8 ) t b(1:n,1:n) = a(1:n,1:n) det = 1.0D+00 do k = 1, n piv = maxloc ( abs ( b(k:n,k) ) ) m = piv(1) + k - 1 if ( m /= k ) then det = - det t = b(m,k) b(m,k) = b(k,k) b(k,k) = t end if det = det * b(k,k) if ( b(k,k) /= 0.0D+00 ) then b(k+1:n,k) = -b(k+1:n,k) / b(k,k) do j = k + 1, n if ( m /= k ) then t = b(m,j) b(m,j) = b(k,j) b(k,j) = t end if b(k+1:n,j) = b(k+1:n,j) + b(k+1:n,k) * b(k,j) end do end if end do return end function r8mat_det_2d ( a ) !*****************************************************************************80 ! !! R8MAT_DET_2D computes the determinant of a 2 by 2 R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The formula for the determinant of a 2 by 2 matrix is ! ! a11 * a22 - a12 * a21. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 March 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A(2,2), the matrix whose determinant is desired. ! ! Output, real ( kind = 8 ) R8MAT_DET_2D, the determinant of the matrix. ! implicit none real ( kind = 8 ) a(2,2) real ( kind = 8 ) r8mat_det_2d r8mat_det_2d = a(1,1) * a(2,2) - a(1,2) * a(2,1) return end function r8mat_det_3d ( a ) !*****************************************************************************80 ! !! R8MAT_DET_3D computes the determinant of a 3 by 3 R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The formula for the determinant of a 3 by 3 matrix is ! ! a11 * a22 * a33 - a11 * a23 * a32 ! + a12 * a23 * a31 - a12 * a21 * a33 ! + a13 * a21 * a32 - a13 * a22 * a31 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 March 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A(3,3), the matrix whose determinant is desired. ! ! Output, real ( kind = 8 ) R8MAT_DET_3D, the determinant of the matrix. ! implicit none real ( kind = 8 ) a(3,3) real ( kind = 8 ) r8mat_det_3d r8mat_det_3d = & a(1,1) * ( a(2,2) * a(3,3) - a(2,3) * a(3,2) ) & + a(1,2) * ( a(2,3) * a(3,1) - a(2,1) * a(3,3) ) & + a(1,3) * ( a(2,1) * a(3,2) - a(2,2) * a(3,1) ) return end function r8mat_det_4d ( a ) !*****************************************************************************80 ! !! R8MAT_DET_4D computes the determinant of a 4 by 4 R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 March 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A(4,4), the matrix whose determinant is desired. ! ! Output, real ( kind = 8 ) R8MAT_DET_4D, the determinant of the matrix. ! implicit none real ( kind = 8 ) a(4,4) real ( kind = 8 ) r8mat_det_4d r8mat_det_4d = & a(1,1) * ( & a(2,2) * ( a(3,3) * a(4,4) - a(3,4) * a(4,3) ) & - a(2,3) * ( a(3,2) * a(4,4) - a(3,4) * a(4,2) ) & + a(2,4) * ( a(3,2) * a(4,3) - a(3,3) * a(4,2) ) ) & - a(1,2) * ( & a(2,1) * ( a(3,3) * a(4,4) - a(3,4) * a(4,3) ) & - a(2,3) * ( a(3,1) * a(4,4) - a(3,4) * a(4,1) ) & + a(2,4) * ( a(3,1) * a(4,3) - a(3,3) * a(4,1) ) ) & + a(1,3) * ( & a(2,1) * ( a(3,2) * a(4,4) - a(3,4) * a(4,2) ) & - a(2,2) * ( a(3,1) * a(4,4) - a(3,4) * a(4,1) ) & + a(2,4) * ( a(3,1) * a(4,2) - a(3,2) * a(4,1) ) ) & - a(1,4) * ( & a(2,1) * ( a(3,2) * a(4,3) - a(3,3) * a(4,2) ) & - a(2,2) * ( a(3,1) * a(4,3) - a(3,3) * a(4,1) ) & + a(2,3) * ( a(3,1) * a(4,2) - a(3,2) * a(4,1) ) ) return end function r8mat_det_5d ( a ) !*****************************************************************************80 ! !! R8MAT_DET_5D computes the determinant of a 5 by 5 R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 March 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A(5,5), the matrix whose determinant is desired. ! ! Output, real ( kind = 8 ) R8MAT_DET_5D, the determinant of the matrix. ! implicit none real ( kind = 8 ) a(5,5) real ( kind = 8 ) b(4,4) integer ( kind = 4 ) i integer ( kind = 4 ) inc integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) r8mat_det_4d real ( kind = 8 ) r8mat_det_5d ! ! Expand the determinant into the sum of the determinants of the ! five 4 by 4 matrices created by dropping row 1, and column k. ! r8mat_det_5d = 0.0D+00 do k = 1, 5 do i = 1, 4 do j = 1, 4 if ( j < k ) then inc = 0 else inc = 1 end if b(i,j) = a(i+1,j+inc) end do end do r8mat_det_5d = r8mat_det_5d & + ( -1 ) ** ( k + 1 ) * a(1,k) * r8mat_det_4d ( b ) end do return end subroutine r8mat_diag_add_scalar ( n, a, s ) !*****************************************************************************80 ! !! R8MAT_DIAG_ADD_SCALAR adds a scalar to the diagonal of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns. ! ! Input/output, real ( kind = 8 ) A(N,N), the N by N matrix to be modified. ! ! Input, real ( kind = 8 ) S, the value to be added to the diagonal ! of the matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i real ( kind = 8 ) s do i = 1, n a(i,i) = a(i,i) + s end do return end subroutine r8mat_diag_add_vector ( n, a, v ) !*****************************************************************************80 ! !! R8MAT_DIAG_ADD_VECTOR adds a vector to the diagonal of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns of ! the matrix. ! ! Input/output, real ( kind = 8 ) A(N,N), the N by N matrix. ! ! Input, real ( kind = 8 ) V(N), the vector to be added to the diagonal of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i real ( kind = 8 ) v(n) do i = 1, n a(i,i) = a(i,i) + v(i) end do return end subroutine r8mat_diag_get_vector ( n, a, v ) !*****************************************************************************80 ! !! R8MAT_DIAG_GET_VECTOR gets the value of the diagonal of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 22 March 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns of ! the matrix. ! ! Input, real ( kind = 8 ) A(N,N), the N by N matrix. ! ! Output, real ( kind = 8 ) V(N), the diagonal entries ! of the matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i real ( kind = 8 ) v(n) do i = 1, n v(i) = a(i,i) end do return end subroutine r8mat_diag_set_scalar ( n, a, s ) !*****************************************************************************80 ! !! R8MAT_DIAG_SET_SCALAR sets the diagonal of an R8MAT to a scalar value. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns. ! ! Input/output, real ( kind = 8 ) A(N,N), the N by N matrix to be modified. ! ! Input, real ( kind = 8 ) S, the value to be assigned to the diagonal ! of the matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i real ( kind = 8 ) s do i = 1, n a(i,i) = s end do return end subroutine r8mat_diag_set_vector ( n, a, v ) !*****************************************************************************80 ! !! R8MAT_DIAG_SET_VECTOR sets the diagonal of an R8MAT to a vector. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns. ! ! Input/output, real ( kind = 8 ) A(N,N), the N by N matrix. ! ! Input, real ( kind = 8 ) V(N), the vector to be assigned to the ! diagonal of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i real ( kind = 8 ) v(n) do i = 1, n a(i,i) = v(i) end do return end subroutine r8mat_diagonal ( n, diag, a ) !*****************************************************************************80 ! !! R8MAT_DIAGONAL returns a diagonal matrix as an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 31 July 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of A. ! ! Input, real ( kind = 8 ) DIAG(N), the diagonal entries. ! ! Output, real ( kind = 8 ) A(N,N), the N by N diagonal matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) diag(n) integer ( kind = 4 ) i a(1:n,1:n) = 0.0D+00 do i = 1, n a(i,i) = diag(i) end do return end function r8mat_diff_frobenius ( m, n, a1, a2 ) !*****************************************************************************80 ! !! R8MAT_DIFF_FROBENIUS returns the Frobenius norm of an R8MAT difference. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The Frobenius norm is defined as ! ! R8MAT_DIFF_FROBENIUS = sqrt ( ! sum ( 1 <= I <= M ) sum ( 1 <= j <= N ) A(I,J) * A(I,J) ) ! ! The matrix Frobenius norm is not derived from a vector norm, but ! is compatible with the vector L2 norm, so that: ! ! r8vec_norm_l2 ( A * x ) <= ! r8mat_diff_frobenius ( A ) * r8vec_norm_l2 ( x ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 March 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows. ! ! Input, integer ( kind = 4 ) N, the number of columns. ! ! Input, real ( kind = 8 ) A1(M,N), A2(M,N), the matrices for whose ! difference the Frobenius norm is desired. ! ! Output, real ( kind = 8 ) R8MAT_DIFF_FROBENIUSE, the Frobenius ! norm of A1 - A2. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a1(m,n) real ( kind = 8 ) a2(m,n) real ( kind = 8 ) r8mat_diff_frobenius r8mat_diff_frobenius = sqrt ( sum ( ( a1(1:m,1:n) - a2(1:m,1:n) ) ** 2 ) ) return end subroutine r8mat_expand_linear ( m, n, x, mfat, nfat, xfat ) !*****************************************************************************80 ! !! R8MAT_EXPAND_LINEAR linearly interpolates new data into an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! In this routine, the expansion is specified by giving the number ! of intermediate values to generate between each pair of original ! data rows and columns. ! ! The interpolation is not actually linear. It uses the functions ! ! 1, x, y, and xy. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns of ! input data. ! ! Input, real ( kind = 8 ) X(M,N), the original data. ! ! Input, integer ( kind = 4 ) MFAT, NFAT, the number of data values ! to interpolate between each row, and each column, of original data values. ! ! Output, real ( kind = 8 ) XFAT(M2,N2), the fattened data, where ! M2 = (M-1)*(MFAT+1)+1, ! N2 = (N-1)*(NFAT+1)+1. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) mfat integer ( kind = 4 ) n integer ( kind = 4 ) nfat integer ( kind = 4 ) i integer ( kind = 4 ) ihi integer ( kind = 4 ) ii integer ( kind = 4 ) iii integer ( kind = 4 ) ip1 integer ( kind = 4 ) j integer ( kind = 4 ) jhi integer ( kind = 4 ) jj integer ( kind = 4 ) jjj integer ( kind = 4 ) jp1 real ( kind = 8 ) s real ( kind = 8 ) t real ( kind = 8 ) x(m,n) real ( kind = 8 ) x00 real ( kind = 8 ) x01 real ( kind = 8 ) x10 real ( kind = 8 ) x11 real ( kind = 8 ) xfat((m-1)*(mfat+1)+1,(n-1)*(nfat+1)+1) do i = 1, m if ( i < m ) then ihi = mfat else ihi = 0 end if do j = 1, n if ( j < n ) then jhi = nfat else jhi = 0 end if if ( i < m ) then ip1 = i + 1 else ip1 = i end if if ( j < n ) then jp1 = j + 1 else jp1 = j end if x00 = x(i,j) x10 = x(ip1,j) x01 = x(i,jp1) x11 = x(ip1,jp1) do ii = 0, ihi s = real ( ii, kind = 8 ) & / real ( ihi + 1, kind = 8 ) do jj = 0, jhi t = real ( jj, kind = 8 ) & / real ( jhi + 1, kind = 8 ) iii = 1 + ( i - 1 ) * ( mfat + 1 ) + ii jjj = 1 + ( j - 1 ) * ( nfat + 1 ) + jj xfat(iii,jjj) = & x00 & + s * ( x10 - x00 ) & + t * ( x01 - x00 ) & + s * t * ( x11 - x10 - x01 + x00 ) end do end do end do end do return end subroutine r8mat_expand_linear2 ( m, n, a, m2, n2, a2 ) !*****************************************************************************80 ! !! R8MAT_EXPAND_LINEAR2 expands an R8MAT by linear interpolation. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! In this version of the routine, the expansion is indicated ! by specifying the dimensions of the expanded array. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns in A. ! ! Input, real ( kind = 8 ) A(M,N), a "small" M by N array. ! ! Input, integer ( kind = 4 ) M2, N2, the number of rows and columns in A2. ! ! Output, real ( kind = 8 ) A2(M2,N2), the expanded array, which ! contains an interpolated version of the data in A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) m2 integer ( kind = 4 ) n integer ( kind = 4 ) n2 real ( kind = 8 ) a(m,n) real ( kind = 8 ) a2(m2,n2) integer ( kind = 4 ) i integer ( kind = 4 ) i1 integer ( kind = 4 ) i2 integer ( kind = 4 ) j integer ( kind = 4 ) j1 integer ( kind = 4 ) j2 real ( kind = 8 ) r real ( kind = 8 ) r1 real ( kind = 8 ) r2 real ( kind = 8 ) s real ( kind = 8 ) s1 real ( kind = 8 ) s2 do i = 1, m2 if ( m2 == 1 ) then r = 0.5D+00 else r = real ( i - 1, kind = 8 ) & / real ( m2 - 1, kind = 8 ) end if i1 = 1 + int ( r * real ( m - 1, kind = 8 ) ) i2 = i1 + 1 if ( m < i2 ) then i1 = m - 1 i2 = m end if r1 = real ( i1 - 1, kind = 8 ) & / real ( m - 1, kind = 8 ) r2 = real ( i2 - 1, kind = 8 ) & / real ( m - 1, kind = 8 ) do j = 1, n2 if ( n2 == 1 ) then s = 0.5D+00 else s = real ( j - 1, kind = 8 ) & / real ( n2 - 1, kind = 8 ) end if j1 = 1 + int ( s * real ( n - 1, kind = 8 ) ) j2 = j1 + 1 if ( n < j2 ) then j1 = n - 1 j2 = n end if s1 = real ( j1 - 1, kind = 8 ) & / real ( n - 1, kind = 8 ) s2 = real ( j2 - 1, kind = 8 ) & / real ( n - 1, kind = 8 ) a2(i,j) = & ( ( r2 - r ) * ( s2 - s ) * a(i1,j1) & + ( r - r1 ) * ( s2 - s ) * a(i2,j1) & + ( r2 - r ) * ( s - s1 ) * a(i1,j2) & + ( r - r1 ) * ( s - s1 ) * a(i2,j2) ) & / ( ( r2 - r1 ) * ( s2 - s1 ) ) end do end do return end subroutine r8mat_flip_cols ( m, n, a, b ) !*****************************************************************************80 ! !! R8MAT_FLIP_COLS reverses the column order of an R8MAT. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 November 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the order of the matrix. ! ! Input, real ( kind = 8 ) A(M,N), the matrix to be flipped. ! ! Output, real ( kind = 8 ) B(M,N), a copy of A, with the columns ! in reverse order. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) b(m,n) b(1:m,n:1:-1) = a(1:m,1:n) return end subroutine r8mat_flip_rows ( m, n, a, b ) !*****************************************************************************80 ! !! R8MAT_FLIP_ROWS reverses the row order of an R8MAT. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 November 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the order of the matrix. ! ! Input, real ( kind = 8 ) A(M,N), the matrix to be flipped. ! ! Output, real ( kind = 8 ) B(M,N), a copy of A, with the rows ! in reverse order. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) b(m,n) b(m:1:-1,1:n) = a(1:m,1:n) return end subroutine r8mat_fs ( n, a, b, info ) !*****************************************************************************80 ! !! R8MAT_FS factors and solves a system with one right hand side. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! This routine differs from R8MAT_FSS in two ways: ! * only one right hand side is allowed; ! * the input matrix A is not modified. ! ! This routine uses partial pivoting, but no pivot vector is required. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 January 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! N must be positive. ! ! Input/output, real ( kind = 8 ) A(N,N). ! On input, A is the coefficient matrix of the linear system. ! On output, A is in unit upper triangular form, and ! represents the U factor of an LU factorization of the ! original coefficient matrix. ! ! Input/output, real ( kind = 8 ) B(N). ! On input, the right hand side of the linear system. ! On output, the solution of the linear systems. ! ! Output, integer ( kind = 4 ) INFO, singularity flag. ! 0, no singularity detected. ! nonzero, the factorization failed on the INFO-th step. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) a2(n,n) real ( kind = 8 ) b(n) integer ( kind = 4 ) i integer ( kind = 4 ) info integer ( kind = 4 ) ipiv integer ( kind = 4 ) jcol real ( kind = 8 ) piv real ( kind = 8 ) row(n) real ( kind = 8 ) t real ( kind = 8 ) temp a2(1:n,1:n) = a(1:n,1:n) info = 0 do jcol = 1, n ! ! Find the maximum element in column I. ! piv = abs ( a2(jcol,jcol) ) ipiv = jcol do i = jcol + 1, n if ( piv < abs ( a2(i,jcol) ) ) then piv = abs ( a2(i,jcol) ) ipiv = i end if end do if ( piv == 0.0D+00 ) then info = jcol write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_FS - Fatal error!' write ( *, '(a,i8)' ) ' Zero pivot on step ', info stop 1 end if ! ! Switch rows JCOL and IPIV, and B. ! if ( jcol /= ipiv ) then row(1:n) = a2(jcol,1:n) a2(jcol,1:n) = a2(ipiv,1:n) a2(ipiv,1:n) = row(1:n) t = b(jcol) b(jcol) = b(ipiv) b(ipiv) = t end if ! ! Scale the pivot row. ! a2(jcol,jcol+1:n) = a2(jcol,jcol+1:n) / a2(jcol,jcol) b(jcol) = b(jcol) / a2(jcol,jcol) a2(jcol,jcol) = 1.0D+00 ! ! Use the pivot row to eliminate lower entries in that column. ! do i = jcol + 1, n if ( a2(i,jcol) /= 0.0D+00 ) then temp = - a2(i,jcol) a2(i,jcol) = 0.0D+00 a2(i,jcol+1:n) = a2(i,jcol+1:n) + temp * a2(jcol,jcol+1:n) b(i) = b(i) + temp * b(jcol) end if end do end do ! ! Back solve. ! do jcol = n, 2, -1 b(1:jcol-1) = b(1:jcol-1) - a2(1:jcol-1,jcol) * b(jcol) end do return end subroutine r8mat_fss ( n, a, nb, b, info ) !*****************************************************************************80 ! !! R8MAT_FSS factors and solves a system with multiple right hand sides. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! This routine does not save the LU factors of the matrix, and hence cannot ! be used to efficiently solve multiple linear systems, or even to ! factor A at one time, and solve a single linear system at a later time. ! ! This routine uses partial pivoting, but no pivot vector is required. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 28 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! N must be positive. ! ! Input/output, real ( kind = 8 ) A(N,N). ! On input, A is the coefficient matrix of the linear system. ! On output, A is in unit upper triangular form, and ! represents the U factor of an LU factorization of the ! original coefficient matrix. ! ! Input, integer ( kind = 4 ) NB, the number of right hand sides. ! ! Input/output, real ( kind = 8 ) B(N,NB). ! On input, the right hand sides of the linear system. ! On output, the solutions of the linear systems. ! ! Output, integer ( kind = 4 ) INFO, singularity flag. ! 0, no singularity detected. ! nonzero, the factorization failed on the INFO-th step. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) nb real ( kind = 8 ) a(n,n) real ( kind = 8 ) b(n,nb) integer ( kind = 4 ) i integer ( kind = 4 ) info integer ( kind = 4 ) ipiv integer ( kind = 4 ) j integer ( kind = 4 ) jcol real ( kind = 8 ) piv real ( kind = 8 ) row(n) real ( kind = 8 ) t(nb) real ( kind = 8 ) temp info = 0 do jcol = 1, n ! ! Find the maximum element in column I. ! piv = abs ( a(jcol,jcol) ) ipiv = jcol do i = jcol + 1, n if ( piv < abs ( a(i,jcol) ) ) then piv = abs ( a(i,jcol) ) ipiv = i end if end do if ( piv == 0.0D+00 ) then info = jcol write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_FSS - Fatal error!' write ( *, '(a,i8)' ) ' Zero pivot on step ', info stop 1 end if ! ! Switch rows JCOL and IPIV, and B. ! if ( jcol /= ipiv ) then row(1:n) = a(jcol,1:n) a(jcol,1:n) = a(ipiv,1:n) a(ipiv,1:n) = row(1:n) t(1:nb) = b(jcol,1:nb) b(jcol,1:nb) = b(ipiv,1:nb) b(ipiv,1:nb) = t(1:nb) end if ! ! Scale the pivot row. ! a(jcol,jcol+1:n) = a(jcol,jcol+1:n) / a(jcol,jcol) b(jcol,1:nb) = b(jcol,1:nb) / a(jcol,jcol) a(jcol,jcol) = 1.0D+00 ! ! Use the pivot row to eliminate lower entries in that column. ! do i = jcol + 1, n if ( a(i,jcol) /= 0.0D+00 ) then temp = - a(i,jcol) a(i,jcol) = 0.0D+00 a(i,jcol+1:n) = a(i,jcol+1:n) + temp * a(jcol,jcol+1:n) b(i,1:nb) = b(i,1:nb) + temp * b(jcol,1:nb) end if end do end do ! ! Back solve. ! do j = 1, nb do jcol = n, 2, -1 b(1:jcol-1,j) = b(1:jcol-1,j) - a(1:jcol-1,jcol) * b(jcol,j) end do end do return end subroutine r8mat_givens_post ( n, a, row, col, g ) !*****************************************************************************80 ! !! R8MAT_GIVENS_POST computes the Givens postmultiplier rotation matrix. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The Givens post-multiplier matrix G(ROW,COL) has the property that ! the (ROW,COL)-th entry of A*G is zero. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 April 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrices A and G. ! ! Input, real ( kind = 8 ) A(N,N), the matrix to be operated upon. ! ! Input, integer ( kind = 4 ) ROW, COL, the row and column of the ! entry of A*G which is to be zeroed out. ! ! Output, real ( kind = 8 ) G(N,N), the Givens rotation matrix. ! G is an orthogonal matrix, that is, the inverse of ! G is the transpose of G. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) col real ( kind = 8 ) g(n,n) integer ( kind = 4 ) row real ( kind = 8 ) theta call r8mat_identity ( n, g ) theta = atan2 ( a(row,col), a(row,row) ) g(row,row) = cos ( theta ) g(row,col) = -sin ( theta ) g(col,row) = sin ( theta ) g(col,col) = cos ( theta ) return end subroutine r8mat_givens_pre ( n, a, row, col, g ) !*****************************************************************************80 ! !! R8MAT_GIVENS_PRE computes the Givens premultiplier rotation matrix. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The Givens premultiplier rotation matrix G(ROW,COL) has the ! property that the (ROW,COL)-th entry of G*A is zero. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 April 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrices A and G. ! ! Input, real ( kind = 8 ) A(N,N), the matrix to be operated upon. ! ! Input, integer ( kind = 4 ) ROW, COL, the row and column of the ! entry of the G*A which is to be zeroed out. ! ! Output, real ( kind = 8 ) G(N,N), the Givens rotation matrix. ! G is an orthogonal matrix, that is, the inverse of ! G is the transpose of G. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) col real ( kind = 8 ) g(n,n) integer ( kind = 4 ) row real ( kind = 8 ) theta call r8mat_identity ( n, g ) theta = atan2 ( a(row,col), a(col,col) ) g(row,row) = cos ( theta ) g(row,col) = -sin ( theta ) g(col,row) = sin ( theta ) g(col,col) = cos ( theta ) return end subroutine r8mat_hess ( fx, n, x, h ) !*****************************************************************************80 ! !! R8MAT_HESS approximates a Hessian matrix via finite differences. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! H(I,J) = d2 F / d X(I) d X(J) ! ! The values returned by this routine will be only approximate. ! In some cases, they will be so poor that they are useless. ! However, one of the best applications of this routine is for ! checking your own Hessian calculations, since as Heraclitus ! said, you'll never get the same result twice when you differentiate ! a complicated expression by hand. ! ! The user function routine, here called "FX", should have the form: ! ! subroutine fx ( n, x, f ) ! integer ( kind = 4 ) n ! real ( kind = 8 ) f ! real ( kind = 8 ) x(n) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, external FX, the name of the user function routine. ! ! Input, integer ( kind = 4 ) N, the number of variables. ! ! Input, real ( kind = 8 ) X(N), the values of the variables. ! ! Output, real ( kind = 8 ) H(N,N), the approximated N by N Hessian matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) eps real ( kind = 8 ) f00 real ( kind = 8 ) fmm real ( kind = 8 ) fmp real ( kind = 8 ) fpm real ( kind = 8 ) fpp external fx real ( kind = 8 ) h(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) s(n) real ( kind = 8 ) x(n) real ( kind = 8 ) xi real ( kind = 8 ) xj ! ! Choose the stepsizes. ! eps = ( epsilon ( eps ) ) ** 0.33D+00 do i = 1, n s(i) = eps * max ( abs ( x(i) ), 1.0D+00 ) end do ! ! Calculate the diagonal elements. ! do i = 1, n xi = x(i) call fx ( n, x, f00 ) x(i) = xi + s(i) call fx ( n, x, fpp ) x(i) = xi - s(i) call fx ( n, x, fmm ) h(i,i) = ( ( fpp - f00 ) + ( fmm - f00 ) ) / s(i) ** 2 x(i) = xi end do ! ! Calculate the off diagonal elements. ! do i = 1, n xi = x(i) do j = i + 1, n xj = x(j) x(i) = xi + s(i) x(j) = xj + s(j) call fx ( n, x, fpp ) x(i) = xi + s(i) x(j) = xj - s(j) call fx ( n, x, fpm ) x(i) = xi - s(i) x(j) = xj + s(j) call fx ( n, x, fmp ) x(i) = xi - s(i) x(j) = xj - s(j) call fx ( n, x, fmm ) h(j,i) = ( ( fpp - fpm ) + ( fmm - fmp ) ) / ( 4.0D+00 * s(i) * s(j) ) h(i,j) = h(j,i) x(j) = xj end do x(i) = xi end do return end subroutine r8mat_house_axh ( n, a, v, ah ) !*****************************************************************************80 ! !! R8MAT_HOUSE_AXH computes A*H where H is a compact Householder matrix. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The Householder matrix H(V) is defined by ! ! H(V) = I - 2 * v * v' / ( v' * v ) ! ! This routine is not particularly efficient. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 March 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of A. ! ! Input, real ( kind = 8 ) A(N,N), the matrix to be postmultiplied. ! ! Input, real ( kind = 8 ) V(N), a vector defining a Householder matrix. ! ! Output, real ( kind = 8 ) AH(N,N), the product A*H. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) ah(n,n) real ( kind = 8 ) ah_temp(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) v(n) real ( kind = 8 ) v_normsq v_normsq = sum ( v(1:n) ** 2 ) ! ! Compute A*H' = A*H ! do i = 1, n do j = 1, n ah_temp(i,j) = a(i,j) do k = 1, n ah_temp(i,j) = ah_temp(i,j) - 2.0D+00 * a(i,k) * v(k) * v(j) / v_normsq end do end do end do ! ! Copy the temporary result into AH. ! Doing it this way means the user can identify the input arguments A and AH. ! ah(1:n,1:n) = ah_temp(1:n,1:n) return end subroutine r8mat_house_form ( n, v, h ) !*****************************************************************************80 ! !! R8MAT_HOUSE_FORM constructs a Householder matrix from its compact form. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! H(v) = I - 2 * v * v' / ( v' * v ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 March 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! ! Input, real ( kind = 8 ) V(N), the vector defining the Householder matrix. ! ! Output, real ( kind = 8 ) H(N,N), the Householder matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) beta real ( kind = 8 ) h(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) v(n) ! ! Compute the L2 norm of V. ! beta = sum ( v(1:n) ** 2 ) ! ! Form the matrix H. ! call r8mat_identity ( n, h ) do i = 1, n do j = 1, n h(i,j) = h(i,j) - 2.0D+00 * v(i) * v(j) / beta end do end do return end subroutine r8mat_house_hxa ( n, a, v, ha ) !*****************************************************************************80 ! !! R8MAT_HOUSE_HXA computes H*A where H is a compact Householder matrix. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The Householder matrix H(V) is defined by ! ! H(V) = I - 2 * v * v' / ( v' * v ) ! ! This routine is not particularly efficient. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 March 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of A. ! ! Input, real ( kind = 8 ) A(N,N), the matrix to be premultiplied. ! ! Input, real ( kind = 8 ) V(N), a vector defining a Householder matrix. ! ! Output, real ( kind = 8 ) HA(N,N), the product H*A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) ha(n,n) real ( kind = 8 ) ha_temp(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) v(n) real ( kind = 8 ) v_normsq v_normsq = sum ( v(1:n) ** 2 ) ! ! Compute A*H' = A*H ! do i = 1, n do j = 1, n ha_temp(i,j) = a(i,j) do k = 1, n ha_temp(i,j) = ha_temp(i,j) - 2.0D+00 * v(i) * v(k) * a(k,j) / v_normsq end do end do end do ! ! Copy the temporary result into HA. ! Doing it this way means the user can identify the input arguments A and HA. ! ha(1:n,1:n) = ha_temp(1:n,1:n) return end subroutine r8mat_house_post ( n, a, row, col, h ) !*****************************************************************************80 ! !! R8MAT_HOUSE_POST computes a Householder post-multiplier matrix. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! H(ROW,COL) has the property that the ROW-th column of ! A*H(ROW,COL) is zero from entry COL+1 to the end. ! ! In the most common case, where a QR factorization is being computed, ! ROW = COL. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 April 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrices. ! ! Input, real ( kind = 8 ) A(N,N), the matrix whose Householder matrix ! is to be computed. ! ! Input, integer ( kind = 4 ) ROW, COL, specify the location of the ! entry of the matrix A which is to be preserved. The entries in ! the same row, but higher column, will be zeroed out if ! A is postmultiplied by H. ! ! Output, real ( kind = 8 ) H(N,N), the Householder matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) a_row(n) integer ( kind = 4 ) col real ( kind = 8 ) h(n,n) integer ( kind = 4 ) row real ( kind = 8 ) v(n) ! ! Set up the vector V. ! a_row(1:col-1) = 0.0D+00 a_row(col:n) = a(row,col:n) call r8vec_house_column ( n, a_row, col, v ) ! ! Form the matrix H(V). ! call r8mat_house_form ( n, v, h ) return end subroutine r8mat_house_pre ( n, a, row, col, h ) !*****************************************************************************80 ! !! R8MAT_HOUSE_PRE computes a Householder pre-multiplier matrix. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! H(ROW,COL) has the property that the COL-th column of ! H(ROW,COL)*A is zero from entry ROW+1 to the end. ! ! In the most common case, where a QR factorization is being computed, ! ROW = COL. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 April 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrices. ! ! Input, real ( kind = 8 ) A(N,N), the matrix whose Householder matrix ! is to be computed. ! ! Input, integer ( kind = 4 ) ROW, COL, specify the location of the ! entry of the matrix A which is to be preserved. The entries in ! the same column, but higher rows, will be zeroed out if A is ! premultiplied by H. ! ! Output, real ( kind = 8 ) H(N,N), the Householder matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) a_col(n) integer ( kind = 4 ) col real ( kind = 8 ) h(n,n) integer ( kind = 4 ) row real ( kind = 8 ) v(n) ! ! Get the vector V. ! a_col(1:row-1) = 0.0D+00 a_col(row:n) = a(row:n,col) call r8vec_house_column ( n, a_col, row, v ) ! ! Form the matrix H(V). ! call r8mat_house_form ( n, v, h ) return end subroutine r8mat_identity ( n, a ) !*****************************************************************************80 ! !! R8MAT_IDENTITY stores the identity matrix in an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 March 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of A. ! ! Output, real ( kind = 8 ) A(N,N), the N by N identity matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i a(1:n,1:n) = 0.0D+00 do i = 1, n a(i,i) = 1.0D+00 end do return end subroutine r8mat_indicator ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_INDICATOR sets up an "indicator" R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The value of each entry suggests its location, as in: ! ! 11 12 13 14 ! 21 22 23 24 ! 31 32 33 34 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 28 May 2008 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows of the matrix. ! M must be positive. ! ! Input, integer ( kind = 4 ) N, the number of columns of the matrix. ! N must be positive. ! ! Output, real ( kind = 8 ) A(M,N), the matrix. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) fac integer ( kind = 4 ) i integer ( kind = 4 ) i4_log_10 integer ( kind = 4 ) j fac = 10 ** ( i4_log_10 ( n ) + 1 ) do i = 1, m do j = 1, n a(i,j) = real ( fac * i + j, kind = 8 ) end do end do return end subroutine r8mat_inverse_2d ( a, b, det ) !*****************************************************************************80 ! !! R8MAT_INVERSE_2D inverts a 2 by 2 R8MAT using Cramer's rule. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! If the determinant is zero, then A is singular, and does not have an ! inverse. In that case, B is simply set to zero, and a ! message is printed. ! ! If the determinant is nonzero, then its value is roughly an estimate ! of how nonsingular the matrix A is. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 November 1998 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A(2,2), the matrix to be inverted. ! ! Output, real ( kind = 8 ) B(2,2), the inverse of the matrix A. ! ! Output, real ( kind = 8 ) DET, the determinant of the matrix A. ! implicit none real ( kind = 8 ) a(2,2) real ( kind = 8 ) b(2,2) real ( kind = 8 ) det real ( kind = 8 ) r8mat_det_2d ! ! Compute the determinant of A. ! det = r8mat_det_2d ( a ) if ( det == 0.0D+00 ) then b(1:2,1:2) = 0.0D+00 else b(1,1) = a(2,2) / det b(1,2) = -a(1,2) / det b(2,1) = -a(2,1) / det b(2,2) = a(1,1) / det end if return end subroutine r8mat_inverse_3d ( a, b, det ) !*****************************************************************************80 ! !! R8MAT_INVERSE_3D inverts a 3 by 3 R8MAT using Cramer's rule. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! If the determinant is zero, then A is singular, and does not have an ! inverse. In that case, B is simply set to zero, and a ! message is printed. ! ! If the determinant is nonzero, then its value is roughly an estimate ! of how nonsingular the matrix A is. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 November 1998 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A(3,3), the matrix to be inverted. ! ! Output, real ( kind = 8 ) B(3,3), the inverse of the matrix A. ! ! Output, real ( kind = 8 ) DET, the determinant of the matrix A. ! implicit none real ( kind = 8 ) a(3,3) real ( kind = 8 ) b(3,3) real ( kind = 8 ) det real ( kind = 8 ) r8mat_det_3d ! ! Compute the determinant of A. ! det = r8mat_det_3d ( a ) ! ! If the determinant is zero, bail out. ! if ( det == 0.0D+00 ) then b(1:3,1:3) = 0.0D+00 return end if ! ! Compute the entries of the inverse matrix using an explicit ! formula. ! b(1,1) = ( a(2,2) * a(3,3) - a(2,3) * a(3,2) ) / det b(1,2) = -( a(1,2) * a(3,3) - a(1,3) * a(3,2) ) / det b(1,3) = ( a(1,2) * a(2,3) - a(1,3) * a(2,2) ) / det b(2,1) = -( a(2,1) * a(3,3) - a(2,3) * a(3,1) ) / det b(2,2) = ( a(1,1) * a(3,3) - a(1,3) * a(3,1) ) / det b(2,3) = -( a(1,1) * a(2,3) - a(1,3) * a(2,1) ) / det b(3,1) = ( a(2,1) * a(3,2) - a(2,2) * a(3,1) ) / det b(3,2) = -( a(1,1) * a(3,2) - a(1,2) * a(3,1) ) / det b(3,3) = ( a(1,1) * a(2,2) - a(1,2) * a(2,1) ) / det return end subroutine r8mat_inverse_4d ( a, b, det ) !*****************************************************************************80 ! !! R8MAT_INVERSE_4D inverts a 4 by 4 R8MAT using Cramer's rule. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! If the determinant is zero, then A is singular, and does not have an ! inverse. In that case, B is simply set to zero, and a ! message is printed. ! ! If the determinant is nonzero, then its value is roughly an estimate ! of how nonsingular the matrix A is. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 July 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A(4,4), the matrix to be inverted. ! ! Output, real ( kind = 8 ) B(4,4), the inverse of the matrix A. ! ! Output, real ( kind = 8 ) DET, the determinant of the matrix A. ! implicit none real ( kind = 8 ) a(4,4) real ( kind = 8 ) b(4,4) real ( kind = 8 ) det real ( kind = 8 ) r8mat_det_4d ! ! Compute the determinant of A. ! det = r8mat_det_4d ( a ) ! ! If the determinant is zero, bail out. ! if ( det == 0.0D+00 ) then b(1:4,1:4) = 0.0D+00 return end if ! ! Compute the entries of the inverse matrix using an explicit formula. ! b(1,1) = +( & + a(2,2) * ( a(3,3) * a(4,4) - a(3,4) * a(4,3) ) & + a(2,3) * ( a(3,4) * a(4,2) - a(3,2) * a(4,4) ) & + a(2,4) * ( a(3,2) * a(4,3) - a(3,3) * a(4,2) ) & ) / det b(2,1) = -( & + a(2,1) * ( a(3,3) * a(4,4) - a(3,4) * a(4,3) ) & + a(2,3) * ( a(3,4) * a(4,1) - a(3,1) * a(4,4) ) & + a(2,4) * ( a(3,1) * a(4,3) - a(3,3) * a(4,1) ) & ) / det b(3,1) = +( & + a(2,1) * ( a(3,2) * a(4,4) - a(3,4) * a(4,2) ) & + a(2,2) * ( a(3,4) * a(4,1) - a(3,1) * a(4,4) ) & + a(2,4) * ( a(3,1) * a(4,2) - a(3,2) * a(4,1) ) & ) / det b(4,1) = -( & + a(2,1) * ( a(3,2) * a(4,3) - a(3,3) * a(4,2) ) & + a(2,2) * ( a(3,3) * a(4,1) - a(3,1) * a(4,3) ) & + a(2,3) * ( a(3,1) * a(4,2) - a(3,2) * a(4,1) ) & ) / det b(1,2) = -( & + a(1,2) * ( a(3,3) * a(4,4) - a(3,4) * a(4,3) ) & + a(1,3) * ( a(3,4) * a(4,2) - a(3,2) * a(4,4) ) & + a(1,4) * ( a(3,2) * a(4,3) - a(3,3) * a(4,2) ) & ) / det b(2,2) = +( & + a(1,1) * ( a(3,3) * a(4,4) - a(3,4) * a(4,3) ) & + a(1,3) * ( a(3,4) * a(4,1) - a(3,1) * a(4,4) ) & + a(1,4) * ( a(3,1) * a(4,3) - a(3,3) * a(4,1) ) & ) / det b(3,2) = -( & + a(1,1) * ( a(3,2) * a(4,4) - a(3,4) * a(4,2) ) & + a(1,2) * ( a(3,4) * a(4,1) - a(3,1) * a(4,4) ) & + a(1,4) * ( a(3,1) * a(4,2) - a(3,2) * a(4,1) ) & ) / det b(4,2) = +( & + a(1,1) * ( a(3,2) * a(4,3) - a(3,3) * a(4,2) ) & + a(1,2) * ( a(3,3) * a(4,1) - a(3,1) * a(4,3) ) & + a(1,3) * ( a(3,1) * a(4,2) - a(3,2) * a(4,1) ) & ) / det b(1,3) = +( & + a(1,2) * ( a(2,3) * a(4,4) - a(2,4) * a(4,3) ) & + a(1,3) * ( a(2,4) * a(4,2) - a(2,2) * a(4,4) ) & + a(1,4) * ( a(2,2) * a(4,3) - a(2,3) * a(4,2) ) & ) / det b(2,3) = -( & + a(1,1) * ( a(2,3) * a(4,4) - a(2,4) * a(4,3) ) & + a(1,3) * ( a(2,4) * a(4,1) - a(2,1) * a(4,4) ) & + a(1,4) * ( a(2,1) * a(4,3) - a(2,3) * a(4,1) ) & ) / det b(3,3) = +( & + a(1,1) * ( a(2,2) * a(4,4) - a(2,4) * a(4,2) ) & + a(1,2) * ( a(2,4) * a(4,1) - a(2,1) * a(4,4) ) & + a(1,4) * ( a(2,1) * a(4,2) - a(2,2) * a(4,1) ) & ) / det b(4,3) = -( & + a(1,1) * ( a(2,2) * a(4,3) - a(2,3) * a(4,2) ) & + a(1,2) * ( a(2,3) * a(4,1) - a(2,1) * a(4,3) ) & + a(1,3) * ( a(2,1) * a(4,2) - a(2,2) * a(4,1) ) & ) / det b(1,4) = -( & + a(1,2) * ( a(2,3) * a(3,4) - a(2,4) * a(3,3) ) & + a(1,3) * ( a(2,4) * a(3,2) - a(2,2) * a(3,4) ) & + a(1,4) * ( a(2,2) * a(3,3) - a(2,3) * a(3,2) ) & ) / det b(2,4) = +( & + a(1,1) * ( a(2,3) * a(3,4) - a(2,4) * a(3,3) ) & + a(1,3) * ( a(2,4) * a(3,1) - a(2,1) * a(3,4) ) & + a(1,4) * ( a(2,1) * a(3,3) - a(2,3) * a(3,1) ) & ) / det b(3,4) = -( & + a(1,1) * ( a(2,2) * a(3,4) - a(2,4) * a(3,2) ) & + a(1,2) * ( a(2,4) * a(3,1) - a(2,1) * a(3,4) ) & + a(1,4) * ( a(2,1) * a(3,2) - a(2,2) * a(3,1) ) & ) / det b(4,4) = +( & + a(1,1) * ( a(2,2) * a(3,3) - a(2,3) * a(3,2) ) & + a(1,2) * ( a(2,3) * a(3,1) - a(2,1) * a(3,3) ) & + a(1,3) * ( a(2,1) * a(3,2) - a(2,2) * a(3,1) ) & ) / det return end function r8mat_is_binary ( m, n, x ) !*****************************************************************************80 ! !! R8MAT_IS_BINARY is true if an R8MAT only contains 0 and 1 entries. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 April 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the dimensions of the array. ! ! Input, real ( kind = 8 ) X(M,N), the array. ! ! Output, logical R8VEC_IS_BINARY, is true if all entries are 0 or 1. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) j logical r8mat_is_binary logical value real ( kind = 8 ) x(m,n) value = .true. do j = 1, n do i = 1, m if ( x(i,j) /= 0.0D+00 .and. x(i,j) /= 1.0D+00 ) then value = .false. exit end if end do end do r8mat_is_binary = value return end subroutine r8mat_is_identity ( n, a, error_frobenius ) !*****************************************************************************80 ! !! R8MAT_IS_IDENTITY determines if an R8MAT is the identity. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The routine returns the Frobenius norm of A - I. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 November 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! ! Input, real ( kind = 8 ) A(N,N), the matrix. ! ! Output, real ( kind = 8 ) ERROR_FROBENIUS, the Frobenius norm ! of the difference matrix A - I, which would be exactly zero ! if A were the identity matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) error_frobenius integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) value value = 0.0D+00 do i = 1, n do j = 1, n if ( i == j ) then value = value + ( a(i,j) - 1.0D+00 ) ** 2 else value = value + a(i,j) ** 2 end if end do end do error_frobenius = sqrt ( error_frobenius ) error_frobenius = value return end function r8mat_is_in_01 ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_IS_IN_01 is TRUE if the entries of an R8MAT are in the range [0,1]. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 October 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, logical ( kind = 4 ) R8MAT_IS_IN_01, is TRUE if every entry of A is ! between 0 and 1. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) logical ( kind = 4 ) r8mat_is_in_01 if ( any ( a(1:m,1:n) < 0.0D+00 .or. 1.0D+00 < a(1:m,1:n) ) ) then r8mat_is_in_01 = .false. else r8mat_is_in_01 = .true. end if return end function r8mat_is_insignificant ( m, n, r, s ) !*****************************************************************************80 ! !! R8MAT_IS_INSIGNIFICANT determines if an R8MAT is insignificant. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the dimension of the matrices. ! ! Input, real ( kind = 8 ) R(M,N), the array to be compared against. ! ! Input, real ( kind = 8 ) S(M,N), the array to be compared. ! ! Output, logical ( kind = 4 ) R8MAT_IS_INSIGNIFICANT, is TRUE if S is ! insignificant compared to R. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) r(m,n) logical ( kind = 4 ) r8mat_is_insignificant real ( kind = 8 ) s(m,n) real ( kind = 8 ) t real ( kind = 8 ) tol logical ( kind = 4 ) value value = .true. do j = 1, n do i = 1, m t = r(i,j) + s(i,j) tol = epsilon ( r(i,j) ) * abs ( r(i,j) ) if ( tol < abs ( r(i,j) - t ) ) then value = .false. exit end if end do end do r8mat_is_insignificant = value return end function r8mat_is_integer ( m, n, x ) !*****************************************************************************80 ! !! R8MAT_IS_INTEGER is true if an R8MAT only contains integer entries. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 August 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the dimensions of the array. ! ! Input, real ( kind = 8 ) X(M,N), the array. ! ! Output, logical R8MAT_IS_INTEGER, is true if all entries are integers. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) j logical r8_is_integer logical r8mat_is_integer logical value real ( kind = 8 ) x(m,n) value = .true. do j = 1, n do i = 1, m if ( .not. r8_is_integer ( x(i,j) ) ) then value = .false. exit end if end do end do r8mat_is_integer = value return end subroutine r8mat_is_nonnegative ( m, n, a, ival ) !*****************************************************************************80 ! !! R8MAT_IS_NONNEGATIVE checks whether an R8MAT is nonnegative. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 October 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the row and column dimensions of ! the matrix. M and N must be positive. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, logical ( kind = 4 ) R8MAT_IS_NONNEGATIVE: ! TRUE, the matrix is nonnegative. ! FALSE, at least one element of A is less than 0. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) logical ( kind = 4 ) ival ival = all ( 0.0D+00 <= a(1:m,1:n) ) return end function r8mat_is_significant ( m, n, r, s ) !*****************************************************************************80 ! !! R8MAT_IS_SIGNIFICANT determines if an R8MAT is relatively significant. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 February 2017 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the dimension of the matrices. ! ! Input, real ( kind = 8 ) R(M,N), the array to be compared against. ! ! Input, real ( kind = 8 ) S(M,N), the array to be compared. ! ! Output, logical ( kind = 4 ) R8MAT_IS_SIGNIFICANT, is TRUE if S is ! significant relative to R. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) r(m,n) logical ( kind = 4 ) r8mat_is_significant real ( kind = 8 ) s(m,n) real ( kind = 8 ) t real ( kind = 8 ) tol logical ( kind = 4 ) value value = .false. do j = 1, n do i = 1, m t = r(i,j) + s(i,j) tol = epsilon ( r(i,j) ) * abs ( r(i,j) ) if ( tol < abs ( r(i,j) - t ) ) then value = .true. exit end if end do end do r8mat_is_significant = value return end subroutine r8mat_is_symmetric ( m, n, a, error_frobenius ) !*****************************************************************************80 ! !! R8MAT_IS_SYMMETRIC checks an R8MAT for symmetry. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 October 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the order of the matrix. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) ERROR_FROBENIUS, measures the ! Frobenius norm of ( A - A' ), which would be zero if the matrix ! were exactly symmetric. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) error_frobenius real ( kind = 8 ) value if ( m /= n ) then value = huge ( value ) else value = sqrt & ( & sum & ( & ( & abs ( a(1:m,1:n) - transpose ( a(1:m,1:n) ) ) & ) ** 2 & ) & ) end if error_frobenius = value return end subroutine r8mat_jac ( m, n, eps, fx, x, fprime ) !*****************************************************************************80 ! !! R8MAT_JAC estimates a dense jacobian matrix of the function FX. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! FPRIME(I,J) = d F(I) / d X(J). ! ! The jacobian is assumed to be dense, and the LINPACK/LAPACK ! double precision general matrix storage mode ("DGE") is used. ! ! Forward differences are used, requiring N+1 function evaluations. ! ! Values of EPS have typically been chosen between ! sqrt ( EPSMCH ) and sqrt ( sqrt ( EPSMCH ) ) where EPSMCH is the ! machine tolerance. ! ! If EPS is too small, then F(X+EPS) will be the same as ! F(X), and the jacobian will be full of zero entries. ! ! If EPS is too large, the finite difference estimate will ! be inaccurate. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of functions. ! ! Input, integer ( kind = 4 ) N, the number of variables. ! ! Input, real ( kind = 8 ) EPS, a tolerance to be used for shifting the ! X values during the finite differencing. No single value ! of EPS will be reliable for all vectors X and functions FX. ! ! Input, external FX, the name of the user written ! routine which evaluates the function at a given point X, of the form: ! subroutine fx ( m, n, x, f ) ! integer ( kind = 4 ) m ! integer ( kind = 4 ) n ! real ( kind = 8 ) f(m) ! real ( kind = 8 ) x(n) ! f(1:m) = ... ! return ! end ! ! Input, real ( kind = 8 ) X(N), the point where the jacobian ! is to be estimated. ! ! Output, real ( kind = 8 ) FPRIME(M,N), the M by N estimated jacobian ! matrix. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) del real ( kind = 8 ) eps real ( kind = 8 ) fprime(m,n) external fx integer ( kind = 4 ) j real ( kind = 8 ) x(n) real ( kind = 8 ) xsave real ( kind = 8 ) work1(m) real ( kind = 8 ) work2(m) ! ! Evaluate the function at the base point, X. ! call fx ( m, n, x, work2 ) ! ! Now, one by one, vary each component J of the base point X, and ! estimate DF(I)/DX(J) = ( F(X+) - F(X) )/ DEL. ! do j = 1, n xsave = x(j) del = eps * ( 1.0D+00 + abs ( x(j) ) ) x(j) = x(j) + del call fx ( m, n, x, work1 ) x(j) = xsave fprime(1:m,j) = ( work1(1:m) - work2(1:m) ) / del end do return end subroutine r8mat_kronecker ( m1, n1, a, m2, n2, b, c ) !*****************************************************************************80 ! !! R8MAT_KRONECKER computes the Kronecker product of two R8MAT's. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! If A is an M1 by N1 array, and B is an M2 by N2 array, then ! the Kronecker product of A and B is an M1*M2 by N1*N2 array ! C(I,J) = A(I1,J1) * B(I2,J2) ! where ! I1 = ( I - 1 ) / M2 + 1 ! I2 = mod ( I - 1, M2 ) + 1 ! J1 = ( J - 1 ) / N2 + 1 ! J2 = mod ( J - 1, N2 ) + 1 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 December 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M1, N1, the order of the first matrix. ! ! Input, real ( kind = 8 ) A(M1,N1), the first matrix. ! ! Input, integer ( kind = 4 ) M2, N2, the order of the second matrix. ! ! Input, real ( kind = 8 ) B(M2,N2), the second matrix. ! ! Output, real ( kind = 8 ) C(M1*M2,N1*N2), the Kronecker product. ! implicit none integer ( kind = 4 ) m1 integer ( kind = 4 ) m2 integer ( kind = 4 ) n1 integer ( kind = 4 ) n2 real ( kind = 8 ) a(m1,n1) real ( kind = 8 ) b(m2,n2) real ( kind = 8 ) c(m1*m2,n1*n2) integer ( kind = 4 ) i integer ( kind = 4 ) i0 integer ( kind = 4 ) i1 integer ( kind = 4 ) i2 integer ( kind = 4 ) j integer ( kind = 4 ) j0 integer ( kind = 4 ) j1 integer ( kind = 4 ) j2 do j1 = 1, n1 do i1 = 1, m1 i0 = ( i1 - 1 ) * m2 j0 = ( j1 - 1 ) * n2 j = j0 do j2 = 1, n2 j = j + 1 i = i0 do i2 = 1, m2 i = i + 1 c(i,j) = a(i1,j1) * b(i2,j2) end do end do end do end do return end subroutine r8mat_l_inverse ( n, a, b ) !*****************************************************************************80 ! !! R8MAT_L_INVERSE inverts a lower triangular R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! A lower triangular matrix is a matrix whose only nonzero entries ! occur on or below the diagonal. ! ! The inverse of a lower triangular matrix is a lower triangular matrix. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, number of rows and columns in the matrix. ! ! Input, real ( kind = 8 ) A(N,N), the lower triangular matrix. ! ! Output, real ( kind = 8 ) B(N,N), the inverse matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) b(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j do j = 1, n do i = 1, n if ( i < j ) then b(i,j) = 0.0D+00 else if ( j == i ) then b(i,j) = 1.0D+00 / a(i,j) else b(i,j) = - dot_product ( a(i,1:i-1), b(1:i-1,j) ) / a(i,i) end if end do end do return end subroutine r8mat_l_print ( m, n, a, title ) !*****************************************************************************80 ! !! R8MAT_L_PRINT prints a lower triangular R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Example: ! ! M = 5, N = 5 ! A = (/ 11, 21, 31, 41, 51, 22, 32, 42, 52, 33, 43, 53, 44, 54, 55 /) ! ! 11 ! 21 22 ! 31 32 33 ! 41 42 43 44 ! 51 52 53 54 55 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 April 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(*), the M by N matrix. Only the lower ! triangular elements are stored, in column major order. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(*) integer ( kind = 4 ) i integer ( kind = 4 ) indx(10) integer ( kind = 4 ) j integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo integer ( kind = 4 ) jmax integer ( kind = 4 ) nn integer ( kind = 4 ) size character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) jmax = min ( n, m ) if ( m <= n ) then size = ( m * ( m + 1 ) ) / 2 else if ( n < m ) then size = ( n * ( n + 1 ) ) / 2 + ( m - n ) * n end if if ( all ( a(1:size) == aint ( a(1:size) ) ) ) then nn = 10 do jlo = 1, jmax, nn jhi = min ( jlo + nn - 1, m, jmax ) write ( *, '(a)' ) ' ' write ( *, '(a8,10i8)' ) ' Col ', ( j, j = jlo, jhi ) write ( *, '(a6)' ) ' Row ' do i = jlo, m jhi = min ( jlo + nn - 1, i, jmax ) do j = jlo, jhi indx(j+1-jlo) = ( j - 1 ) * m + i - ( j * ( j - 1 ) ) / 2 end do write ( *, '(i8,10i8)' ) i, int ( a(indx(1:jhi+1-jlo)) ) end do end do else if ( maxval ( abs ( a(1:size) ) ) < 1000000.0D+00 ) then nn = 5 do jlo = 1, jmax, nn jhi = min ( jlo + nn - 1, m - 1, jmax ) write ( *, '(a)' ) ' ' write ( *, '(8x,5(i8,6x))' ) ( j, j = jlo, jhi ) write ( *, '(a)' ) ' ' do i = jlo, m jhi = min ( jlo + nn - 1, i, jmax ) do j = jlo, jhi indx(j+1-jlo) = ( j - 1 ) * m + i - ( j * ( j - 1 ) ) / 2 end do write ( *, '(i8,5f14.6)' ) i, a(indx(1:jhi+1-jlo)) end do end do else nn = 5 do jlo = 1, jmax, nn jhi = min ( jlo + nn - 1, m - 1, jmax ) write ( *, '(a)' ) ' ' write ( *, '(8x,5(i8,6x))' ) ( j, j = jlo, jhi ) write ( *, '(a)' ) ' ' do i = jlo, m jhi = min ( jlo + nn - 1, i, jmax ) do j = jlo, jhi indx(j+1-jlo) = ( j - 1 ) * m + i - ( j * ( j - 1 ) ) / 2 end do write ( *, '(i8,5g14.6)' ) i, a(indx(1:jhi+1-jlo)) end do end do end if return end subroutine r8mat_l_solve ( n, a, b, x ) !*****************************************************************************80 ! !! R8MAT_L_SOLVE solves a lower triangular linear system. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns of ! the matrix A. ! ! Input, real ( kind = 8 ) A(N,N), the N by N lower triangular matrix. ! ! Input, real ( kind = 8 ) B(N), the right hand side of the linear system. ! ! Output, real ( kind = 8 ) X(N), the solution of the linear system. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) b(n) integer ( kind = 4 ) i real ( kind = 8 ) x(n) ! ! Solve L * x = b. ! do i = 1, n x(i) = ( b(i) - dot_product ( a(i,1:i-1), x(1:i-1) ) ) / a(i,i) end do return end subroutine r8mat_l1_inverse ( n, a, b ) !*****************************************************************************80 ! !! R8MAT_L1_INVERSE inverts a unit lower triangular R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! A unit lower triangular matrix is a matrix with only 1's on the main ! diagonal, and only 0's above the main diagonal. ! ! The inverse of a unit lower triangular matrix is also ! a unit lower triangular matrix. ! ! This routine can invert a matrix in place, that is, with no extra ! storage. If the matrix is stored in A, then the call ! ! call r8mat_l1_inverse ( n, a, a ) ! ! will result in A being overwritten by its inverse. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 December 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, number of rows and columns in the matrix. ! ! Input, real ( kind = 8 ) A(N,N), the unit lower triangular matrix. ! ! Output, real ( kind = 8 ) B(N,N), the inverse matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) b(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j do i = 1, n do j = 1, n if ( i < j ) then b(i,j) = 0.0D+00 else if ( j == i ) then b(i,j) = 1.0D+00 else b(i,j) = -dot_product ( a(i,1:i-1), b(1:i-1,j) ) end if end do end do return end subroutine r8mat_lt_solve ( n, a, b, x ) !*****************************************************************************80 ! !! R8MAT_LT_SOLVE solves a transposed lower triangular linear system. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Given the lower triangular matrix A, the linear system to be solved is: ! ! A' * x = b ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns ! of the matrix. ! ! Input, real ( kind = 8 ) A(N,N), the N by N lower triangular matrix. ! ! Input, real ( kind = 8 ) B(N), the right hand side of the linear system. ! ! Output, real ( kind = 8 ) X(N), the solution of the linear system. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) b(n) integer ( kind = 4 ) i real ( kind = 8 ) x(n) ! ! Solve L'*x = b. ! do i = n, 1, -1 x(i) = ( b(i) - dot_product ( x(i+1:n), a(i+1:n,i) ) ) / a(i,i) end do return end subroutine r8mat_lu ( m, n, a, l, p, u ) !*****************************************************************************80 ! !! R8MAT_LU computes the LU factorization of a rectangular R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The routine is given an M by N matrix A, and produces ! ! L, an M by M unit lower triangular matrix, ! U, an M by N upper triangular matrix, and ! P, an M by M permutation matrix P, ! ! so that ! ! A = P' * L * U. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 April 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix to be factored. ! ! Output, real ( kind = 8 ) L(M,M), the M by M unit lower triangular factor. ! ! Output, real ( kind = 8 ) P(M,M), the M by M permutation matrix. ! ! Output, real ( kind = 8 ) U(M,N), the M by N upper triangular factor. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i integer ( kind = 4 ) ipiv integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) l(m,m) real ( kind = 8 ) p(m,m) real ( kind = 8 ) pivot real ( kind = 8 ) t real ( kind = 8 ) u(m,n) ! ! Initialize: ! ! U:=A ! L:=Identity ! P:=Identity ! u(1:m,1:n) = a(1:m,1:n) call r8mat_identity ( m, l ) p(1:m,1:m) = l(1:m,1:m) ! ! On step J, find the pivot row, IPIV, and the pivot value PIVOT. ! do j = 1, min ( m - 1, n ) pivot = 0.0D+00 ipiv = 0 do i = j, m if ( pivot < abs ( u(i,j) ) ) then pivot = abs ( u(i,j) ) ipiv = i end if end do ! ! Unless IPIV is zero, swap rows J and IPIV. ! if ( ipiv /= 0 ) then do k = 1, n t = u(j,k) u(j,k) = u(ipiv,k) u(ipiv,k) = t t = l(j,k) l(j,k) = l(ipiv,k) l(ipiv,k) = t t = p(j,k) p(j,k) = p(ipiv,k) p(ipiv,k) = t end do ! ! Zero out the entries in column J, from row J+1 to M. ! do i = j + 1, m if ( u(i,j) /= 0.0D+00 ) then l(i,j) = u(i,j) / u(j,j) u(i,j) = 0.0D+00 u(i,j+1:n) = u(i,j+1:n) - l(i,j) * u(j,j+1:n) end if end do end if end do return end function r8mat_max ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_MAX returns the maximum entry of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix. ! ! Output, real ( kind = 8 ) R8MAT_MAX, the maximum entry of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) r8mat_max r8mat_max = maxval ( a(1:m,1:n) ) return end subroutine r8mat_max_columns ( m, n, a, max_col ) !*****************************************************************************80 ! !! R8MAT_MAX_COLUMNS returns the column maximums of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix. ! ! Output, real ( kind = 8 ) MAX_COL(N), the column maximums, ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) j real ( kind = 8 ) max_col(n) do j = 1, n max_col(j) = maxval ( a(1:m,j) ) end do return end subroutine r8mat_max_rows ( m, n, a, max_row ) !*****************************************************************************80 ! !! R8MAT_MAX_ROW returns the row maximums of an R8MAT. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix. ! ! Output, real ( kind = 8 ) MAX_ROW(M), the row maximums. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i real ( kind = 8 ) max_row(m) do i = 1, m max_row(i) = maxval ( a(i,1:n) ) end do return end subroutine r8mat_max_index ( m, n, a, i, j ) !*****************************************************************************80 ! !! R8MAT_MAX_INDEX returns the location of the maximum entry of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix. ! ! Output, integer ( kind = 4 ) I, J, the indices of the maximum entry of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i integer ( kind = 4 ) ii integer ( kind = 4 ) j integer ( kind = 4 ) jj i = -1 j = -1 do jj = 1, n do ii = 1, m if ( ii == 1 .and. jj == 1 ) then i = ii j = jj else if ( a(i,j) < a(ii,jj) ) then i = ii j = jj end if end do end do return end function r8mat_maxcol_minrow ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_MAXCOL_MINROW gets the maximum column minimum row of an M by N R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! R8MAT_MAXCOL_MINROW = max ( 1 <= I <= N ) ( min ( 1 <= J <= M ) A(I,J) ) ! ! For a given matrix, R8MAT_MAXCOL_MINROW <= R8MAT_MINROW_MAXCOL. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) R8MAT_MAXCOL_MINROW, the maximum column ! minimum row entry of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i real ( kind = 8 ) r8mat_maxcol_minrow real ( kind = 8 ) r8mat_minrow r8mat_maxcol_minrow = 0.0D+00 do i = 1, m r8mat_minrow = minval ( a(i,1:n) ) if ( i == 1 ) then r8mat_maxcol_minrow = r8mat_minrow else r8mat_maxcol_minrow = max ( r8mat_maxcol_minrow, r8mat_minrow ) end if end do return end function r8mat_maxrow_mincol ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_MAXROW_MINCOL gets the maximum row minimum column of an M by N R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! R8MAT_MAXROW_MINCOL = max ( 1 <= J <= N ) ( min ( 1 <= I <= M ) A(I,J) ) ! ! For a given matrix, R8MAT_MAXROW_MINCOL <= R8MAT_MINCOL_MAXROW. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) R8MAT_MAXROW_MINCOL, the maximum row ! minimum column entry of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) j real ( kind = 8 ) r8mat_maxrow_mincol real ( kind = 8 ) r8mat_mincol r8mat_maxrow_mincol = 0.0D+00 do j = 1, n r8mat_mincol = minval ( a(1:m,j) ) if ( j == 1 ) then r8mat_maxrow_mincol = r8mat_mincol else r8mat_maxrow_mincol = max ( r8mat_maxrow_mincol, r8mat_mincol ) end if end do return end function r8mat_mean ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_MEAN returns the mean of an M by N R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 03 September 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) R8MAT_MEAN, the mean of all the entries. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) r8mat_mean r8mat_mean = sum ( a(1:m,1:n) ) / real ( m * n, kind = 8 ) return end subroutine r8mat_mean_columns ( m, n, a, means ) !*****************************************************************************80 ! !! R8MAT_MEAN_COLUMNS returns the column means of an M by N R8MAT. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) MEANS(N), the column means. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) j real ( kind = 8 ) means(n) do j = 1, n means(j) = sum ( a(1:m,j) ) / real ( m, kind = 8 ) end do return end subroutine r8mat_mean_rows ( m, n, a, means ) !*****************************************************************************80 ! !! R8MAT_MEAN_ROWS returns the row means of an M by N R8MAT. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) MEANS(M), the row means. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i real ( kind = 8 ) means(m) do i = 1, m means(i) = sum ( a(i,1:n) ) / real ( n, kind = 8 ) end do return end function r8mat_min ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_MIN returns the minimum entry of an M by N R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) R8MAT_MIN, the minimum entry of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) r8mat_min r8mat_min = minval ( a(1:m,1:n) ) return end subroutine r8mat_min_columns ( m, n, a, min_col ) !*****************************************************************************80 ! !! R8MAT_MIN_COLUMNS returns the column minimums of an R8MAT. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix. ! ! Output, real ( kind = 8 ) MIN_COL(N), the column minimums, ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) j real ( kind = 8 ) min_col(n) do j = 1, n min_col(j) = minval ( a(1:m,j) ) end do return end subroutine r8mat_min_rows ( m, n, a, min_row ) !*****************************************************************************80 ! !! R8MAT_MIN_ROW returns the row minimums of an R8MAT. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix. ! ! Output, real ( kind = 8 ) MIN_ROW(M), the row minimums. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i real ( kind = 8 ) min_row(m) do i = 1, m min_row(i) = minval ( a(i,1:n) ) end do return end subroutine r8mat_min_index ( m, n, a, i, j ) !*****************************************************************************80 ! !! R8MAT_MIN_INDEX returns the location of the minimum entry of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix. ! ! Output, integer ( kind = 4 ) I, J, the indices of the minimum entry of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i integer ( kind = 4 ) ii integer ( kind = 4 ) j integer ( kind = 4 ) jj i = -1 j = -1 do jj = 1, n do ii = 1, m if ( ii == 1 .and. jj == 1 ) then i = ii j = jj else if ( a(ii,jj) < a(i,j) ) then i = ii j = jj end if end do end do return end function r8mat_mincol_maxrow ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_MINCOL_MAXROW gets the minimum column maximum row of an M by N R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! R8MAT_MINCOL_MAXROW = min ( 1 <= I <= N ) ( max ( 1 <= J <= M ) A(I,J) ) ! ! For a given matrix, R8MAT_MAXROW_MINCOL <= R8MAT_MINCOL_MAXROW. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) R8MAT_MINCOL_MAXROW, the minimum column ! maximum row entry of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i real ( kind = 8 ) r8mat_mincol_maxrow real ( kind = 8 ) r8mat_maxrow r8mat_mincol_maxrow = 0.0D+00 do i = 1, m r8mat_maxrow = maxval ( a(i,1:n) ) if ( i == 1 ) then r8mat_mincol_maxrow = r8mat_maxrow else r8mat_mincol_maxrow = min ( r8mat_mincol_maxrow, r8mat_maxrow ) end if end do return end function r8mat_minrow_maxcol ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_MINROW_MAXCOL gets the minimum row maximum column of an M by N R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! R8MAT_MINROW_MAXCOL = min ( 1 <= J <= N ) ( max ( 1 <= I <= M ) A(I,J) ) ! ! For a given matrix, R8MAT_MAXCOL_MINROW <= R8MAT_MINROW_MAXCOL. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) R8MAT_MINROW_MAXCOL, the minimum row ! maximum column entry of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) j real ( kind = 8 ) r8mat_minrow_maxcol real ( kind = 8 ) r8mat_maxcol r8mat_minrow_maxcol = 0.0D+00 do j = 1, n r8mat_maxcol = maxval ( a(1:m,j) ) if ( j == 1 ) then r8mat_minrow_maxcol = r8mat_maxcol else r8mat_minrow_maxcol = min ( r8mat_minrow_maxcol, r8mat_maxcol ) end if end do return end subroutine r8mat_minvm ( n1, n2, a, b, c ) !*****************************************************************************80 ! !! R8MAT_MINVM computes inverse(A) * B for R8MAT's. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 28 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N1, N2, the order of the matrices. ! ! Input, real ( kind = 8 ) A(N1,N1), B(N1,N2), the matrices. ! ! Output, real ( kind = 8 ) C(N1,N2), the result, C = inverse(A) * B. ! implicit none integer ( kind = 4 ) n1 integer ( kind = 4 ) n2 real ( kind = 8 ) a(n1,n1) real ( kind = 8 ) alu(n1,n1) real ( kind = 8 ) b(n1,n2) real ( kind = 8 ) c(n1,n2) integer ( kind = 4 ) info alu(1:n1,1:n1) = a(1:n1,1:n1) c(1:n1,1:n2) = b(1:n1,1:n2) call r8mat_fss ( n1, alu, n2, c, info ) if ( info /= 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_MINVM - Fatal error!' write ( *, '(a)' ) ' The matrix A was numerically singular.' stop 1 end if return end subroutine r8mat_mm ( n1, n2, n3, a, b, c ) !*****************************************************************************80 ! !! R8MAT_MM multiplies two R8MAT's. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! In FORTRAN90, this operation is more efficiently done by the ! command: ! ! C(1:N1,1:N3) = MATMUL ( A(1:N1,1;N2), B(1:N2,1:N3) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N1, N2, N3, the order of the matrices. ! ! Input, real ( kind = 8 ) A(N1,N2), B(N2,N3), the matrices to multiply. ! ! Output, real ( kind = 8 ) C(N1,N3), the product matrix C = A * B. ! implicit none integer ( kind = 4 ) n1 integer ( kind = 4 ) n2 integer ( kind = 4 ) n3 real ( kind = 8 ) a(n1,n2) real ( kind = 8 ) b(n2,n3) real ( kind = 8 ) c(n1,n3) c(1:n1,1:n3) = matmul ( a(1:n1,1:n2), b(1:n2,1:n3) ) return end subroutine r8mat_mmt ( n1, n2, n3, a, b, c ) !*****************************************************************************80 ! !! R8MAT_MMT computes C = A * B' for two R8MAT's. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! In FORTRAN90, this operation is more efficiently done by the ! command: ! ! C(1:N1,1:N3) = matmul ( A(1:N1,1;N2) ), transpose ( B(1:N3,1:N2) ) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 November 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N1, N2, N3, the order of the matrices. ! ! Input, real ( kind = 8 ) A(N1,N2), B(N3,N2), the matrices to multiply. ! ! Output, real ( kind = 8 ) C(N1,N3), the product matrix C = A * B. ! implicit none integer ( kind = 4 ) n1 integer ( kind = 4 ) n2 integer ( kind = 4 ) n3 real ( kind = 8 ) a(n1,n2) real ( kind = 8 ) b(n3,n2) real ( kind = 8 ) c(n1,n3) c(1:n1,1:n3) = matmul ( & a(1:n1,1:n2), & transpose ( b(1:n3,1:n2) ) & ) return end subroutine r8mat_mtm ( n1, n2, n3, a, b, c ) !*****************************************************************************80 ! !! R8MAT_MTM computes C = A' * B for two R8MAT's. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! In FORTRAN90, this operation is more efficiently done by the ! command: ! ! C(1:N1,1:N3) = matmul ( transpose ( A(1:N2,1;N1) ), B(1:N2,1:N3) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 September 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N1, N2, N3, the order of the matrices. ! ! Input, real ( kind = 8 ) A(N2,N1), B(N2,N3), the matrices to multiply. ! ! Output, real ( kind = 8 ) C(N1,N3), the product matrix C = A * B. ! implicit none integer ( kind = 4 ) n1 integer ( kind = 4 ) n2 integer ( kind = 4 ) n3 real ( kind = 8 ) a(n2,n1) real ( kind = 8 ) b(n2,n3) real ( kind = 8 ) c(n1,n3) c(1:n1,1:n3) = matmul ( transpose ( a(1:n2,1:n1) ), b(1:n2,1:n3) ) return end subroutine r8mat_mtv ( m, n, a, x, y ) !*****************************************************************************80 ! !! R8MAT_MTV multiplies a transposed matrix times a vector ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns of ! the matrix. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix. ! ! Input, real ( kind = 8 ) X(M), the vector to be multiplied by A. ! ! Output, real ( kind = 8 ) Y(N), the product A'*X. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) x(m) real ( kind = 8 ) y(n) y(1:n) = matmul ( transpose ( a(1:m,1:n) ), x(1:m) ) return end subroutine r8mat_mv ( m, n, a, x, y ) !*****************************************************************************80 ! !! R8MAT_MV multiplies a matrix times a vector. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! In FORTRAN90, this operation can be more efficiently carried ! out by the command ! ! Y(1:M) = MATMUL ( A(1:M,1:N), X(1:N) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns of ! the matrix. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix. ! ! Input, real ( kind = 8 ) X(N), the vector to be multiplied by A. ! ! Output, real ( kind = 8 ) Y(M), the product A*X. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) x(n) real ( kind = 8 ) y(m) y(1:m) = matmul ( a(1:m,1:n), x(1:n) ) return end subroutine r8mat_nint ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_NINT rounds the entries of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input/output, real ( kind = 8 ) A(M,N), the matrix to be NINT'ed. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) a(1:m,1:n) = real ( nint ( a(1:m,1:n) ), kind = 8 ) return end function r8mat_nonzeros ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_NONZEROS counts the nonzeros in an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 31 August 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, integer ( kind = 4 ) R8MAT_NONZEROS, the number of nonzeros. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) r8mat_nonzeros integer ( kind = 4 ) value value = 0 do j = 1, n do i = 1, m if ( a(i,j) /= 0.0D+00 ) then value = value + 1 end if end do end do r8mat_nonzeros = value return end function r8mat_norm_eis ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_NORM_EIS returns the EISPACK norm of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The EISPACK norm is defined as: ! ! R8MAT_NORM_EIS = ! sum ( 1 <= I <= M ) sum ( 1 <= J <= N ) abs ( A(I,J) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix whose EISPACK norm is desired. ! ! Output, real ( kind = 8 ) R8MAT_NORM_EIS, the EISPACK norm of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) r8mat_norm_eis r8mat_norm_eis = sum ( abs ( a(1:m,1:n) ) ) return end function r8mat_norm_fro ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_NORM_FRO returns the Frobenius norm of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The Frobenius norm is defined as ! ! R8MAT_NORM_FRO = sqrt ( ! sum ( 1 <= I <= M ) sum ( 1 <= j <= N ) A(I,J) * A(I,J) ) ! ! The matrix Frobenius norm is not derived from a vector norm, but ! is compatible with the vector L2 norm, so that: ! ! r8vec_norm_l2 ( A * x ) <= r8mat_norm_fro ( A ) * r8vec_norm_l2 ( x ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 March 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix whose Frobenius ! norm is desired. ! ! Output, real ( kind = 8 ) R8MAT_NORM_FRO, the Frobenius norm of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) r8mat_norm_fro r8mat_norm_fro = sqrt ( sum ( a(1:m,1:n) ** 2 ) ) return end function r8mat_norm_fro_affine ( m, n, a1, a2 ) !*****************************************************************************80 ! !! R8MAT_NORM_FRO_AFFINE returns the Frobenius norm of an R8MAT difference. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The Frobenius norm is defined as ! ! R8MAT_NORM_FRO = sqrt ( ! sum ( 1 <= I <= M ) sum ( 1 <= j <= N ) A(I,J) * A(I,J) ) ! ! The matrix Frobenius norm is not derived from a vector norm, but ! is compatible with the vector L2 norm, so that: ! ! r8vec_norm_l2 ( A * x ) <= r8mat_norm_fro ( A ) * r8vec_norm_l2 ( x ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 March 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows. ! ! Input, integer ( kind = 4 ) N, the number of columns. ! ! Input, real ( kind = 8 ) A1(M,N), A2(M,N), the matrices for whose ! difference the Frobenius norm is desired. ! ! Output, real ( kind = 8 ) R8MAT_NORM_FRO_AFFINE, the Frobenius ! norm of A1 - A2. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a1(m,n) real ( kind = 8 ) a2(m,n) real ( kind = 8 ) r8mat_norm_fro_affine r8mat_norm_fro_affine = sqrt ( sum ( ( a1(1:m,1:n) - a2(1:m,1:n) ) ** 2 ) ) return end function r8mat_norm_l1 ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_NORM_L1 returns the matrix L1 norm of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The matrix L1 norm is defined as: ! ! R8MAT_NORM_L1 = max ( 1 <= J <= N ) ! sum ( 1 <= I <= M ) abs ( A(I,J) ). ! ! The matrix L1 norm is derived from the vector L1 norm, and ! satisifies: ! ! r8vec_norm_l1 ( A * x ) <= r8mat_norm_l1 ( A ) * r8vec_norm_l1 ( x ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix whose L1 norm is desired. ! ! Output, real ( kind = 8 ) R8MAT_NORM_L1, the L1 norm of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) col_sum integer ( kind = 4 ) j real ( kind = 8 ) r8mat_norm_l1 r8mat_norm_l1 = 0.0D+00 do j = 1, n col_sum = sum ( abs ( a(1:m,j) ) ) r8mat_norm_l1 = max ( r8mat_norm_l1, col_sum ) end do return end function r8mat_norm_l2 ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_NORM_L2 returns the matrix L2 norm of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The matrix L2 norm is defined as: ! ! R8MAT_NORM_L2 = sqrt ( max ( 1 <= I <= M ) LAMBDA(I) ) ! ! where LAMBDA contains the eigenvalues of A * A'. ! ! The matrix L2 norm is derived from the vector L2 norm, and ! satisifies: ! ! r8vec_norm_l2 ( A * x ) <= r8mat_norm_l2 ( A ) * r8vec_norm_l2 ( x ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 March 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix whose L2 norm is desired. ! ! Output, real ( kind = 8 ) R8MAT_NORM_L2, the L2 norm of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) b(m,m) real ( kind = 8 ) diag(m) real ( kind = 8 ) r8mat_norm_l2 ! ! Compute B = A * A'. ! b(1:m,1:m) = matmul ( a(1:m,1:n), transpose ( a(1:m,1:n) ) ) ! ! Diagonalize B. ! call r8mat_symm_jacobi ( m, b ) ! ! Find the maximum eigenvalue, and take its square root. ! call r8mat_diag_get_vector ( m, b, diag ) r8mat_norm_l2 = sqrt ( maxval ( diag(1:m) ) ) return end function r8mat_norm_li ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_NORM_LI returns the matrix L-oo norm of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The matrix L-oo norm is defined as: ! ! R8MAT_NORM_LI = max ( 1 <= I <= M ) sum ( 1 <= J <= N ) abs ( A(I,J) ). ! ! The matrix L-oo norm is derived from the vector L-oo norm, ! and satisifies: ! ! r8vec_norm_li ( A * x ) <= r8mat_norm_li ( A ) * r8vec_norm_li ( x ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix whose L-oo ! norm is desired. ! ! Output, real ( kind = 8 ) R8MAT_NORM_LI, the L-oo norm of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i real ( kind = 8 ) r8mat_norm_li real ( kind = 8 ) row_sum r8mat_norm_li = 0.0D+00 do i = 1, m row_sum = sum ( abs ( a(i,1:n) ) ) r8mat_norm_li = max ( r8mat_norm_li, row_sum ) end do return end subroutine r8mat_normal_01 ( m, n, seed, r ) !*****************************************************************************80 ! !! R8MAT_NORMAL_01 returns a unit pseudonormal R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 November 2010 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Paul Bratley, Bennett Fox, Linus Schrage, ! A Guide to Simulation, ! Second Edition, ! Springer, 1987, ! ISBN: 0387964673, ! LC: QA76.9.C65.B73. ! ! Bennett Fox, ! Algorithm 647: ! Implementation and Relative Efficiency of Quasirandom ! Sequence Generators, ! ACM Transactions on Mathematical Software, ! Volume 12, Number 4, December 1986, pages 362-376. ! ! Pierre L'Ecuyer, ! Random Number Generation, ! in Handbook of Simulation, ! edited by Jerry Banks, ! Wiley, 1998, ! ISBN: 0471134031, ! LC: T57.62.H37. ! ! Peter Lewis, Allen Goodman, James Miller, ! A Pseudo-Random Number Generator for the System/360, ! IBM Systems Journal, ! Volume 8, 1969, pages 136-143. ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns ! in the array. ! ! Input/output, integer ( kind = 4 ) SEED, the "seed" value, which ! should NOT be 0. On output, SEED has been updated. ! ! Output, real ( kind = 8 ) R(M,N), the array of pseudonormal values. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) seed real ( kind = 8 ) r(m,n) call r8vec_normal_01 ( m * n, seed, r ) return end subroutine r8mat_nullspace ( m, n, a, nullspace_size, nullspace ) !*****************************************************************************80 ! !! R8MAT_NULLSPACE computes the nullspace of a matrix. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Let A be an MxN matrix. ! ! If X is an N-vector, and A*X = 0, then X is a null vector of A. ! ! The set of all null vectors of A is called the nullspace of A. ! ! The 0 vector is always in the null space. ! ! If the 0 vector is the only vector in the nullspace of A, then A ! is said to have maximum column rank. (Because A*X=0 can be regarded ! as a linear combination of the columns of A). In particular, if A ! is square, and has maximum column rank, it is nonsingular. ! ! The dimension of the nullspace is the number of linearly independent ! vectors that span the nullspace. If A has maximum column rank, ! its nullspace has dimension 0. ! ! This routine uses the reduced row echelon form of A to determine ! a set of NULLSPACE_SIZE independent null vectors. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 October 2008 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns of ! the matrix A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix to be analyzed. ! ! Input, integer ( kind = 4 ) NULLSPACE_SIZE, the size of the nullspace. ! ! Output, real ( kind = 8 ) NULLSPACE(N,NULLSPACE_SIZE), vectors that ! span the nullspace. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) nullspace_size real ( kind = 8 ) a(m,n) integer ( kind = 4 ) col(n) real ( kind = 8 ) det integer ( kind = 4 ) i integer ( kind = 4 ) i2 integer ( kind = 4 ) j integer ( kind = 4 ) j2 real ( kind = 8 ) nullspace(n,nullspace_size) integer ( kind = 4 ) row(m) real ( kind = 8 ) rref(m,n) ! ! Make a copy of A. ! rref(1:m,1:n) = a(1:m,1:n) ! ! Get the reduced row echelon form of A. ! call r8mat_rref ( m, n, rref, det ) ! ! Note in ROW the columns of the leading nonzeros. ! COL(J) = +J if there is a leading 1 in that column, and -J otherwise. ! row(1:m) = 0 do j = 1, n col(j) = - j end do do i = 1, m do j = 1, n if ( rref(i,j) == 1.0D+00 ) then row(i) = j col(j) = j exit end if end do end do nullspace(1:n,1:nullspace_size) = 0.0D+00 j2 = 0 ! ! If column J does not contain a leading 1, then it contains ! information about a null vector. ! do j = 1, n if ( col(j) < 0 ) then j2 = j2 + 1 do i = 1, m if ( rref(i,j) /= 0.0D+00 ) then i2 = row(i) nullspace(i2,j2) = - rref(i,j) end if end do nullspace(j,j2) = 1.0D+00 end if end do return end subroutine r8mat_nullspace_size ( m, n, a, nullspace_size ) !*****************************************************************************80 ! !! R8MAT_NULLSPACE_SIZE computes the size of the nullspace of a matrix. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Let A be an MxN matrix. ! ! If X is an N-vector, and A*X = 0, then X is a null vector of A. ! ! The set of all null vectors of A is called the nullspace of A. ! ! The 0 vector is always in the null space. ! ! If the 0 vector is the only vector in the nullspace of A, then A ! is said to have maximum column rank. (Because A*X=0 can be regarded ! as a linear combination of the columns of A). In particular, if A ! is square, and has maximum column rank, it is nonsingular. ! ! The dimension of the nullspace is the number of linearly independent ! vectors that span the nullspace. If A has maximum column rank, ! its nullspace has dimension 0. ! ! This routine ESTIMATES the dimension of the nullspace. Cases of ! singularity that depend on exact arithmetic will probably be missed. ! ! The nullspace will be estimated by counting the leading 1's in the ! reduced row echelon form of A, and subtracting this from N. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 April 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns of ! the matrix A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix to be analyzed. ! ! Output, integer ( kind = 4 ) NULLSPACE_SIZE, the estimated size ! of the nullspace. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) det integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) leading integer ( kind = 4 ) nullspace_size real ( kind = 8 ) rref(m,n) ! ! Get the reduced row echelon form of A. ! rref(1:m,1:n) = a(1:m,1:n) call r8mat_rref ( m, n, rref, det ) ! ! Count the leading 1's in A. ! leading = 0 do i = 1, m do j = 1, n if ( rref(i,j) == 1.0D+00 ) then leading = leading + 1 exit end if end do end do nullspace_size = n - leading return end subroutine r8mat_orth_uniform ( n, seed, q ) !*****************************************************************************80 ! !! R8MAT_ORTH_UNIFORM returns a random orthogonal R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Thanks to Eugene Petrov, B I Stepanov Institute of Physics, ! National Academy of Sciences of Belarus, for convincingly ! pointing out the severe deficiencies of an earlier version of ! this routine. ! ! Essentially, the computation involves saving the Q factor of the ! QR factorization of a matrix whose entries are normally distributed. ! However, it is only necessary to generate this matrix a column at ! a time, since it can be shown that when it comes time to annihilate ! the subdiagonal elements of column K, these (transformed) elements of ! column K are still normally distributed random values. Hence, there ! is no need to generate them at the beginning of the process and ! transform them K-1 times. ! ! For computational efficiency, the individual Householder transformations ! could be saved, as recommended in the reference, instead of being ! accumulated into an explicit matrix format. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 February 2015 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Pete Stewart, ! Efficient Generation of Random Orthogonal Matrices With an Application ! to Condition Estimators, ! SIAM Journal on Numerical Analysis, ! Volume 17, Number 3, June 1980, pages 403-409. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of A. ! ! Input/output, integer ( kind = 4 ) SEED, a seed for the random ! number generator. ! ! Output, real ( kind = 8 ) Q(N,N), the orthogonal matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a_col(n) integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) q(n,n) real ( kind = 8 ) r8_normal_01 integer ( kind = 4 ) seed real ( kind = 8 ) v(n) ! ! Start with Q = the identity matrix. ! do i = 1, n do j = 1, n if ( i == j ) then q(i,j) = 1.0D+00 else q(i,j) = 0.0D+00 end if end do end do ! ! Now behave as though we were computing the QR factorization of ! some other random matrix A. Generate the N elements of the first column, ! compute the Householder matrix H1 that annihilates the subdiagonal elements, ! and set Q := Q * H1' = Q * H. ! ! On the second step, generate the lower N-1 elements of the second column, ! compute the Householder matrix H2 that annihilates them, ! and set Q := Q * H2' = Q * H2 = H1 * H2. ! ! On the N-1 step, generate the lower 2 elements of column N-1, ! compute the Householder matrix HN-1 that annihilates them, and ! and set Q := Q * H(N-1)' = Q * H(N-1) = H1 * H2 * ... * H(N-1). ! This is our random orthogonal matrix. ! do j = 1, n - 1 ! ! Set the vector that represents the J-th column to be annihilated. ! a_col(1:j-1) = 0.0D+00 do i = j, n a_col(i) = r8_normal_01 ( seed ) end do ! ! Compute the vector V that defines a Householder transformation matrix ! H(V) that annihilates the subdiagonal elements of A. ! call r8vec_house_column ( n, a_col, j, v ) ! ! Postmultiply the matrix Q by H'(V) = H(V). ! call r8mat_house_axh ( n, q, v, q ) end do return end subroutine r8mat_plot ( m, n, a, title ) !*****************************************************************************80 ! !! R8MAT_PLOT "plots" an R8MAT, with an optional title. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo character r8mat_plot_symbol character ( len = 70 ) string character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) do jlo = 1, n, 70 jhi = min ( jlo + 70-1, n ) write ( *, '(a)' ) ' ' write ( *, '(8x,2x,70i1)' ) ( mod ( j, 10 ), j = jlo, jhi ) write ( *, '(a)' ) ' ' do i = 1, m do j = jlo, jhi string(j+1-jlo:j+1-jlo) = r8mat_plot_symbol ( a(i,j) ) end do write ( *, '(i8,2x,a)' ) i, string(1:jhi+1-jlo) end do end do return end function r8mat_plot_symbol ( r ) !*****************************************************************************80 ! !! R8MAT_PLOT_SYMBOL returns a symbol for an element of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) R, a value whose symbol is desired. ! ! Output, character R8MAT_PLOT_SYMBOL, is ! '-' if R is negative, ! '0' if R is zero, ! '+' if R is positive. ! implicit none character r8mat_plot_symbol real ( kind = 8 ) r if ( r < 0.0D+00 ) then r8mat_plot_symbol = '-' else if ( r == 0.0D+00 ) then r8mat_plot_symbol = '0' else if ( 0.0D+00 < r ) then r8mat_plot_symbol = '+' end if return end subroutine r8mat_poly_char ( n, a, p ) !*****************************************************************************80 ! !! R8MAT_POLY_CHAR computes the characteristic polynomial of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 March 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix A. ! ! Input, real ( kind = 8 ) A(N,N), the N by N matrix. ! ! Output, real ( kind = 8 ) P(0:N), the coefficients of the characteristic ! polynomial of A. P(N) contains the coefficient of X^N ! (which will be 1), P(I) contains the coefficient of X^I, ! and P(0) contains the constant term. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) order real ( kind = 8 ) p(0:n) real ( kind = 8 ) r8mat_trace real ( kind = 8 ) trace real ( kind = 8 ) work1(n,n) real ( kind = 8 ) work2(n,n) ! ! Initialize WORK1 to the identity matrix. ! call r8mat_identity ( n, work1 ) p(n) = 1.0D+00 do order = n - 1, 0, -1 ! ! Work2 = A * WORK1. ! work2(1:n,1:n) = matmul ( a(1:n,1:n), work1(1:n,1:n) ) ! ! Take the trace. ! trace = r8mat_trace ( n, work2 ) ! ! P(ORDER) = -Trace ( WORK2 ) / ( N - ORDER ) ! p(order) = -trace / real ( n - order, kind = 8 ) ! ! WORK1 := WORK2 + P(ORDER) * Identity. ! work1(1:n,1:n) = work2(1:n,1:n) do i = 1, n work1(i,i) = work1(i,i) + p(order) end do end do return end subroutine r8mat_power ( n, a, npow, b ) !*****************************************************************************80 ! !! R8MAT_POWER computes a nonnegative power of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The algorithm is: ! ! B = I ! do NPOW times: ! B = A * B ! end ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 April 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of A. ! ! Input, real ( kind = 8 ) A(N,N), the matrix to be raised to a power. ! ! Input, integer ( kind = 4 ) NPOW, the power to which A is to be raised. ! NPOW must be nonnegative. ! ! Output, real ( kind = 8 ) B(N,N), the value of A^NPOW. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) b(n,n) integer ( kind = 4 ) ipow integer ( kind = 4 ) npow if ( npow < 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_POWER - Fatal error!' write ( *, '(a)' ) ' Input value of NPOW < 0.' write ( *, '(a,i8)' ) ' NPOW = ', npow stop 1 end if call r8mat_identity ( n, b ) do ipow = 1, npow b(1:n,1:n) = matmul ( a(1:n,1:n), b(1:n,1:n) ) end do return end subroutine r8mat_power_method ( n, a, r, v ) !*****************************************************************************80 ! !! R8MAT_POWER_METHOD applies the power method to an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! If the power method has not converged, then calling the routine ! again immediately with the output from the previous call will ! continue the iteration. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 February 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of A. ! ! Input, real ( kind = 8 ) A(N,N), the matrix. ! ! Output, real ( kind = 8 ) R, the estimated eigenvalue. ! ! Input/output, real ( kind = 8 ) V(N), on input, an estimate ! for the eigenvector. On output, an improved estimate for the ! eigenvector. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) av(n) real ( kind = 8 ) eps integer ( kind = 4 ) it real ( kind = 8 ), parameter :: it_eps = 0.0001D+00 integer ( kind = 4 ), parameter :: it_max = 100 integer ( kind = 4 ), parameter :: it_min = 10 integer ( kind = 4 ) j real ( kind = 8 ) r real ( kind = 8 ) r2 real ( kind = 8 ) r_old real ( kind = 8 ) v(n) eps = sqrt ( epsilon ( 1.0D+00 ) ) r = sqrt ( sum ( v(1:n) ** 2 ) ) if ( r == 0.0D+00 ) then v(1:n) = 1.0D+00 r = sqrt ( real ( n, kind = 8 ) ) end if v(1:n) = v(1:n) / r do it = 1, it_max av(1:n) = matmul ( a(1:n,1:n), v(1:n) ) r_old = r r = sqrt ( sum ( av(1:n) ** 2 ) ) if ( it_min < it ) then if ( abs ( r - r_old ) <= it_eps * ( 1.0D+00 + abs ( r ) ) ) then exit end if end if v(1:n) = av(1:n) if ( r /= 0.0D+00 ) then v(1:n) = v(1:n) / r end if ! ! Perturb V a bit, to avoid cases where the initial guess is exactly ! the eigenvector of a smaller eigenvalue. ! if ( it < it_max / 2 ) then j = 1 + mod ( it - 1, n ) v(j) = v(j) + eps * ( 1.0D+00 + abs ( v(j) ) ) r2 = sqrt ( sum ( v(1:n) ** 2 ) ) v(1:n) = v(1:n) / r2 end if end do return end subroutine r8mat_print ( m, n, a, title ) !*****************************************************************************80 ! !! R8MAT_PRINT prints an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 September 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) character ( len = * ) title call r8mat_print_some ( m, n, a, 1, 1, m, n, title ) return end subroutine r8mat_print_some ( m, n, a, ilo, jlo, ihi, jhi, title ) !*****************************************************************************80 ! !! R8MAT_PRINT_SOME prints some of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 September 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) A(M,N), an M by N matrix to be printed. ! ! Input, integer ( kind = 4 ) ILO, JLO, the first row and column to print. ! ! Input, integer ( kind = 4 ) IHI, JHI, the last row and column to print. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ), parameter :: incx = 5 integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) character ( len = 14 ) ctemp(incx) integer ( kind = 4 ) i integer ( kind = 4 ) i2hi integer ( kind = 4 ) i2lo integer ( kind = 4 ) ihi integer ( kind = 4 ) ilo integer ( kind = 4 ) inc integer ( kind = 4 ) j integer ( kind = 4 ) j2 integer ( kind = 4 ) j2hi integer ( kind = 4 ) j2lo integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) if ( m <= 0 .or. n <= 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) ' (None)' return end if do j2lo = max ( jlo, 1 ), min ( jhi, n ), incx j2hi = j2lo + incx - 1 j2hi = min ( j2hi, n ) j2hi = min ( j2hi, jhi ) inc = j2hi + 1 - j2lo write ( *, '(a)' ) ' ' do j = j2lo, j2hi j2 = j + 1 - j2lo write ( ctemp(j2), '(i8,6x)' ) j end do write ( *, '('' Col '',5a14)' ) ctemp(1:inc) write ( *, '(a)' ) ' Row' write ( *, '(a)' ) ' ' i2lo = max ( ilo, 1 ) i2hi = min ( ihi, m ) do i = i2lo, i2hi do j2 = 1, inc j = j2lo - 1 + j2 if ( a(i,j) == real ( int ( a(i,j) ), kind = 8 ) ) then write ( ctemp(j2), '(f8.0,6x)' ) a(i,j) else write ( ctemp(j2), '(g14.6)' ) a(i,j) end if end do write ( *, '(i5,a,5a14)' ) i, ':', ( ctemp(j), j = 1, inc ) end do end do return end subroutine r8mat_print2 ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_PRINT2 prints an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 August 2002 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows of A. ! ! Input, integer ( kind = 4 ) N, the number of columns of A. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix to be printed. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) amax real ( kind = 8 ) amin integer ( kind = 4 ) i character ( len = 10 ) iform integer ( kind = 4 ) ihi integer ( kind = 4 ) ilo logical ( kind = 4 ) integ integer ( kind = 4 ) j integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo integer ( kind = 4 ) lmax integer ( kind = 4 ) npline real ( kind = 8 ) r8_log_10 ! ! Check if all entries are integral. ! integ = .true. do i = 1, m do j = 1, n if ( integ ) then if ( a(i,j) /= real ( int ( a(i,j) ), kind = 8 ) ) then integ = .false. end if end if end do end do ! ! Find the maximum and minimum entries. ! amax = maxval ( a(1:m,1:n) ) amin = minval ( a(1:m,1:n) ) ! ! Use the information about the maximum size of an entry to ! compute an intelligent format for use with integer entries. ! ! Later, we might also do this for real matrices. ! lmax = int ( r8_log_10 ( amax ) ) if ( integ ) then npline = 79 / ( lmax + 3 ) write ( iform, '(''('',i2,''I'',i2,'')'')' ) npline, lmax+3 else npline = 5 iform = ' ' end if ! ! Print a scalar quantity. ! if ( m == 1 .and. n == 1 ) then if ( integ ) then write ( *, iform ) int ( a(1,1) ) else write ( *, '(2x,g14.6)' ) a(1,1) end if ! ! Column vector of length M, ! else if ( n == 1 ) then do ilo = 1, m, npline ihi = min ( ilo+npline-1, m ) if ( integ ) then write ( *, iform ) ( int ( a(i,1) ), i = ilo, ihi ) else write ( *, '(2x,5g14.6)' ) a(ilo:ihi,1) end if end do ! ! Row vector of length N, ! else if ( m == 1 ) then do jlo = 1, n, npline jhi = min ( jlo+npline-1, n ) if ( integ ) then write ( *, iform ) int ( a(1,jlo:jhi) ) else write ( *, '(2x,5g14.6)' ) a(1,jlo:jhi) end if end do ! ! M by N Array ! else do jlo = 1, n, npline jhi = min ( jlo+npline-1, n ) if ( npline < n ) then write ( *, '(a)' ) ' ' write ( *, '(a,i8,a,i8)' ) 'Matrix columns ', jlo, ' to ', jhi write ( *, '(a)' ) ' ' end if do i = 1, m if ( integ ) then write ( *, iform ) int ( a(i,jlo:jhi) ) else write ( *, '(2x,5g14.6)' ) a(i,jlo:jhi) end if end do end do end if return end function r8mat_product_elementwise ( m, n, a, b ) !*****************************************************************************80 ! !! R8MAT_PRODUCT_ELEMENTWISE returns the elementwise produce to two R8MAT's. ! ! Example: ! ! A = [ 1, 2, 3; B = [ 1, 3, 5; product = 86 ! 4, 5, 6 ] 2, 4, 6 ] ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 31 March 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows. ! ! Input, integer ( kind = 4 ) N, the number of columns. ! ! Input, real ( kind = 8 ) A(M,N), B(M,N), the two matrices. ! ! Output, real ( kind = 8 ) I4MAT_PRODUCT_ELEMENTWISE, the elementwise ! product of A and B. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) b(m,n) integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) r8mat_product_elementwise real ( kind = 8 ) value value = 0.0D+00 do j = 1, n do i = 1, m value = value + a(i,j) * b(i,j) end do end do r8mat_product_elementwise = value return end subroutine r8mat_ref ( m, n, a, det ) !*****************************************************************************80 ! !! R8MAT_REF computes the row echelon form of a matrix. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! A matrix is in row echelon form if: ! ! * The first nonzero entry in each row is 1. ! ! * The leading 1 in a given row occurs in a column to ! the right of the leading 1 in the previous row. ! ! * Rows which are entirely zero must occur last. ! ! Example: ! ! Input matrix: ! ! 1.0 3.0 0.0 2.0 6.0 3.0 1.0 ! -2.0 -6.0 0.0 -2.0 -8.0 3.0 1.0 ! 3.0 9.0 0.0 0.0 6.0 6.0 2.0 ! -1.0 -3.0 0.0 1.0 0.0 9.0 3.0 ! ! Output matrix: ! ! 1.0 3.0 0.0 2.0 6.0 3.0 1.0 ! 0.0 0.0 0.0 1.0 2.0 4.5 1.5 ! 0.0 0.0 0.0 0.0 0.0 1.0 0.3 ! 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 April 2018 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Charles Cullen, ! An Introduction to Numerical Linear Algebra, ! PWS Publishing Company, 1994, ! ISBN: 978-0534936903, ! LC: QA185.D37.C85. ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns of ! the matrix A. ! ! Input/output, real ( kind = 8 ) A(M,N). On input, the matrix to be ! analyzed. On output, the REF form of the matrix. ! ! Output, real ( kind = 8 ) DET, the pseudo-determinant. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) det integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) lead integer ( kind = 4 ) r real ( kind = 8 ) r8_epsilon real ( kind = 8 ) temp real ( kind = 8 ) tol det = 1.0D+00 tol = r8_epsilon ( ) * sum ( abs ( a(1:m,1:n) ) ) lead = 1 do r = 1, m if ( n < lead ) then exit end if i = r do while ( abs ( a(i,lead) ) <= tol ) i = i + 1 if ( m < i ) then i = r lead = lead + 1 if ( n < lead ) then lead = -1 exit end if end if end do if ( lead < 0 ) then exit end if do j = 1, n temp = a(i,j) a(i,j) = a(r,j) a(r,j) = temp end do det = det * a(r,lead) a(r,1:n) = a(r,1:n) / a(r,lead) do i = r + 1, m a(i,1:n) = a(i,1:n) - a(i,lead) * a(r,1:n) end do lead = lead + 1 end do return end function r8mat_rms ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_RMS returns the RMS norm of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The matrix RMS norm is defined as: ! ! R8MAT_RMS = sqrt ( ! sum ( 1 <= I <= M ) sum ( 1 <= J <= N ) A(I,J)^2 / M / N ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 December 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the dimensions of the matrix. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) R8MAT_RMS, the RMS norm of A. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) r8mat_rms r8mat_rms = sqrt ( sum ( a(1:m,1:n) ** 2 ) & / real ( m, kind = 8 ) / real ( n, kind = 8 ) ) return end subroutine r8mat_row_copy ( m, n, i, v, a ) !*****************************************************************************80 ! !! R8MAT_ROW_COPY copies a vector into a row of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 June 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the order of the matrix. ! ! Input, integer ( kind = 4 ) I, the index of the row. ! 1 <= I <= M. ! ! Input, real ( kind = 8 ) V(N), the row to be copied. ! ! Input/output, real ( kind = 8 ) A(M,N), the matrix into which ! the row is to be copied. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i real ( kind = 8 ) v(n) a(i,1:n) = v(1:n) return end subroutine r8mat_row_set ( i, r, m, n, a ) !*****************************************************************************80 ! !! R8MAT_ROW_SET copies a vector into a row of an R8MAT. ! ! Discussion: ! ! Because I try to avoid using "leading dimensions", which allow ! a user to set aside too much space for an array, but then ! still put things in the right place, I need to use a routine ! like this when I occasionally have to deal with arrays that ! are not "tight". ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 February 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) I, the row index. ! ! Input, real ( kind = 8 ) R(N), the vector. ! ! Input, integer ( kind = 4 ) M, N, the number of rows and ! columns of the matrix. ! ! Input/output, real ( kind = 8 ) A(M,N), the matrix to be updated. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i real ( kind = 8 ) r(n) a(i,1:n) = r(1:n) return end subroutine r8mat_rref ( m, n, a, det ) !*****************************************************************************80 ! !! R8MAT_RREF computes the reduced row echelon form of a matrix. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! A matrix is in row echelon form if: ! ! * The first nonzero entry in each row is 1. ! ! * The leading 1 in a given row occurs in a column to ! the right of the leading 1 in the previous row. ! ! * Rows which are entirely zero must occur last. ! ! The matrix is in reduced row echelon form if, in addition to ! the first three conditions, it also satisfies: ! ! * Each column containing a leading 1 has no other nonzero entries. ! ! Example: ! ! Input matrix: ! ! 1.0 3.0 0.0 2.0 6.0 3.0 1.0 ! -2.0 -6.0 0.0 -2.0 -8.0 3.0 1.0 ! 3.0 9.0 0.0 0.0 6.0 6.0 2.0 ! -1.0 -3.0 0.0 1.0 0.0 9.0 3.0 ! ! Output matrix: ! ! 1.0 3.0 0.0 0.0 2.0 0.0 0.0 ! 0.0 0.0 0.0 1.0 2.0 0.0 0.0 ! 0.0 0.0 0.0 0.0 0.0 1.0 0.3 ! 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 April 2018 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Charles Cullen, ! An Introduction to Numerical Linear Algebra, ! PWS Publishing Company, 1994, ! ISBN: 978-0534936903, ! LC: QA185.D37.C85. ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns of ! the matrix A. ! ! Input/output, real ( kind = 8 ) A(M,N). On input, the matrix to be ! analyzed. On output, the RREF form of the matrix. ! ! Output, real ( kind = 8 ) DET, the pseudo-determinant. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) det integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) lead integer ( kind = 4 ) r real ( kind = 8 ) r8_epsilon real ( kind = 8 ) temp real ( kind = 8 ) tol det = 1.0D+00 tol = r8_epsilon ( ) * sum ( abs ( a(1:m,1:n) ) ) lead = 1 do r = 1, m if ( n < lead ) then exit end if i = r do while ( abs ( a(i,lead) ) <= tol ) i = i + 1 if ( m < i ) then i = r lead = lead + 1 if ( n < lead ) then lead = -1 exit end if end if end do if ( lead < 0 ) then exit end if do j = 1, n temp = a(i,j) a(i,j) = a(r,j) a(r,j) = temp end do det = det * a(r,lead) a(r,1:n) = a(r,1:n) / a(r,lead) do i = 1, m if ( i /= r ) then a(i,1:n) = a(i,1:n) - a(i,lead) * a(r,1:n) end if end do lead = lead + 1 end do return end subroutine r8mat_scale ( m, n, s, a ) !*****************************************************************************80 ! !! R8MAT_SCALE multiplies an R8MAT by a scalar. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 December 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) S, the scale factor. ! ! Input/output, real ( kind = 8 ) A(M,N), the matrix to be scaled. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) s a(1:m,1:n) = a(1:m,1:n) * s return end subroutine r8mat_scale_01 ( m, n, x, xs ) !*****************************************************************************80 ! !! R8MAT_SCALE_01 scales an R8MAT to [0,1]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) X(M,N), the array to be scaled. ! ! Output, real ( kind = 8 ) XS(M,N), the scaled array. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) j real ( kind = 8 ) x(m,n) real ( kind = 8 ) xmax(n) real ( kind = 8 ) xmin(n) real ( kind = 8 ) xs(m,n) xs(1:m,1:n) = x(1:m,1:n) call r8mat_max_columns ( m, n, x, xmax ) call r8mat_min_columns ( m, n, x, xmin ) do j = 1, n if ( 0.0D+00 < xmax(j) - xmin(j) ) then xs(1:m,j) = ( xs(1:m,j) - xmin(j) ) / ( xmax(j) - xmin(j) ) else xs(1:m,j) = 0.5D+00 end if end do return end subroutine r8mat_scale_ab ( m, n, x, a, b, xs ) !*****************************************************************************80 ! !! R8MAT_SCALE_AB scales an R8MAT to [A,B]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) X(M,N), the array to be scaled. ! ! Input, real ( kind = 8 ) A, B, the new limits. ! ! Output, real ( kind = 8 ) XS(M,N), the scaled array. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b integer ( kind = 4 ) j real ( kind = 8 ) x(m,n) real ( kind = 8 ) xmax(n) real ( kind = 8 ) xmin(n) real ( kind = 8 ) xs(m,n) xs(1:m,1:n) = x(1:m,1:n) call r8mat_max_columns ( m, n, x, xmax ) call r8mat_min_columns ( m, n, x, xmin ) do j = 1, n if ( 0.0D+00 < xmax(j) - xmin(j) ) then xs(1:m,j) = a + ( b - a ) * ( xs(1:m,j) - xmin(j) ) / ( xmax(j) - xmin(j) ) else xs(1:m,j) = ( a + b ) / 2.0D+00 end if end do return end subroutine r8mat_solve ( n, rhs_num, a, info ) !*****************************************************************************80 ! !! R8MAT_SOLVE uses Gauss-Jordan elimination to solve an N by N linear system. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 August 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! ! Input, integer ( kind = 4 ) RHS_NUM, the number of right hand sides. ! RHS_NUM must be at least 0. ! ! Input/output, real ( kind = 8 ) A(N,N+RHS_NUM), contains in rows and ! columns 1 to N the coefficient matrix, and in columns N+1 through ! N+RHS_NUM, the right hand sides. On output, the coefficient matrix ! area has been destroyed, while the right hand sides have ! been overwritten with the corresponding solutions. ! ! Output, integer ( kind = 4 ) INFO, singularity flag. ! 0, the matrix was not singular, the solutions were computed; ! J, factorization failed on step J, and the solutions could not ! be computed. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) rhs_num real ( kind = 8 ) a(n,n+rhs_num) real ( kind = 8 ) apivot real ( kind = 8 ) factor integer ( kind = 4 ) i integer ( kind = 4 ) info integer ( kind = 4 ) ipivot integer ( kind = 4 ) j real ( kind = 8 ) t(n+rhs_num) info = 0 do j = 1, n ! ! Choose a pivot row. ! ipivot = j apivot = a(j,j) do i = j + 1, n if ( abs ( apivot ) < abs ( a(i,j) ) ) then apivot = a(i,j) ipivot = i end if end do if ( apivot == 0.0D+00 ) then info = j return end if ! ! The pivot row moves into the J-th row. ! if ( ipivot /= j ) then t( 1:n+rhs_num) = a(ipivot,1:n+rhs_num) a(ipivot,1:n+rhs_num) = a(j, 1:n+rhs_num) a(j, 1:n+rhs_num) = t( 1:n+rhs_num) end if ! ! A(J,J) becomes 1. ! a(j,j) = 1.0D+00 a(j,j+1:n+rhs_num) = a(j,j+1:n+rhs_num) / apivot ! ! A(I,J) becomes 0. ! do i = 1, n if ( i /= j ) then factor = a(i,j) a(i,j) = 0.0D+00 a(i,j+1:n+rhs_num) = a(i,j+1:n+rhs_num) - factor * a(j,j+1:n+rhs_num) end if end do end do return end subroutine r8mat_solve_2d ( a, b, det, x ) !*****************************************************************************80 ! !! R8MAT_SOLVE_2D solves a 2 by 2 linear system using Cramer's rule. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! If the determinant DET is returned as zero, then the matrix A is ! singular, and does not have an inverse. In that case, X is ! returned as the zero vector. ! ! If DET is nonzero, then its value is roughly an estimate ! of how nonsingular the matrix A is. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 November 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A(2,2), the matrix. ! ! Input, real ( kind = 8 ) B(2), the right hand side. ! ! Output, real ( kind = 8 ) DET, the determinant of the matrix A. ! ! Output, real ( kind = 8 ) X(2), the solution of the system, ! if DET is nonzero. ! implicit none real ( kind = 8 ) a(2,2) real ( kind = 8 ) b(2) real ( kind = 8 ) det real ( kind = 8 ) x(2) ! ! Compute the determinant. ! det = a(1,1) * a(2,2) - a(1,2) * a(2,1) ! ! If the determinant is zero, bail out. ! if ( det == 0.0D+00 ) then x(1:2) = 0.0D+00 return end if ! ! Compute the solution. ! x(1) = ( a(2,2) * b(1) - a(1,2) * b(2) ) / det x(2) = ( -a(2,1) * b(1) + a(1,1) * b(2) ) / det return end subroutine r8mat_solve_3d ( a, b, det, x ) !*****************************************************************************80 ! !! R8MAT_SOLVE_3D solves a 3 by 3 linear system using Cramer's rule. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! If the determinant DET is returned as zero, then the matrix A is ! singular, and does not have an inverse. In that case, X is ! returned as the zero vector. ! ! If DET is nonzero, then its value is roughly an estimate ! of how nonsingular the matrix A is. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 December 2006 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A(3,3), the matrix. ! ! Input, real ( kind = 8 ) B(3), the right hand side. ! ! Output, real ( kind = 8 ) DET, the determinant of the matrix A. ! ! Output, real ( kind = 8 ) X(3), the solution of the system, ! if DET is nonzero. ! implicit none real ( kind = 8 ) a(3,3) real ( kind = 8 ) b(3) real ( kind = 8 ) det real ( kind = 8 ) x(3) ! ! Compute the determinant. ! det = a(1,1) * ( a(2,2) * a(3,3) - a(2,3) * a(3,2) ) & + a(1,2) * ( a(2,3) * a(3,1) - a(2,1) * a(3,3) ) & + a(1,3) * ( a(2,1) * a(3,2) - a(2,2) * a(3,1) ) ! ! If the determinant is zero, bail out. ! if ( det == 0.0D+00 ) then x(1:3) = 0.0D+00 return end if ! ! Compute the solution. ! x(1) = ( ( a(2,2) * a(3,3) - a(2,3) * a(3,2) ) * b(1) & - ( a(1,2) * a(3,3) - a(1,3) * a(3,2) ) * b(2) & + ( a(1,2) * a(2,3) - a(1,3) * a(2,2) ) * b(3) ) / det x(2) = ( - ( a(2,1) * a(3,3) - a(2,3) * a(3,1) ) * b(1) & + ( a(1,1) * a(3,3) - a(1,3) * a(3,1) ) * b(2) & - ( a(1,1) * a(2,3) - a(1,3) * a(2,1) ) * b(3) ) / det x(3) = ( ( a(2,1) * a(3,2) - a(2,2) * a(3,1) ) * b(1) & - ( a(1,1) * a(3,2) - a(1,2) * a(3,1) ) * b(2) & + ( a(1,1) * a(2,2) - a(1,2) * a(2,1) ) * b(3) ) / det return end subroutine r8mat_solve2 ( n, a, b, x, ierror ) !*****************************************************************************80 ! !! R8MAT_SOLVE2 computes the solution of an N by N linear system. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The linear system may be represented as ! ! A*X = B ! ! If the linear system is singular, but consistent, then the routine will ! still produce a solution. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 October 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of equations. ! ! Input/output, real ( kind = 8 ) A(N,N). ! On input, A is the coefficient matrix to be inverted. ! On output, A has been overwritten. ! ! Input/output, real ( kind = 8 ) B(N). ! On input, B is the right hand side of the system. ! On output, B has been overwritten. ! ! Output, real ( kind = 8 ) X(N), the solution of the linear system. ! ! Output, integer ( kind = 4 ) IERROR. ! 0, no error detected. ! 1, consistent singularity. ! 2, inconsistent singularity. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) amax real ( kind = 8 ) b(n) integer ( kind = 4 ) i integer ( kind = 4 ) ierror integer ( kind = 4 ) imax integer ( kind = 4 ) ipiv(n) integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) x(n) ierror = 0 ipiv(1:n) = 0 x(1:n) = 0.0D+00 ! ! Process the matrix. ! do k = 1, n ! ! In column K: ! Seek the row IMAX with the properties that: ! IMAX has not already been used as a pivot; ! A(IMAX,K) is larger in magnitude than any other candidate. ! amax = 0.0D+00 imax = 0 do i = 1, n if ( ipiv(i) == 0 ) then if ( amax < abs ( a(i,k) ) ) then imax = i amax = abs ( a(i,k) ) end if end if end do ! ! If you found a pivot row IMAX, then, ! eliminate the K-th entry in all rows that have not been used for pivoting. ! if ( imax /= 0 ) then ipiv(imax) = k a(imax,k+1:n) = a(imax,k+1:n) / a(imax,k) b(imax) = b(imax) / a(imax,k) a(imax,k) = 1.0D+00 do i = 1, n if ( ipiv(i) == 0 ) then a(i,k+1:n) = a(i,k+1:n) - a(i,k) * a(imax,k+1:n) b(i) = b(i) - a(i,k) * b(imax) a(i,k) = 0.0D+00 end if end do end if end do ! ! Now, every row with nonzero IPIV begins with a 1, and ! all other rows are all zero. Begin solution. ! do j = n, 1, -1 imax = 0 do k = 1, n if ( ipiv(k) == j ) then imax = k end if end do if ( imax == 0 ) then x(j) = 0.0D+00 if ( b(j) == 0.0D+00 ) then ierror = 1 write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_SOLVE2 - Warning:' write ( *, '(a,i8)' ) ' Consistent singularity, equation = ', j else ierror = 2 write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_SOLVE2 - Error:' write ( *, '(a,i8)' ) ' Inconsistent singularity, equation = ', j end if else x(j) = b(imax) do i = 1, n if ( i /= imax ) then b(i) = b(i) - a(i,j) * x(j) end if end do end if end do return end subroutine r8mat_standardize ( m, n, x, xs ) !*****************************************************************************80 ! !! R8MAT_STANDARDIZE standardizes an R8MAT. ! ! Discussion: ! ! The output array will have columns of 0 mean and unit standard deviation. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) X(M,N), the array to be standardized. ! ! Output, real ( kind = 8 ) XS(M,N), the standardized array. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) j real ( kind = 8 ) mu(n) real ( kind = 8 ) sigma(n) real ( kind = 8 ) x(m,n) real ( kind = 8 ) xs(m,n) call r8mat_mean_columns ( m, n, x, mu ) call r8mat_std_columns ( m, n, x, sigma ) do j = 1, n if ( sigma(j) /= 0.0D+00 ) then xs(1:m,j) = ( x(1:m,j) - mu(j) ) / sigma(j) else xs(1:m,j) = 0.0D+00 end if end do return end subroutine r8mat_std_columns ( m, n, a, stds ) !*****************************************************************************80 ! !! R8MAT_STD_COLUMNS returns the column standard deviations of an M by N R8MAT. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) R8MAT_STD_COLUMNS(N), the column stds. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) j real ( kind = 8 ) stds(n) do j = 1, n call r8vec_std_sample ( m, a(1:m,j), stds(j) ) end do return end subroutine r8mat_std_rows ( m, n, a, stds ) !*****************************************************************************80 ! !! R8MAT_STD_ROWS returns the row standard deviations of an M by N R8MAT. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(M,N), the matrix. ! ! Output, real ( kind = 8 ) R8MAT_STD_ROWS(M), the row stds. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) integer ( kind = 4 ) i real ( kind = 8 ) stds(m) do i = 1, m call r8vec_std_sample ( n, a(i,1:n), stds(i) ) end do return end function r8mat_sum ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_SUM returns the sum of the entries of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! In FORTRAN90, this facility is offered by the built in ! SUM function: ! ! R8MAT_SUM ( M, N, A ) = SUM ( A(1:M,1:N) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 January 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, real ( kind = 8 ) R8MAT_SUM, the sum of the entries. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) r8mat_sum r8mat_sum = sum ( a(1:m,1:n) ) return end subroutine r8mat_symm_eigen ( n, x, q, a ) !*****************************************************************************80 ! !! R8MAT_SYMM_EIGEN returns a symmetric matrix with given eigensystem. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The user must supply the desired eigenvalue vector, and the desired ! eigenvector matrix. The eigenvector matrix must be orthogonal. A ! suitable random orthogonal matrix can be generated by R8MAT_ORTH_UNIFORM. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 October 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of A. ! ! Input, real ( kind = 8 ) X(N), the desired eigenvalues for the matrix. ! ! Input, real ( kind = 8 ) Q(N,N), the eigenvector matrix of A. ! ! Output, real ( kind = 8 ) A(N,N), a symmetric matrix with ! eigenvalues X and eigenvectors the columns of Q. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) q(n,n) real ( kind = 8 ) x(n) ! ! Set A = Q * Lambda * Q'. ! a(1:n,1:n) = 0.0D+00 do i = 1, n do j = 1, n do k = 1, n a(i,j) = a(i,j) + q(i,k) * x(k) * q(j,k) end do end do end do return end subroutine r8mat_symm_jacobi ( n, a ) !*****************************************************************************80 ! !! R8MAT_SYMM_JACOBI applies Jacobi eigenvalue iteration to a symmetric matrix. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! This code was modified so that it treats as zero the off-diagonal ! elements that are sufficiently close to, but not exactly, zero. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 June 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of A. ! ! Input/output, real ( kind = 8 ) A(N,N), a symmetric N by N matrix. ! On output, the matrix has been overwritten by an approximately ! diagonal matrix, with the eigenvalues on the diagonal. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) c real ( kind = 8 ) r8mat_norm_fro real ( kind = 8 ), parameter :: eps = 0.00001D+00 integer ( kind = 4 ) i integer ( kind = 4 ) it integer ( kind = 4 ), parameter :: it_max = 100 integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) norm_fro real ( kind = 8 ) s real ( kind = 8 ) sum2 real ( kind = 8 ) t real ( kind = 8 ) t1 real ( kind = 8 ) t2 real ( kind = 8 ) u norm_fro = r8mat_norm_fro ( n, n, a ) it = 0 do it = it + 1 do i = 1, n do j = 1, i - 1 if ( eps * norm_fro < abs ( a(i,j) ) + abs ( a(j,i) ) ) then u = ( a(j,j) - a(i,i) ) / ( a(i,j) + a(j,i) ) t = sign ( 1.0D+00, u ) / ( abs ( u ) + sqrt ( u * u + 1.0D+00 ) ) c = 1.0D+00 / sqrt ( t * t + 1.0D+00 ) s = t * c ! ! A -> A * Q. ! do k = 1, n t1 = a(i,k) t2 = a(j,k) a(i,k) = t1 * c - t2 * s a(j,k) = t1 * s + t2 * c end do ! ! A -> QT * A ! do k = 1, n t1 = a(k,i) t2 = a(k,j) a(k,i) = c * t1 - s * t2 a(k,j) = s * t1 + c * t2 end do end if end do end do ! ! Test the size of the off-diagonal elements. ! sum2 = 0.0D+00 do i = 1, n do j = 1, i - 1 sum2 = sum2 + abs ( a(i,j) ) end do end do if ( sum2 <= eps * ( norm_fro + 1.0D+00 ) ) then exit end if if ( it_max <= it ) then exit end if end do return end subroutine r8mat_to_r8cmat ( lda, m, n, a1, a2 ) !*****************************************************************************80 ! !! R8MAT_TO_R8CMAT transfers data from an R8MAT to an R8CMAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, ! accessible as a vector: ! (I,J) -> (I+J*M). ! or as a doubly-dimensioned array, if declared A(M,N): ! (I,J) -> A(I,J) ! ! An R8CMAT is an MxN array of R8's, stored with a leading dimension LD, ! accessible as a vector: ! (I,J) -> (I+J*LD). ! or as a doubly-dimensioned array, if declared A(LD,N): ! (I,J) -> A(I,J) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 March 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) LDA, the leading dimension of A2. ! ! Input, integer ( kind = 4 ) M, the number of rows of data. ! M <= LDA. ! ! Input, integer ( kind = 4 ) N, the number of columns of data. ! ! Input, real ( kind = 8 ) A1(M,N), the matrix to be copied. ! ! Output, real ( kind = 8 ) A2(LDA,N), contains a copy of the ! information in A1, in the MxN submatrix. ! implicit none integer ( kind = 4 ) lda integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a1(m,n) real ( kind = 8 ) a2(lda,n) a2(1:m,1:n) = a1(1:m,1:n) return end subroutine r8mat_to_r8plu ( n, a, pivot, lu, info ) !*****************************************************************************80 ! !! R8MAT_TO_R8PLU factors a general R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! This routine is a simplified version of the LINPACK routine DGEFA. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 December 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, ! LINPACK User's Guide, ! SIAM, 1979, ! ISBN13: 978-0-898711-72-1. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! N must be positive. ! ! Input, real ( kind = 8 ) A(N,N), the matrix to be factored. ! ! Output, integer ( kind = 4 ) PIVOT(N), a vector of pivot indices. ! ! Output, real ( kind = 8 ) LU(N,N), an upper triangular matrix U and ! the multipliers L which were used to obtain it. The factorization ! can be written A = L * U, where L is a product of permutation and ! unit lower triangular matrices and U is upper triangular. ! ! Output, integer ( kind = 4 ) INFO, singularity flag. ! 0, no singularity detected. ! nonzero, the factorization failed on the INFO-th step. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) info integer ( kind = 4 ) pivot(n) integer ( kind = 4 ) j integer ( kind = 4 ) k integer ( kind = 4 ) l real ( kind = 8 ) lu(n,n) real ( kind = 8 ) temp lu(1:n,1:n) = a(1:n,1:n) info = 0 do k = 1, n - 1 ! ! Find L, the index of the pivot row. ! l = k do i = k + 1, n if ( abs ( lu(l,k) ) < abs ( lu(i,k) ) ) then l = i end if end do pivot(k) = l ! ! If the pivot index is zero, the algorithm has failed. ! if ( lu(l,k) == 0.0D+00 ) then info = k write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_TO_R8PLU - Fatal error!' write ( *, '(a,i8)' ) ' Zero pivot on step ', info return end if ! ! Interchange rows L and K if necessary. ! if ( l /= k ) then temp = lu(l,k) lu(l,k) = lu(k,k) lu(k,k) = temp end if ! ! Normalize the values that lie below the pivot entry A(K,K). ! lu(k+1:n,k) = -lu(k+1:n,k) / lu(k,k) ! ! Row elimination with column indexing. ! do j = k + 1, n if ( l /= k ) then temp = lu(l,j) lu(l,j) = lu(k,j) lu(k,j) = temp end if lu(k+1:n,j) = lu(k+1:n,j) + lu(k+1:n,k) * lu(k,j) end do end do pivot(n) = n if ( lu(n,n) == 0.0D+00 ) then info = n write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_TO_R8PLU - Fatal error!' write ( *, '(a,i8)' ) ' Zero pivot on step ', info end if return end function r8mat_trace ( n, a ) !*****************************************************************************80 ! !! R8MAT_TRACE computes the trace of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The trace of a square matrix is the sum of the diagonal elements. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 July 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix A. ! ! Input, real ( kind = 8 ) A(N,N), the matrix whose trace is desired. ! ! Output, real ( kind = 8 ) R8MAT_TRACE, the trace of the matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i real ( kind = 8 ) r8mat_trace r8mat_trace = 0.0D+00 do i = 1, n r8mat_trace = r8mat_trace + a(i,i) end do return end subroutine r8mat_transpose ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_TRANSPOSE transposes an R8MAT. ! ! Discussion: ! ! An R8MAT is an M by N array of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 April 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) A(M,N), an M by N matrix. ! ! Output, real ( kind = 8 ) A(N,M), the transposed matrix. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m*n) real ( kind = 8 ), allocatable :: b(:) integer ( kind = 4 ) i integer ( kind = 4 ) j allocate ( b(m*n) ) do j = 1, n do i = 1, m b(j+(i-1)*n)= a(i+(j-1)*m) end do end do do j = 1, n do i = 1, m a(j+(i-1)*n) = b(j+(i-1)*n) end do end do deallocate ( b ) return end subroutine r8mat_transpose_in_place ( n, a ) !*****************************************************************************80 ! !! R8MAT_TRANSPOSE_IN_PLACE transposes an R8MAT in place. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 June 2008 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns ! of the matrix A. ! ! Input/output, real ( kind = 8 ) A(N,N), the matrix to be transposed. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) t do j = 1, n do i = 1, j - 1 t = a(i,j) a(i,j) = a(j,i) a(j,i) = t end do end do return end subroutine r8mat_transpose_new ( m, n, a, at ) !*****************************************************************************80 ! !! R8MAT_TRANSPOSE_NEW makes a transposed copy of an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! FORTRAN90 provides the transpose ( ) function which should be preferred ! over this routine. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 June 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns ! of the matrix A. ! ! Input, real ( kind = 8 ) A(N,N), the matrix to be transposed. ! ! Output, real ( kind = 8 ) AT(N,M), the matrix to be transposed. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) at(n,m) at = transpose ( a ) return end subroutine r8mat_transpose_print ( m, n, a, title ) !*****************************************************************************80 ! !! R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 June 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) A(M,N), an M by N matrix to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) character ( len = * ) title call r8mat_transpose_print_some ( m, n, a, 1, 1, m, n, title ) return end subroutine r8mat_transpose_print_some ( m, n, a, ilo, jlo, ihi, jhi, title ) !*****************************************************************************80 ! !! R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 September 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, real ( kind = 8 ) A(M,N), an M by N matrix to be printed. ! ! Input, integer ( kind = 4 ) ILO, JLO, the first row and column to print. ! ! Input, integer ( kind = 4 ) IHI, JHI, the last row and column to print. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ), parameter :: incx = 5 integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) character ( len = 14 ) ctemp(incx) integer ( kind = 4 ) i integer ( kind = 4 ) i2 integer ( kind = 4 ) i2hi integer ( kind = 4 ) i2lo integer ( kind = 4 ) ihi integer ( kind = 4 ) ilo integer ( kind = 4 ) inc integer ( kind = 4 ) j integer ( kind = 4 ) j2hi integer ( kind = 4 ) j2lo integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) if ( m <= 0 .or. n <= 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) ' (None)' return end if do i2lo = max ( ilo, 1 ), min ( ihi, m ), incx i2hi = i2lo + incx - 1 i2hi = min ( i2hi, m ) i2hi = min ( i2hi, ihi ) inc = i2hi + 1 - i2lo write ( *, '(a)' ) ' ' do i = i2lo, i2hi i2 = i + 1 - i2lo write ( ctemp(i2), '(i8,6x)' ) i end do write ( *, '('' Row '',5a14)' ) ctemp(1:inc) write ( *, '(a)' ) ' Col' write ( *, '(a)' ) ' ' j2lo = max ( jlo, 1 ) j2hi = min ( jhi, n ) do j = j2lo, j2hi do i2 = 1, inc i = i2lo - 1 + i2 write ( ctemp(i2), '(g14.6)' ) a(i,j) end do write ( *, '(i5,a,5a14)' ) j, ':', ( ctemp(i), i = 1, inc ) end do end do return end subroutine r8mat_u_inverse ( n, a, b ) !*****************************************************************************80 ! !! R8MAT_U_INVERSE inverts an upper triangular R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! An upper triangular matrix is a matrix whose only nonzero entries ! occur on or above the diagonal. ! ! The inverse of an upper triangular matrix is an upper triangular matrix. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, number of rows and columns in the matrix. ! ! Input, real ( kind = 8 ) A(N,N), the upper triangular matrix. ! ! Output, real ( kind = 8 ) B(N,N), the inverse matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) b(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j do j = n, 1, -1 do i = n, 1, -1 if ( j < i ) then b(i,j) = 0.0D+00 else if ( i == j ) then b(i,j) = 1.0D+00 / a(i,j) else b(i,j) = - dot_product ( a(i,i+1:j), b(i+1:j,j) ) / a(i,i) end if end do end do return end subroutine r8mat_u_solve ( n, a, b, x ) !*****************************************************************************80 ! !! R8MAT_U_SOLVE solves an upper triangular linear system. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 October 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns of ! the matrix A. ! ! Input, real ( kind = 8 ) A(N,N), the N by N upper triangular matrix. ! ! Input, real ( kind = 8 ) B(N), the right hand side of the linear system. ! ! Output, real ( kind = 8 ) X(N), the solution of the linear system. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) b(n) integer ( kind = 4 ) i real ( kind = 8 ) x(n) ! ! Solve U * x = b. ! do i = n, 1, -1 x(i) = ( b(i) - dot_product ( a(i,i+1:n), x(i+1:n) ) ) / a(i,i) end do return end subroutine r8mat_u1_inverse ( n, a, b ) !*****************************************************************************80 ! !! R8MAT_U1_INVERSE inverts a unit upper triangular R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! A unit upper triangular matrix is a matrix with only 1's on the main ! diagonal, and only 0's below the main diagonal. ! ! The inverse of a unit upper triangular matrix is also ! a unit upper triangular matrix. ! ! This routine can invert a matrix in place, that is, with no extra ! storage. If the matrix is stored in A, then the call ! ! call r8mat_u1_inverse ( n, a, a ) ! ! will result in A being overwritten by its inverse. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 December 2004 ! ! Author: ! ! Original FORTRAN77 version by Albert Nijenhuis, Herbert Wilf. ! FORTRAN90 version by John Burkardt, ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, number of rows and columns in the matrix. ! ! Input, real ( kind = 8 ) A(N,N), the unit upper triangular matrix. ! ! Output, real ( kind = 8 ) B(N,N), the inverse matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) b(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j do j = n, 1, -1 do i = n, 1, -1 if ( j < i ) then b(i,j) = 0.0D+00 else if ( i == j ) then b(i,j) = 1.0D+00 else b(i,j) = -dot_product ( a(i,i+1:j), b(i+1:j,j) ) end if end do end do return end subroutine r8mat_uniform_01 ( m, n, seed, r ) !*****************************************************************************80 ! !! R8MAT_UNIFORM_01 fills an R8MAT with unit pseudorandom numbers. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 August 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Paul Bratley, Bennett Fox, Linus Schrage, ! A Guide to Simulation, ! Springer Verlag, pages 201-202, 1983. ! ! Bennett Fox, ! Algorithm 647: ! Implementation and Relative Efficiency of Quasirandom ! Sequence Generators, ! ACM Transactions on Mathematical Software, ! Volume 12, Number 4, pages 362-376, 1986. ! ! Peter Lewis, Allen Goodman, James Miller, ! A Pseudo-Random Number Generator for the System/360, ! IBM Systems Journal, ! Volume 8, pages 136-143, 1969. ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns in ! the array. ! ! Input/output, integer ( kind = 4 ) SEED, the "seed" value, which ! should NOT be 0. On output, SEED has been updated. ! ! Output, real ( kind = 8 ) R(M,N), the array of pseudorandom values. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ), parameter :: i4_huge = 2147483647 integer ( kind = 4 ) j integer ( kind = 4 ) k integer ( kind = 4 ) seed real ( kind = 8 ) r(m,n) do j = 1, n do i = 1, m k = seed / 127773 seed = 16807 * ( seed - k * 127773 ) - k * 2836 if ( seed < 0 ) then seed = seed + i4_huge end if r(i,j) = real ( seed, kind = 8 ) * 4.656612875D-10 end do end do return end subroutine r8mat_uniform_ab ( m, n, a, b, seed, r ) !*****************************************************************************80 ! !! R8MAT_UNIFORM_AB returns a scaled pseudorandom R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! A <= R(I,J) <= B. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 31 May 2007 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Paul Bratley, Bennett Fox, Linus Schrage, ! A Guide to Simulation, ! Second Edition, ! Springer, 1987, ! ISBN: 0387964673, ! LC: QA76.9.C65.B73. ! ! Bennett Fox, ! Algorithm 647: ! Implementation and Relative Efficiency of Quasirandom ! Sequence Generators, ! ACM Transactions on Mathematical Software, ! Volume 12, Number 4, December 1986, pages 362-376. ! ! Pierre L'Ecuyer, ! Random Number Generation, ! in Handbook of Simulation, ! edited by Jerry Banks, ! Wiley, 1998, ! ISBN: 0471134031, ! LC: T57.62.H37. ! ! Peter Lewis, Allen Goodman, James Miller, ! A Pseudo-Random Number Generator for the System/360, ! IBM Systems Journal, ! Volume 8, Number 2, 1969, pages 136-143. ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns ! in the array. ! ! Input, real ( kind = 8 ) A, B, the lower and upper limits. ! ! Input/output, integer ( kind = 4 ) SEED, the "seed" value, which ! should NOT be 0. On output, SEED has been updated. ! ! Output, real ( kind = 8 ) R(M,N), the array of pseudorandom values. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b integer ( kind = 4 ) i integer ( kind = 4 ), parameter :: i4_huge = 2147483647 integer ( kind = 4 ) j integer ( kind = 4 ) k integer ( kind = 4 ) seed real ( kind = 8 ) r(m,n) if ( seed == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_UNIFORM_AB - Fatal error!' write ( *, '(a)' ) ' Input value of SEED = 0.' stop 1 end if do j = 1, n do i = 1, m k = seed / 127773 seed = 16807 * ( seed - k * 127773 ) - k * 2836 if ( seed < 0 ) then seed = seed + i4_huge end if r(i,j) = a + ( b - a ) * real ( seed, kind = 8 ) * 4.656612875D-10 end do end do return end subroutine r8mat_uniform_abvec ( m, n, a, b, seed, r ) !*****************************************************************************80 ! !! R8MAT_UNIFORM_ABVEC returns a scaled pseudorandom R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! A(I) <= R(I,J) <= B(I) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 October 2012 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Paul Bratley, Bennett Fox, Linus Schrage, ! A Guide to Simulation, ! Second Edition, ! Springer, 1987, ! ISBN: 0387964673, ! LC: QA76.9.C65.B73. ! ! Bennett Fox, ! Algorithm 647: ! Implementation and Relative Efficiency of Quasirandom ! Sequence Generators, ! ACM Transactions on Mathematical Software, ! Volume 12, Number 4, December 1986, pages 362-376. ! ! Pierre L'Ecuyer, ! Random Number Generation, ! in Handbook of Simulation, ! edited by Jerry Banks, ! Wiley, 1998, ! ISBN: 0471134031, ! LC: T57.62.H37. ! ! Peter Lewis, Allen Goodman, James Miller, ! A Pseudo-Random Number Generator for the System/360, ! IBM Systems Journal, ! Volume 8, Number 2, 1969, pages 136-143. ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns ! in the array. ! ! Input, real ( kind = 8 ) A(M), B(M), the lower and upper limits. ! ! Input/output, integer ( kind = 4 ) SEED, the "seed" value, which ! should NOT be 0. On output, SEED has been updated. ! ! Output, real ( kind = 8 ) R(M,N), the array of pseudorandom values. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m) real ( kind = 8 ) b(m) integer ( kind = 4 ) i integer ( kind = 4 ), parameter :: i4_huge = 2147483647 integer ( kind = 4 ) j integer ( kind = 4 ) k integer ( kind = 4 ) seed real ( kind = 8 ) r(m,n) if ( seed == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8MAT_UNIFORM_ABVEC - Fatal error!' write ( *, '(a)' ) ' Input value of SEED = 0.' stop 1 end if do j = 1, n do i = 1, m k = seed / 127773 seed = 16807 * ( seed - k * 127773 ) - k * 2836 if ( seed < 0 ) then seed = seed + i4_huge end if r(i,j) = a(i) + ( b(i) - a(i) ) * real ( seed, kind = 8 ) & * 4.656612875D-10 end do end do return end subroutine r8mat_ut_solve ( n, a, b, x ) !*****************************************************************************80 ! !! R8MAT_UT_SOLVE solves a transposed upper triangular linear system. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Given the upper triangular matrix A, the linear system to be solved is: ! ! A' * x = b ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 October 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of rows and columns ! of the matrix. ! ! Input, real ( kind = 8 ) A(N,N), the N by N upper triangular matrix. ! ! Input, real ( kind = 8 ) B(N), the right hand side of the linear system. ! ! Output, real ( kind = 8 ) X(N), the solution of the linear system. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) real ( kind = 8 ) b(n) integer ( kind = 4 ) i real ( kind = 8 ) x(n) ! ! Solve U' * x = b. ! do i = 1, n x(i) = ( b(i) - dot_product ( x(1:i-1), a(1:i-1,i) ) ) / a(i,i) end do return end subroutine r8mat_vand2 ( n, x, a ) !*****************************************************************************80 ! !! R8MAT_VAND2 returns the N by N row Vandermonde matrix A. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! The row Vandermonde matrix returned by this routine reads "across" ! rather than down. In particular, each row begins with a 1, followed by ! some value X, followed by successive powers of X. ! ! Formula: ! ! A(I,J) = X(I)^(J-1) ! ! Properties: ! ! A is nonsingular if, and only if, the X values are distinct. ! ! The determinant of A is ! ! det(A) = product ( 2 <= I <= N ) ( ! product ( 1 <= J <= I-1 ) ( ( X(I) - X(J) ) ) ). ! ! The matrix A is generally ill-conditioned. ! ! Example: ! ! N = 5, X = (2, 3, 4, 5, 6) ! ! 1 2 4 8 16 ! 1 3 9 27 81 ! 1 4 16 64 256 ! 1 5 25 125 625 ! 1 6 36 216 1296 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 March 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix desired. ! ! Input, real ( kind = 8 ) X(N), the values that define A. ! ! Output, real ( kind = 8 ) A(N,N), the N by N row Vandermonde matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) x(n) do i = 1, n do j = 1, n if ( j == 1 .and. x(i) == 0.0D+00 ) then a(i,j) = 1.0D+00 else a(i,j) = x(i) ** ( j - 1 ) end if end do end do return end function r8mat_vtmv ( m, n, x, a, y ) !*****************************************************************************80 ! !! R8MAT_VTMV multiplies computes the scalar x' * A * y. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 June 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns of ! the matrix. ! ! Input, real ( kind = 8 ) X(N), the first vector factor. ! ! Input, real ( kind = 8 ) A(M,N), the M by N matrix. ! ! Input, real ( kind = 8 ) Y(M), the second vector factor. ! ! Output, real ( kind = 8 ) R8MAT_VTMV, the value of X' * A * Y. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) real ( kind = 8 ) r8mat_vtmv real ( kind = 8 ) x(m) real ( kind = 8 ) y(n) r8mat_vtmv = dot_product ( x(1:m), matmul ( a(1:m,1:n), y(1:n) ) ) return end subroutine r8mat_zeros ( m, n, a ) !*****************************************************************************80 ! !! R8MAT_ZEROS zeroes an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 July 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Output, real ( kind = 8 ) A(M,N), the matrix of zeroes. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(m,n) a(1:m,1:n) = 0.0D+00 return end subroutine r8plu_det ( n, pivot, lu, det ) !*****************************************************************************80 ! !! R8PLU_DET computes the determinant of an R8PLU matrix. ! ! Discussion: ! ! The matrix should have been factored by R8MAT_TO_R8PLU. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 December 2004 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart, ! LINPACK User's Guide, ! SIAM, 1979, ! ISBN13: 978-0-898711-72-1. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! N must be positive. ! ! Input, integer ( kind = 4 ) PIVOT(N), the pivot vector computed ! by R8MAT_TO_R8PLU. ! ! Input, real ( kind = 8 ) LU(N,N), the LU factors computed ! by R8MAT_TO_R8PLU. ! ! Output, real ( kind = 8 ) DET, the determinant of the matrix. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) det integer ( kind = 4 ) i real ( kind = 8 ) lu(n,n) integer ( kind = 4 ) pivot(n) det = 1.0D+00 do i = 1, n det = det * lu(i,i) if ( pivot(i) /= i ) then det = -det end if end do return end subroutine r8plu_inverse ( n, pivot, lu, a_inverse ) !*****************************************************************************80 ! !! R8PLU_INVERSE computes the inverse of an R8PLU matrix. ! ! Discussion: ! ! The matrix should have been factored by R8MAT_TO_R8PLU. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix A. ! ! Input, integer ( kind = 4 ) PIVOT(N), the pivot vector from ! R8MAT_TO_R8PLU. ! ! Input, real ( kind = 8 ) LU(N,N), the LU factors computed by ! R8MAT_TO_R8PLU. ! ! Output, real ( kind = 8 ) A_INVERSE(N,N), the inverse of the original ! matrix A that was factored by R8MAT_TO_R8PLU. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a_inverse(n,n) integer ( kind = 4 ) i real ( kind = 8 ) lu(n,n) integer ( kind = 4 ) pivot(n) integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) temp real ( kind = 8 ) work(n) a_inverse(1:n,1:n) = lu(1:n,1:n) ! ! Compute Inverse(U). ! do k = 1, n a_inverse(k,k) = 1.0D+00 / a_inverse(k,k) a_inverse(1:k-1,k) = -a_inverse(1:k-1,k) * a_inverse(k,k) do j = k + 1, n temp = a_inverse(k,j) a_inverse(k,j) = 0.0D+00 a_inverse(1:k,j) = a_inverse(1:k,j) + temp * a_inverse(1:k,k) end do end do ! ! Form Inverse(U) * Inverse(L). ! do k = n - 1, 1, -1 work(k+1:n) = a_inverse(k+1:n,k) a_inverse(k+1:n,k) = 0.0D+00 do j = k + 1, n a_inverse(1:n,k) = a_inverse(1:n,k) + a_inverse(1:n,j) * work(j) end do if ( pivot(k) /= k ) then do i = 1, n temp = a_inverse(i,k) a_inverse(i,k) = a_inverse(i,pivot(k)) a_inverse(i,pivot(k)) = temp end do end if end do return end subroutine r8plu_mul ( n, pivot, lu, x, b ) !*****************************************************************************80 ! !! R8PLU_MUL computes A * x using the PLU factors of A. ! ! Discussion: ! ! It is assumed that R8MAT_TO_R8PLU has computed the PLU factors of ! the matrix A. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! N must be positive. ! ! Input, integer ( kind = 4 ) PIVOT(N), the pivot vector computed ! by R8MAT_TO_R8PLU. ! ! Input, real ( kind = 8 ) LU(N,N), the matrix factors computed by ! R8MAT_TO_R8PLU. ! ! Input, real ( kind = 8 ) X(N), the vector to be multiplied. ! ! Output, real ( kind = 8 ) B(N), the result of the multiplication. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) b(n) integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) lu(n,n) integer ( kind = 4 ) pivot(n) real ( kind = 8 ) temp real ( kind = 8 ) x(n) b(1:n) = x(1:n) ! ! Y = U * X. ! do j = 1, n b(1:j-1) = b(1:j-1) + lu(1:j-1,j) * b(j) b(j) = lu(j,j) * b(j) end do ! ! B = PL * Y = PL * U * X = A * x. ! do j = n - 1, 1, -1 b(j+1:n) = b(j+1:n) - lu(j+1:n,j) * b(j) k = pivot(j) if ( k /= j ) then temp = b(k) b(k) = b(j) b(j) = temp end if end do return end subroutine r8plu_solve ( n, pivot, lu, b, x ) !*****************************************************************************80 ! !! R8PLU_SOLVE solves a linear system A*x=b from the PLU factors. ! ! Discussion: ! ! The PLU factors should have been computed by R8MAT_TO_R8PLU. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! ! Input, integer ( kind = 4 ) PIVOT(N), the pivot vector from R8MAT_TO_R8PLU. ! ! Input, real ( kind = 8 ) LU(N,N), the LU factors from R8MAT_TO_R8PLU. ! ! Input, real ( kind = 8 ) B(N), the right hand side vector. ! ! Output, real ( kind = 8 ) X(N), the solution vector. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) b(n) integer ( kind = 4 ) pivot(n) integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) lu(n,n) real ( kind = 8 ) temp real ( kind = 8 ) x(n) ! ! Solve PL * Y = B. ! x(1:n) = b(1:n) do k = 1, n - 1 j = pivot(k) if ( j /= k ) then temp = x(j) x(j) = x(k) x(k) = temp end if x(k+1:n) = x(k+1:n) + lu(k+1:n,k) * x(k) end do ! ! Solve U * X = Y. ! do k = n, 1, -1 x(k) = x(k) / lu(k,k) x(1:k-1) = x(1:k-1) - lu(1:k-1,k) * x(k) end do return end subroutine r8plu_to_r8mat ( n, pivot, lu, a ) !*****************************************************************************80 ! !! R8PLU_TO_R8MAT recovers the matrix A that was factored by R8MAT_TO_R8PLU. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! N must be positive. ! ! Input, integer ( kind = 4 ) PIVOT(N), the pivot vector computed ! by R8MAT_TO_R8PLU. ! ! Input, real ( kind = 8 ) LU(N,N), the matrix factors computed by ! R8MAT_TO_R8PLU. ! ! Output, real ( kind = 8 ) A(N,N), the matrix whose factors are ! represented by LU and PIVOT. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n,n) integer ( kind = 4 ) i real ( kind = 8 ) lu(n,n) integer ( kind = 4 ) pivot(n) integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) temp a(1:n,1:n) = 0.0D+00 do i = 1, n a(i,i) = 1.0D+00 end do do j = 1, n do i = 1, n a(1:i-1,j) = a(1:i-1,j) + lu(1:i-1,i) * a(i,j) a(i,j) = lu(i,i) * a(i,j) end do ! ! B = PL * Y = PL * U * X = A * x. ! do i = n - 1, 1, -1 a(i+1:n,j) = a(i+1:n,j) - lu(i+1:n,i) * a(i,j) k = pivot(i) if ( k /= i ) then temp = a(k,j) a(k,j) = a(i,j) a(i,j) = temp end if end do end do return end function r8r8_compare ( x1, y1, x2, y2 ) !*****************************************************************************80 ! !! R8R8_COMPARE compares two R8R8's. ! ! Discussion: ! ! An R8R8 is simply a pair of R8 values, stored separately. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 May 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X1, Y1, the first vector. ! ! Input, real ( kind = 8 ) X2, Y2, the second vector. ! ! Output, integer ( kind = 4 ) R8R8_COMPARE: ! -1, (X1,Y1) < (X2,Y2); ! 0, (X1,Y1) = (X2,Y2); ! +1, (X1,Y1) > (X2,Y2). ! implicit none integer ( kind = 4 ) compare integer ( kind = 4 ) r8r8_compare real ( kind = 8 ) x1 real ( kind = 8 ) x2 real ( kind = 8 ) y1 real ( kind = 8 ) y2 if ( x1 < x2 ) then compare = -1 else if ( x2 < x1 ) then compare = +1 else if ( y1 < y2 ) then compare = -1 else if ( y2 < y1 ) then compare = +1 else compare = 0 end if r8r8_compare = compare return end subroutine r8r8_print ( a1, a2, title ) !*****************************************************************************80 ! !! R8R8_PRINT prints an R8R8. ! ! Discussion: ! ! An R8R8 is simply a pair of R8R8's, stored separately. ! ! A format is used which suggests a coordinate pair: ! ! Example: ! ! Center : ( 1.23, 7.45 ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 October 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) A1, A2, the coordinates of the vector. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none real ( kind = 8 ) a1 real ( kind = 8 ) a2 character ( len = * ) title if ( 0 < len_trim ( title ) ) then write ( *, '( 2x, a, a4, g14.6, a1, g14.6, a1 )' ) & trim ( title ), ' : (', a1, ',', a2, ')' else write ( *, '( 2x, a1, g14.6, a1, g14.6, a1 )' ) '(', a1, ',', a2, ')' end if return end function r8r8r8_compare ( x1, y1, z1, x2, y2, z2 ) !*****************************************************************************80 ! !! R8R8R8_COMPARE compares two R8R8R8's. ! ! Discussion: ! ! An R8R8R8 is simply 3 R8 values, stored as scalars. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 May 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X1, Y1, Z1, the first vector. ! ! Input, real ( kind = 8 ) X2, Y2, Z2, the second vector. ! ! Output, integer ( kind = 4 ) R8R8R8_COMPARE: ! -1, (X1,Y1,Z1) < (X2,Y2,Z2); ! 0, (X1,Y1,Z1) = (X2,Y2,Z2); ! +1, (X1,Y1,Z1) > (X2,Y2,Z2). ! implicit none integer ( kind = 4 ) compare integer ( kind = 4 ) r8r8r8_compare real ( kind = 8 ) x1 real ( kind = 8 ) x2 real ( kind = 8 ) y1 real ( kind = 8 ) y2 real ( kind = 8 ) z1 real ( kind = 8 ) z2 if ( x1 < x2 ) then compare = -1 else if ( x2 < x1 ) then compare = +1 else if ( y1 < y2 ) then compare = -1 else if ( y2 < y1 ) then compare = +1 else if ( z1 < z2 ) then compare = -1 else if ( z2 < z1 ) then compare = +1 else compare = 0 end if r8r8r8_compare = compare return end subroutine r8r8r8vec_index_insert_unique ( n_max, n, x, y, z, indx, & xval, yval, zval, ival, ierror ) !*****************************************************************************80 ! !! R8R8R8VEC_INDEX_INSERT_UNIQUE inserts unique R8R8R in an indexed sorted list. ! ! Discussion: ! ! An R8R8R8VEC is set of N R8R8R8 items. ! ! An R8R8R8 is simply 3 R8 values, stored as scalars. ! ! If the input value does not occur in the current list, it is added, ! and N, X, Y, Z and INDX are updated. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N_MAX, the maximum size of the list. ! ! Input/output, integer ( kind = 4 ) N, the size of the list. ! ! Input/output, real ( kind = 8 ) X(N), Y(N), Z(N), the R8R8R8 vector. ! ! Input/output, integer ( kind = 4 ) INDX(N), the sort index of the list. ! ! Input, real ( kind = 8 ) XVAL, YVAL, ZVAL, the value to be inserted ! if it is not already in the list. ! ! Output, integer ( kind = 4 ) IVAL, the index in X, Y, Z corresponding ! to the value XVAL, YVAL, ZVAL. ! ! Output, integer ( kind = 4 ) IERROR, 0 for no error, 1 if an error ! occurred. ! implicit none integer ( kind = 4 ) n_max integer ( kind = 4 ) equal integer ( kind = 4 ) ierror integer ( kind = 4 ) indx(n_max) integer ( kind = 4 ) ival integer ( kind = 4 ) less integer ( kind = 4 ) more integer ( kind = 4 ) n real ( kind = 8 ) x(n_max) real ( kind = 8 ) xval real ( kind = 8 ) y(n_max) real ( kind = 8 ) yval real ( kind = 8 ) z(n_max) real ( kind = 8 ) zval ierror = 0 if ( n <= 0 ) then if ( n_max <= 0 ) then ierror = 1 write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8R8R8VEC_INDEX_INSERT_UNIQUE - Fatal error!' write ( *, '(a)' ) ' Not enough space to store new data.' return end if n = 1 x(1) = xval y(1) = yval z(1) = zval indx(1) = 1 ival = 1 return end if ! ! Does ( XVAL, YVAL, ZVAL ) already occur in ( X, Y, Z)? ! call r8r8r8vec_index_search ( n, x, y, z, indx, xval, yval, zval, & less, equal, more ) if ( equal == 0 ) then if ( n_max <= n ) then ierror = 1 write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8R8R8VEC_INDEX_INSERT_UNIQUE - Fatal error!' write ( *, '(a)' ) ' Not enough space to store new data.' return end if x(n+1) = xval y(n+1) = yval z(n+1) = zval ival = n + 1 indx(n+1:more+1:-1) = indx(n:more:-1) indx(more) = n + 1 n = n + 1 else ival = indx(equal) end if return end subroutine r8r8r8vec_index_search ( n, x, y, z, indx, xval, yval, & zval, less, equal, more ) !*****************************************************************************80 ! !! R8R8R8VEC_INDEX_SEARCH searches for R8R8R8 value in an indexed sorted list. ! ! Discussion: ! ! An R8R8R8VEC is set of N R8R8R8 items. ! ! An R8R8R8 is simply 3 R8 values, stored as scalars. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 May 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the list. ! ! Input, real ( kind = 8 ) X(N), Y(N), Z(N), the list. ! ! Input, integer ( kind = 4 ) INDX(N), the sort index of the list. ! ! Input, real ( kind = 8 ) XVAL, YVAL, ZVAL, the value to be sought. ! ! Output, integer ( kind = 4 ) LESS, EQUAL, MORE, the indexes in INDX of the ! entries of X that are just less than, equal to, and just greater ! than XVAL. If XVAL does not occur in X, then EQUAL is zero. ! If XVAL is the minimum entry of X, then LESS is 0. If XVAL ! is the greatest entry of X, then MORE is N+1. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) compare integer ( kind = 4 ) r8r8r8_compare integer ( kind = 4 ) equal integer ( kind = 4 ) hi integer ( kind = 4 ) indx(n) integer ( kind = 4 ) less integer ( kind = 4 ) lo integer ( kind = 4 ) mid integer ( kind = 4 ) more real ( kind = 8 ) x(n) real ( kind = 8 ) xhi real ( kind = 8 ) xlo real ( kind = 8 ) xmid real ( kind = 8 ) xval real ( kind = 8 ) y(n) real ( kind = 8 ) yhi real ( kind = 8 ) ylo real ( kind = 8 ) ymid real ( kind = 8 ) yval real ( kind = 8 ) z(n) real ( kind = 8 ) zhi real ( kind = 8 ) zlo real ( kind = 8 ) zmid real ( kind = 8 ) zval if ( n <= 0 ) then less = 0 equal = 0 more = 0 return end if lo = 1 hi = n xlo = x(indx(lo)) ylo = y(indx(lo)) zlo = z(indx(lo)) xhi = x(indx(hi)) yhi = y(indx(hi)) zhi = z(indx(hi)) compare = r8r8r8_compare ( xval, yval, zval, xlo, ylo, zlo ) if ( compare == -1 ) then less = 0 equal = 0 more = 1 return else if ( compare == 0 ) then less = 0 equal = 1 more = 2 return end if compare = r8r8r8_compare ( xval, yval, zval, xhi, yhi, zhi ) if ( compare == 1 ) then less = n equal = 0 more = n + 1 return else if ( compare == 0 ) then less = n - 1 equal = n more = n + 1 return end if do if ( lo + 1 == hi ) then less = lo equal = 0 more = hi return end if mid = ( lo + hi ) / 2 xmid = x(indx(mid)) ymid = y(indx(mid)) zmid = z(indx(mid)) compare = r8r8r8_compare ( xval, yval, zval, xmid, ymid, zmid ) if ( compare == 0 ) then equal = mid less = mid - 1 more = mid + 1 return else if ( compare == -1 ) then hi = mid else if ( compare == +1 ) then lo = mid end if end do return end subroutine r8r8vec_index_insert_unique ( n_max, n, x, y, indx, xval, yval, & ival, ierror ) !*****************************************************************************80 ! !! R8R8VEC_INDEX_INSERT_UNIQUE inserts a unique R8R8 in an indexed sorted list. ! ! Discussion: ! ! An R8R8VEC is set of N R8R8 items. ! ! An R8R8 is simply 2 R8 values, stored as scalars. ! ! If the input value does not occur in the current list, it is added, ! and N, X, Y and INDX are updated. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N_MAX, the maximum size of the list. ! ! Input/output, integer ( kind = 4 ) N, the size of the list. ! ! Input/output, real ( kind = 8 ) X(N), Y(N), the list of R8R8 vectors. ! ! Input/output, integer ( kind = 4 ) INDX(N), the sort index of the list. ! ! Input, real ( kind = 8 ) XVAL, YVAL, the value to be inserted if it is ! not already in the list. ! ! Output, integer ( kind = 4 ) IVAL, the index in X, Y corresponding to the ! value XVAL, YVAL. ! ! Output, integer ( kind = 4 ) IERROR, 0 for no error, 1 if an ! error occurred. ! implicit none integer ( kind = 4 ) n_max integer ( kind = 4 ) equal integer ( kind = 4 ) ierror integer ( kind = 4 ) indx(n_max) integer ( kind = 4 ) ival integer ( kind = 4 ) less integer ( kind = 4 ) more integer ( kind = 4 ) n real ( kind = 8 ) x(n_max) real ( kind = 8 ) xval real ( kind = 8 ) y(n_max) real ( kind = 8 ) yval ierror = 0 if ( n <= 0 ) then if ( n_max <= 0 ) then ierror = 1 write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8R8VEC_INDEX_INSERT_UNIQUE - Fatal error!' write ( *, '(a)' ) ' Not enough space to store new data.' return end if n = 1 x(1) = xval y(1) = yval indx(1) = 1 ival = 1 return end if ! ! Does ( XVAL, YVAL ) already occur in ( X, Y )? ! call r8r8vec_index_search ( n, x, y, indx, xval, yval, less, equal, more ) if ( equal == 0 ) then if ( n_max <= n ) then ierror = 1 write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8R8VEC_INDEX_INSERT_UNIQUE - Fatal error!' write ( *, '(a)' ) ' Not enough space to store new data.' return end if x(n+1) = xval y(n+1) = yval ival = n + 1 indx(n+1:more+1:-1) = indx(n:more:-1) indx(more) = n + 1 n = n + 1 else ival = indx(equal) end if return end subroutine r8r8vec_index_search ( n, x, y, indx, xval, yval, less, equal, & more ) !*****************************************************************************80 ! !! R8R8VEC_INDEX_SEARCH searches for an R8R8 in an indexed sorted list. ! ! Discussion: ! ! An R8R8VEC is set of N R8R8 items. ! ! An R8R8 is simply 2 R8 values, stored as scalars. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 May 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the current list. ! ! Input, real ( kind = 8 ) X(N), Y(N), the list. ! ! Input, integer ( kind = 4 ) INDX(N), the sort index of the list. ! ! Input, real ( kind = 8 ) XVAL, YVAL, the value to be sought. ! ! Output, integer ( kind = 4 ) LESS, EQUAL, MORE, the indexes in INDX of the ! entries of X that are just less than, equal to, and just greater ! than XVAL. If XVAL does not occur in X, then EQUAL is zero. ! If XVAL is the minimum entry of X, then LESS is 0. If XVAL ! is the greatest entry of X, then MORE is N+1. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) compare integer ( kind = 4 ) r8r8_compare integer ( kind = 4 ) equal integer ( kind = 4 ) hi integer ( kind = 4 ) indx(n) integer ( kind = 4 ) less integer ( kind = 4 ) lo integer ( kind = 4 ) mid integer ( kind = 4 ) more real ( kind = 8 ) x(n) real ( kind = 8 ) xhi real ( kind = 8 ) xlo real ( kind = 8 ) xmid real ( kind = 8 ) xval real ( kind = 8 ) y(n) real ( kind = 8 ) yhi real ( kind = 8 ) ylo real ( kind = 8 ) ymid real ( kind = 8 ) yval if ( n <= 0 ) then less = 0 equal = 0 more = 0 return end if lo = 1 hi = n xlo = x(indx(lo)) ylo = y(indx(lo)) xhi = x(indx(hi)) yhi = y(indx(hi)) compare = r8r8_compare ( xval, yval, xlo, ylo ) if ( compare == -1 ) then less = 0 equal = 0 more = 1 return else if ( compare == 0 ) then less = 0 equal = 1 more = 2 return end if compare = r8r8_compare ( xval, yval, xhi, yhi ) if ( compare == 1 ) then less = n equal = 0 more = n + 1 return else if ( compare == 0 ) then less = n - 1 equal = n more = n + 1 return end if do if ( lo + 1 == hi ) then less = lo equal = 0 more = hi return end if mid = ( lo + hi ) / 2 xmid = x(indx(mid)) ymid = y(indx(mid)) compare = r8r8_compare ( xval, yval, xmid, ymid ) if ( compare == 0 ) then equal = mid less = mid - 1 more = mid + 1 return else if ( compare == -1 ) then hi = mid else if ( compare == +1 ) then lo = mid end if end do return end subroutine r8rows_to_i4mat ( m, n, r8rows, r8mat ) !*****************************************************************************80 ! !! R8ROWS_TO_R8MAT converts a row-major vector to an R8MAT. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, in column major order. ! ! I am frustrated that the FORTRAN standard for initializing an array ! forces me to enter a table of data by columns, so that I have to ! transpose the information, which is confusing to me and any reader. ! ! This function allows me to declare a vector of the right type and length, ! fill it with data that I can display row-wise, and then have the ! data copied into a FORTRAN array. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 September 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, N, the number of rows and columns. ! ! Input, integer ( kind = 4 ) R8ROWS(M*N), the data. stored rowwise ! in a vector. ! ! Output, integer ( kind = 4 ) R8MAT(M,N), a copy of the data, stored ! columnwise in an array. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) r8mat(m,n) real ( kind = 8 ) r8rows(m*n) k = 0 do i = 1, m do j = 1, n k = k + 1 r8mat(i,j) = r8rows(k) end do end do return end subroutine r8slmat_print ( m, n, a, title ) !*****************************************************************************80 ! !! R8SLMAT_PRINT prints a strict lower triangular R8MAT. ! ! Example: ! ! M = 5, N = 5 ! A = (/ 21, 31, 41, 51, 32, 42, 52, 43, 53, 54 /) ! ! 21 ! 31 32 ! 41 42 43 ! 51 52 53 54 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 June 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of rows in A. ! ! Input, integer ( kind = 4 ) N, the number of columns in A. ! ! Input, real ( kind = 8 ) A(*), the M by N matrix. Only the strict ! lower triangular elements are stored, in column major order. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(*) integer ( kind = 4 ) i integer ( kind = 4 ) indx(10) integer ( kind = 4 ) j integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo integer ( kind = 4 ) jmax integer ( kind = 4 ) nn integer ( kind = 4 ) size character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) jmax = min ( n, m - 1 ) if ( m-1 <= n ) then size = ( m * ( m - 1 ) ) / 2 else if ( n < m-1 ) then size = ( n * ( n - 1 ) ) / 2 + ( m - n - 1 ) * n end if if ( all ( a(1:size) == aint ( a(1:size) ) ) ) then nn = 10 do jlo = 1, jmax, nn jhi = min ( jlo + nn - 1, m - 1, jmax ) write ( *, '(a)' ) ' ' write ( *, '(a8,10i8)' ) ' Col', ( j, j = jlo, jhi ) write ( *, '(a8)' ) ' Row' do i = jlo + 1, m jhi = min ( jlo + nn - 1, i - 1, jmax ) do j = jlo, jhi indx(j+1-jlo) = ( j - 1 ) * m + i - ( j * ( j + 1 ) ) / 2 end do write ( *, '(2x,i8,10i8)' ) i, int ( a(indx(1:jhi+1-jlo)) ) end do end do else if ( maxval ( abs ( a(1:size) ) ) < 1000000.0D+00 ) then nn = 5 do jlo = 1, jmax, nn jhi = min ( jlo + nn - 1, m - 1, jmax ) write ( *, '(a)' ) ' ' write ( *, '(a10,5(i8,6x))' ) ' Col', ( j, j = jlo, jhi ) write ( *, '(a10)' ) ' Row' do i = jlo + 1, m jhi = min ( jlo + nn - 1, i - 1, jmax ) do j = jlo, jhi indx(j+1-jlo) = ( j - 1 ) * m + i - ( j * ( j + 1 ) ) / 2 end do write ( *, '(2x,i8,5f14.6)' ) i, a(indx(1:jhi+1-jlo)) end do end do else nn = 5 do jlo = 1, jmax, nn jhi = min ( jlo + nn - 1, m - 1, jmax ) write ( *, '(a)' ) ' ' write ( *, '(a10,5(i8,6x))' ) ' Col', ( j, j = jlo, jhi ) write ( *, '(a10)' ) ' Row' do i = jlo + 1, m jhi = min ( jlo + nn - 1, i - 1, jmax ) do j = jlo, jhi indx(j+1-jlo) = ( j - 1 ) * m + i - ( j * ( j + 1 ) ) / 2 end do write ( *, '(2x,i8,5g14.6)' ) i, a(indx(1:jhi+1-jlo)) end do end do end if return end subroutine r8vec_01_to_ab ( n, a, amax, amin ) !*****************************************************************************80 ! !! R8VEC_01_TO_AB shifts and rescales an R8VEC to lie within given bounds. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! On input, A contains the original data, which is presumed to lie ! between 0 and 1. However, it is not necessary that this be so. ! ! On output, A has been shifted and rescaled so that all entries which ! on input lay in [0,1] now lie between AMIN and AMAX. Other entries will ! be mapped in a corresponding way. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 April 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of data values. ! ! Input/output, real ( kind = 8 ) A(N), the vector to be rescaled. ! ! Input, real ( kind = 8 ) AMAX, AMIN, the maximum and minimum values ! allowed for A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) amax real ( kind = 8 ) amax2 real ( kind = 8 ) amax3 real ( kind = 8 ) amin real ( kind = 8 ) amin2 real ( kind = 8 ) amin3 if ( amax == amin ) then a(1:n) = amin return end if amax2 = max ( amax, amin ) amin2 = min ( amax, amin ) amin3 = minval ( a(1:n) ) amax3 = maxval ( a(1:n) ) if ( amax3 /= amin3 ) then a(1:n) = ( ( amax3 - a(1:n) ) * amin2 & + ( a(1:n) - amin3 ) * amax2 ) & / ( amax3 - amin3 ) else a(1:n) = 0.5D+00 * ( amax2 + amin2 ) end if return end function r8vec_amax ( n, a ) !*****************************************************************************80 ! !! R8VEC_AMAX returns the maximum absolute value in an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 January 2015 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, real ( kind = 8 ) R8VEC_AMAX, the value of the entry ! of largest magnitude. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_amax real ( kind = 8 ) value value = maxval ( abs ( a(1:n) ) ) r8vec_amax = value return end subroutine r8vec_amax_index ( n, a, amax_index ) !*****************************************************************************80 ! !! R8VEC_AMAX_INDEX returns the index of the maximum absolute value in an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 January 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, integer ( kind = 4 ) AMAX_INDEX, the index of the entry of ! largest magnitude. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) amax integer ( kind = 4 ) amax_index integer ( kind = 4 ) i if ( n <= 0 ) then amax_index = -1 else amax_index = 1 amax = abs ( a(1) ) do i = 2, n if ( amax < abs ( a(i) ) ) then amax_index = i amax = abs ( a(i) ) end if end do end if return end function r8vec_amin ( n, a ) !*****************************************************************************80 ! !! R8VEC_AMIN returns the minimum absolute value in an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, real ( kind = 8 ) R8VEC_AMIN, the value of the entry ! of smallest magnitude. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_amin real ( kind = 8 ) value value = minval ( abs ( a(1:n) ) ) r8vec_amin = value return end subroutine r8vec_amin_index ( n, a, amin_index ) !*****************************************************************************80 ! !! R8VEC_AMIN_INDEX returns the index of the minimum absolute value in an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 January 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, integer ( kind = 4 ) AMIN_INDEX, the index of the entry of ! smallest magnitude. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) amin integer ( kind = 4 ) amin_index integer ( kind = 4 ) i if ( n <= 0 ) then amin_index = 0 else amin_index = 1 amin = abs ( a(1) ) do i = 2, n if ( abs ( a(i) ) < amin ) then amin_index = i amin = abs ( a(i) ) end if end do end if return end subroutine r8vec_any_normal ( dim_num, v1, v2 ) !*****************************************************************************80 ! !! R8VEC_ANY_NORMAL returns some normal vector to V1. ! ! Discussion: ! ! If DIM_NUM < 2, then no normal vector can be returned. ! ! If V1 is the zero vector, then any unit vector will do. ! ! No doubt, there are better, more robust algorithms. But I will take ! just about ANY reasonable unit vector that is normal to V1. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 August 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) DIM_NUM, the spatial dimension. ! ! Input, real ( kind = 8 ) V1(DIM_NUM), the vector. ! ! Output, real ( kind = 8 ) V2(DIM_NUM), a vector that is ! normal to V2, and has unit Euclidean length. ! implicit none integer ( kind = 4 ) dim_num integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) r8vec_norm real ( kind = 8 ) v1(dim_num) real ( kind = 8 ) v2(dim_num) real ( kind = 8 ) vj real ( kind = 8 ) vk if ( dim_num < 2 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_ANY_NORMAL - Fatal error!' write ( *, '(a)' ) ' Called with DIM_NUM < 2.' stop 1 end if if ( r8vec_norm ( dim_num, v1 ) == 0.0D+00 ) then v2(1) = 1.0D+00 v2(2:dim_num) = 0.0D+00 return end if ! ! Seek the largest entry in V1, VJ = V1(J), and the ! second largest, VK = V1(K). ! ! Since V1 does not have zero norm, we are guaranteed that ! VJ, at least, is not zero. ! j = - 1 vj = 0.0D+00 k = - 1 vk = 0.0D+00 do i = 1, dim_num if ( abs ( vk ) < abs ( v1(i) ) .or. k < 1 ) then if ( abs ( vj ) < abs ( v1(i) ) .or. j < 1 ) then k = j vk = vj j = i vj = v1(i) else k = i vk = v1(i) end if end if end do ! ! Setting V2 to zero, except that V2(J) = -VK, and V2(K) = VJ, ! will just about do the trick. ! v2(1:dim_num) = 0.0D+00 v2(j) = - vk / sqrt ( vk * vk + vj * vj ) v2(k) = vj / sqrt ( vk * vk + vj * vj ) return end function r8vec_asum ( n, a ) !*****************************************************************************80 ! !! R8VEC_ASUM returns the sum of the absolute values of the entries of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 January 2015 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, real ( kind = 8 ) R8VEC_ASUM, the sum of the absolute ! values of the entries. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_asum r8vec_asum = sum ( abs ( a(1:n) ) ) return end subroutine r8vec_bin ( n, x, bin_num, bin_min, bin_max, bin, bin_limit ) !*****************************************************************************80 ! !! R8VEC_BIN computes bins based on a given R8VEC. ! ! Discussion: ! ! The user specifies minimum and maximum bin values, BIN_MIN and ! BIN_MAX, and the number of bins, BIN_NUM. This determines a ! "bin width": ! ! H = ( BIN_MAX - BIN_MIN ) / BIN_NUM ! ! so that bin I will count all entries X(J) such that ! ! BIN_LIMIT(I-1) <= X(J) < BIN_LIMIT(I). ! ! The array X does NOT have to be sorted. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 July 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of X. ! ! Input, real ( kind = 8 ) X(N), an (unsorted) array to be binned. ! ! Input, integer ( kind = 4 ) BIN_NUM, the number of bins. Two extra bins, ! #0 and #BIN_NUM+1, count extreme values. ! ! Input, real ( kind = 8 ) BIN_MIN, BIN_MAX, define the range and size ! of the bins. BIN_MIN and BIN_MAX must be distinct. ! Normally, BIN_MIN < BIN_MAX, and the documentation will assume ! this, but proper results will be computed if BIN_MIN > BIN_MAX. ! ! Output, integer ( kind = 4 ) BIN(0:BIN_NUM+1). ! BIN(0) counts entries of X less than BIN_MIN. ! BIN(BIN_NUM+1) counts entries greater than or equal to BIN_MAX. ! For 1 <= I <= BIN_NUM, BIN(I) counts the entries X(J) such that ! BIN_LIMIT(I-1) <= X(J) < BIN_LIMIT(I). ! where H is the bin spacing. ! ! Output, real ( kind = 8 ) BIN_LIMIT(0:BIN_NUM), the "limits" of the bins. ! BIN(I) counts the number of entries X(J) such that ! BIN_LIMIT(I-1) <= X(J) < BIN_LIMIT(I). ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) bin_num integer ( kind = 4 ) bin(0:bin_num+1) real ( kind = 8 ) bin_limit(0:bin_num) real ( kind = 8 ) bin_max real ( kind = 8 ) bin_min integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) t real ( kind = 8 ) x(n) if ( bin_max == bin_min ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_BIN - Fatal error!' write ( *, '(a)' ) ' BIN_MIN = BIN_MAX.' stop 1 end if bin(0:bin_num+1) = 0 do i = 1, n t = ( x(i) - bin_min ) / ( bin_max - bin_min ) if ( t < 0.0D+00 ) then j = 0 else if ( 1.0D+00 <= t ) then j = bin_num + 1 else j = 1 + int ( real ( bin_num, kind = 8 ) * t ) end if bin(j) = bin(j) + 1 end do ! ! Compute the bin limits. ! do i = 0, bin_num bin_limit(i) = ( real ( bin_num - i, kind = 8 ) * bin_min & + real ( i, kind = 8 ) * bin_max ) & / real ( bin_num, kind = 8 ) end do return end subroutine r8vec_binary_next ( n, bvec ) !*****************************************************************************80 ! !! R8VEC_BINARY_NEXT generates the next binary vector. ! ! Discussion: ! ! The vectors have the order ! ! (0,0,...,0), ! (0,0,...,1), ! ... ! (1,1,...,1) ! ! and the "next" vector after (1,1,...,1) is (0,0,...,0). That is, ! we allow wrap around. ! ! Example: ! ! N = 3 ! ! Input Output ! ----- ------ ! 0 0 0 => 0 0 1 ! 0 0 1 => 0 1 0 ! 0 1 0 => 0 1 1 ! 0 1 1 => 1 0 0 ! 1 0 0 => 1 0 1 ! 1 0 1 => 1 1 0 ! 1 1 0 => 1 1 1 ! 1 1 1 => 0 0 0 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 March 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of the vectors. ! ! Input, real ( kind = 8 ) BVEC(N), the vector whose successor is desired. ! ! Output, real ( kind = 8 ) BVEC(N), the successor to the input vector. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i real ( kind = 8 ) bvec(n) do i = n, 1, -1 if ( bvec(i) == 0.0D+00 ) then bvec(i) = 1.0D+00 return end if bvec(i) = 0.0D+00 end do return end subroutine r8vec_blend ( n, t1, x1, t2, x2, x ) !*****************************************************************************80 ! !! R8VEC_BLEND performs weighted interpolation of two R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The formula used is: ! ! x(i) = t * x1(i) + (1-t) * x2(i) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 March 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in each vector. ! ! Input, real ( kind = 8 ) T1, the weight factor for vector 1. ! ! Input, real ( kind = 8 ) X1(N), the first vector. ! ! Input, real ( kind = 8 ) T2, the weight factor for vector 2. ! ! Input, real ( kind = 8 ) X2(N), the second vector. ! ! Output, real ( kind = 8 ) X(N), the interpolated or extrapolated value. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) t1 real ( kind = 8 ) t2 real ( kind = 8 ) x(n) real ( kind = 8 ) x1(n) real ( kind = 8 ) x2(n) x(1:n) = t1 * x1(1:n) + t2 * x2(1:n) return end subroutine r8vec_bracket ( n, x, xval, left, right ) !*****************************************************************************80 ! !! R8VEC_BRACKET searches a sorted R8VEC for successive brackets of a value. ! ! Discussion: ! ! This is an inefficient implementation that uses linear search. ! ! An R8VEC is a vector of R8's. ! ! If the values in the vector are thought of as defining intervals ! on the real line, then this routine searches for the interval ! nearest to or containing the given value. ! ! It is always true that RIGHT = LEFT+1. ! ! If XVAL < X(1), then LEFT = 1, RIGHT = 2, and ! XVAL < X(1) < X(2); ! If X(1) <= XVAL < X(N), then ! X(LEFT) <= XVAL < X(RIGHT); ! If X(N) <= XVAL, then LEFT = N-1, RIGHT = N, and ! X(LEFT) <= X(RIGHT) <= XVAL. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 April 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, length of input array. ! ! Input, real ( kind = 8 ) X(N), an array that has been sorted into ! ascending order. ! ! Input, real ( kind = 8 ) XVAL, a value to be bracketed. ! ! Output, integer ( kind = 4 ) LEFT, RIGHT, the results of the search. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) left integer ( kind = 4 ) right real ( kind = 8 ) x(n) real ( kind = 8 ) xval do i = 2, n - 1 if ( xval < x(i) ) then left = i - 1 right = i return end if end do left = n - 1 right = n return end subroutine r8vec_bracket2 ( n, x, xval, start, left, right ) !*****************************************************************************80 ! !! R8VEC_BRACKET2 searches a sorted R8VEC for successive brackets of a value. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! If the values in the vector are thought of as defining intervals ! on the real line, then this routine searches for the interval ! containing the given value. ! ! R8VEC_BRACKET2 is a variation on R8VEC_BRACKET. It seeks to reduce ! the search time by allowing the user to suggest an interval that ! probably contains the value. The routine will look in that interval ! and the intervals to the immediate left and right. If this does ! not locate the point, a binary search will be carried out on ! appropriate subportion of the sorted array. ! ! In the most common case, 1 <= LEFT < LEFT + 1 = RIGHT <= N, ! and X(LEFT) <= XVAL <= X(RIGHT). ! ! Special cases: ! Value is less than all data values: ! LEFT = -1, RIGHT = 1, and XVAL < X(RIGHT). ! Value is greater than all data values: ! LEFT = N, RIGHT = -1, and X(LEFT) < XVAL. ! Value is equal to a data value: ! LEFT = RIGHT, and X(LEFT) = X(RIGHT) = XVAL. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 February 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, length of the input array. ! ! Input, real ( kind = 8 ) X(N), an array that has been sorted into ! ascending order. ! ! Input, real ( kind = 8 ) XVAL, a value to be bracketed by entries of X. ! ! Input, integer ( kind = 4 ) START, between 1 and N, specifies that XVAL ! is likely to be in the interval: ! [ X(START), X(START+1) ] ! or, if not in that interval, then either ! [ X(START+1), X(START+2) ] ! or ! [ X(START-1), X(START) ]. ! ! Output, integer ( kind = 4 ) LEFT, RIGHT, the results of the search. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) high integer ( kind = 4 ) left integer ( kind = 4 ) low integer ( kind = 4 ) right integer ( kind = 4 ) start real ( kind = 8 ) x(n) real ( kind = 8 ) xval ! ! Check. ! if ( n < 1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_BRACKET2 - Fatal error!' write ( *, '(a)' ) ' N < 1.' stop 1 end if if ( start < 1 .or. n < start ) then start = ( n + 1 ) / 2 end if ! ! XVAL = X(START)? ! if ( x(start) == xval ) then left = start right = start return ! ! X(START) < XVAL? ! else if ( x(start) < xval ) then ! ! X(START) = X(N) < XVAL < oo? ! if ( n < start + 1 ) then left = start right = -1 return ! ! XVAL = X(START+1)? ! else if ( xval == x(start+1) ) then left = start + 1 right = start + 1 return ! ! X(START) < XVAL < X(START+1)? ! else if ( xval < x(start+1) ) then left = start right = start + 1 return ! ! X(START+1) = X(N) < XVAL < oo? ! else if ( n < start + 2 ) then left = start + 1 right = -1 return ! ! XVAL = X(START+2)? ! else if ( xval == x(start+2) ) then left = start + 2 right = start + 2 return ! ! X(START+1) < XVAL < X(START+2)? ! else if ( xval < x(start+2) ) then left = start + 1 right = start + 2 return ! ! Binary search for XVAL in [ X(START+2), X(N) ], ! where XVAL is guaranteed to be greater than X(START+2). ! else low = start + 2 high = n call r8vec_bracket ( high + 1 - low, x(low), xval, left, right ) left = left + low - 1 right = right + low - 1 end if ! ! -oo < XVAL < X(START) = X(1). ! else if ( start == 1 ) then left = -1 right = start return ! ! XVAL = X(START-1)? ! else if ( xval == x(start-1) ) then left = start - 1 right = start - 1 return ! ! X(START-1) < XVAL < X(START)? ! else if ( x(start-1) <= xval ) then left = start - 1 right = start return ! ! Binary search for XVAL in [ X(1), X(START-1) ], ! where XVAL is guaranteed to be less than X(START-1). ! else low = 1 high = start - 1 call r8vec_bracket ( high + 1 - low, x(1), xval, left, right ) end if return end subroutine r8vec_bracket3 ( n, t, tval, left ) !*****************************************************************************80 ! !! R8VEC_BRACKET3 finds the interval containing or nearest a given value. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The routine always returns the index LEFT of the sorted array ! T with the property that either ! * T is contained in the interval [ T(LEFT), T(LEFT+1) ], or ! * T < T(LEFT) = T(1), or ! * T > T(LEFT+1) = T(N). ! ! The routine is useful for interpolation problems, where ! the abscissa must be located within an interval of data ! abscissas for interpolation, or the "nearest" interval ! to the (extreme) abscissa must be found so that extrapolation ! can be carried out. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 April 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, length of the input array. ! ! Input, real ( kind = 8 ) T(N), an array that has been sorted ! into ascending order. ! ! Input, real ( kind = 8 ) TVAL, a value to be bracketed by entries of T. ! ! Input/output, integer ( kind = 4 ) LEFT. ! On input, if 1 <= LEFT <= N-1, LEFT is taken as a suggestion for the ! interval [ T(LEFT), T(LEFT+1) ] in which TVAL lies. This interval ! is searched first, followed by the appropriate interval to the left ! or right. After that, a binary search is used. ! On output, LEFT is set so that the interval [ T(LEFT), T(LEFT+1) ] ! is the closest to TVAL; it either contains TVAL, or else TVAL ! lies outside the interval [ T(1), T(N) ]. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) high integer ( kind = 4 ) left integer ( kind = 4 ) low integer ( kind = 4 ) mid real ( kind = 8 ) t(n) real ( kind = 8 ) tval ! ! Check the input data. ! if ( n < 2 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_BRACKET3 - Fatal error!' write ( *, '(a)' ) ' N must be at least 2.' stop 1 end if ! ! If LEFT is not between 1 and N-1, set it to the middle value. ! if ( left < 1 .or. n - 1 < left ) then left = ( n + 1 ) / 2 end if ! ! CASE 1: TVAL < T(LEFT): ! Search for TVAL in [T(I), T(I+1)] for intervals I = 1 to LEFT-1. ! if ( tval < t(left) ) then if ( left == 1 ) then return else if ( left == 2 ) then left = 1 return else if ( t(left-1) <= tval ) then left = left - 1 return else if ( tval <= t(2) ) then left = 1 return end if ! ! ...Binary search for TVAL in [T(I), T(I+1)] for intervals I = 2 to LEFT-2. ! low = 2 high = left - 2 do if ( low == high ) then left = low return end if mid = ( low + high + 1 ) / 2 if ( t(mid) <= tval ) then low = mid else high = mid - 1 end if end do ! ! CASE2: T(LEFT+1) < TVAL: ! Search for TVAL in [T(I),T(I+1)] for intervals I = LEFT+1 to N-1. ! else if ( t(left+1) < tval ) then if ( left == n - 1 ) then return else if ( left == n - 2 ) then left = left + 1 return else if ( tval <= t(left+2) ) then left = left + 1 return else if ( t(n-1) <= tval ) then left = n - 1 return end if ! ! ...Binary search for TVAL in [T(I), T(I+1)] for intervals I = LEFT+2 to N-2. ! low = left + 2 high = n - 2 do if ( low == high ) then left = low return end if mid = ( low + high + 1 ) / 2 if ( t(mid) <= tval ) then low = mid else high = mid - 1 end if end do ! ! CASE3: T(LEFT) <= TVAL <= T(LEFT+1): ! T is in [T(LEFT), T(LEFT+1)], as the user said it might be. ! else end if return end subroutine r8vec_bracket4 ( nt, t, ns, s, left ) !*****************************************************************************80 ! !! R8VEC_BRACKET4 finds the nearest interval to each of a vector of values. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The routine always returns the index LEFT of the sorted array ! T with the property that either ! * T is contained in the interval [ T(LEFT), T(LEFT+1) ], or ! * T < T(LEFT) = T(1), or ! * T > T(LEFT+1) = T(NT). ! ! The routine is useful for interpolation problems, where ! the abscissa must be located within an interval of data ! abscissas for interpolation, or the "nearest" interval ! to the (extreme) abscissa must be found so that extrapolation ! can be carried out. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 April 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) NT, length of the input array. ! ! Input, real ( kind = 8 ) T(NT), an array that has been sorted ! into ascending order. ! ! Input, integer ( kind = 4 ) NS, the number of points to be bracketed. ! ! Input, real ( kind = 8 ) S(NS), values to be bracketed by entries of T. ! ! Output, integer ( kind = 4 ) LEFT(NS). ! LEFT(I) is set so that the interval [ T(LEFT(I)), T(LEFT(I)+1) ] ! is the closest to S(I); it either contains S(I), or else S(I) ! lies outside the interval [ T(1), T(NT) ]. ! implicit none integer ( kind = 4 ) ns integer ( kind = 4 ) nt integer ( kind = 4 ) high integer ( kind = 4 ) i integer ( kind = 4 ) left(ns) integer ( kind = 4 ) low integer ( kind = 4 ) mid real ( kind = 8 ) s(ns) real ( kind = 8 ) t(nt) ! ! Check the input data. ! if ( nt < 2 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_BRACKET4 - Fatal error!' write ( *, '(a)' ) ' NT must be at least 2.' stop 1 end if do i = 1, ns left(i) = ( nt + 1 ) / 2 ! ! CASE 1: S < T(LEFT): ! Search for S in [T(I), T(I+1)] for intervals I = 1 to LEFT-1. ! if ( s(i) < t(left(i)) ) then if ( left(i) == 1 ) then cycle else if ( left(i) == 2 ) then left(i) = 1 cycle else if ( t(left(i)-1) <= s(i) ) then left(i) = left(i) - 1 cycle else if ( s(i) <= t(2) ) then left(i) = 1 cycle end if ! ! ...Binary search for S in [T(I), T(I+1)] for intervals I = 2 to LEFT-2. ! low = 2 high = left(i) - 2 do if ( low == high ) then left(i) = low exit end if mid = ( low + high + 1 ) / 2 if ( t(mid) <= s(i) ) then low = mid else high = mid - 1 end if end do ! ! CASE2: T(LEFT+1) < S: ! Search for S in [T(I),T(I+1)] for intervals I = LEFT+1 to N-1. ! else if ( t(left(i)+1) < s(i) ) then if ( left(i) == nt - 1 ) then cycle else if ( left(i) == nt - 2 ) then left(i) = left(i) + 1 cycle else if ( s(i) <= t(left(i)+2) ) then left(i) = left(i) + 1 cycle else if ( t(nt-1) <= s(i) ) then left(i) = nt - 1 cycle end if ! ! ...Binary search for S in [T(I), T(I+1)] for intervals I = LEFT+2 to NT-2. ! low = left(i) + 2 high = nt - 2 do if ( low == high ) then left(i) = low exit end if mid = ( low + high + 1 ) / 2 if ( t(mid) <= s(i) ) then low = mid else high = mid - 1 end if end do ! ! CASE3: T(LEFT) <= S <= T(LEFT+1): ! S is in [T(LEFT), T(LEFT+1)]. ! else end if end do return end function r8vec_bracket5 ( nd, xd, xi ) !*****************************************************************************80 ! !! R8VEC_BRACKET5 brackets data between successive entries of a sorted R8VEC. ! ! Discussion: ! ! We assume XD is sorted. ! ! If XI is contained in the interval [XD(1),XD(N)], then the returned ! value B indicates that XI is contained in [ XD(B), XD(B+1) ]. ! ! If XI is not contained in the interval [XD(1),XD(N)], then B = -1. ! ! This code implements a version of binary search which is perhaps more ! understandable than the usual ones. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 October 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) ND, the number of data values. ! ! Input, real ( kind = 8 ) XD(N), the sorted data. ! ! Input, real ( kind = 8 ) XD, the query value. ! ! Output, integer ( kind = 4 ) R8VEC_BRACKET5, the bracket information. ! implicit none integer ( kind = 4 ) nd integer ( kind = 4 ) b integer ( kind = 4 ) l integer ( kind = 4 ) m integer ( kind = 4 ) r integer ( kind = 4 ) r8vec_bracket5 real ( kind = 8 ) xd(nd) real ( kind = 8 ) xi if ( xi < xd(1) .or. xd(nd) < xi ) then b = -1 else l = 1 r = nd do while ( l + 1 < r ) m = ( l + r ) / 2 if ( xi < xd(m) ) then r = m else l = m end if end do b = l end if r8vec_bracket5 = b return end subroutine r8vec_bracket6 ( nd, xd, ni, xi, b ) !*****************************************************************************80 ! !! R8VEC_BRACKET6 brackets data between successive entries of a sorted R8VEC. ! ! Discussion: ! ! We assume XD is sorted. ! ! If XI(I) is contained in the interval [XD(1),XD(N)], then the value of ! B(I) indicates that XI(I) is contained in [ XD(B(I)), XD(B(I)+1) ]. ! ! If XI(I) is not contained in the interval [XD(1),XD(N)], then B(I) = -1. ! ! This code implements a version of binary search which is perhaps more ! understandable than the usual ones. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 October 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) ND, the number of data values. ! ! Input, real ( kind = 8 ) XD(N), the sorted data. ! ! Input, integer ( kind = 4 ) NI, the number of inquiry values. ! ! Input, real ( kind = 8 ) XD(NI), the query values. ! ! Output, integer ( kind = 4 ) B(NI), the bracket information. ! implicit none integer ( kind = 4 ) nd integer ( kind = 4 ) ni integer ( kind = 4 ) b(ni) integer ( kind = 4 ) i integer ( kind = 4 ) l integer ( kind = 4 ) m integer ( kind = 4 ) r real ( kind = 8 ) xd(nd) real ( kind = 8 ) xi(ni) do i = 1, ni if ( xi(i) < xd(1) .or. xd(nd) < xi(i) ) then b(i) = -1 else l = 1 r = nd do while ( l + 1 < r ) m = ( l + r ) / 2 if ( xi(i) < xd(m) ) then r = m else l = m end if end do b(i) = l end if end do return end subroutine r8vec_ceiling ( n, r8vec, ceilingvec ) !*****************************************************************************80 ! !! R8VEC_CEILING rounds "up" (towards +oo) entries of an R8VEC. ! ! Example: ! ! R8 Value ! ! -1.1 -1.0 ! -1.0 -1.0 ! -0.9 0.0 ! 0.0 0.0 ! 5.0 5.0 ! 5.1 6.0 ! 5.9 6.0 ! 6.0 6.0 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries. ! ! Input, real ( kind = 8 ) R8VEC(N), the vector. ! ! Output, real ( kind = 8 ) CEILINGVEC(N), the rounded values. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) ceilingvec(n) integer ( kind = 4 ) i real ( kind = 8 ) r8vec(n) real ( kind = 8 ) value do i = 1, n value = real ( int ( r8vec(i) ), kind = 8 ) if ( value < r8vec(i) ) then value = value + 1.0D+00 end if ceilingvec(i) = value end do return end subroutine r8vec_cheby_extreme ( n, a, b, x ) !*****************************************************************************80 ! !! R8VEC_CHEBY_EXTREME creates Chebyshev Extreme values in [A,B]. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 September 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A, B, the first and last entries. ! ! Output, real ( kind = 8 ) X(N), a vector of Chebyshev spaced data. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b real ( kind = 8 ) c integer ( kind = 4 ) i real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) theta real ( kind = 8 ) x(n) if ( n == 1 ) then x(1) = ( a + b ) / 2.0D+00 else do i = 1, n theta = real ( n - i, kind = 8 ) * r8_pi / real ( n - 1, kind = 8 ) c = cos ( theta ) if ( mod ( n, 2 ) == 1 ) then if ( 2 * i - 1 == n ) then c = 0.0D+00 end if end if x(i) = ( ( 1.0D+00 - c ) * a & + ( 1.0D+00 + c ) * b ) & / 2.0D+00 end do end if return end subroutine r8vec_cheby_zero ( n, a, b, x ) !*****************************************************************************80 ! !! R8VEC_CHEBY_ZERO creates Chebyshev Zero values in [A,B]. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 September 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A, B, the first and last entries. ! ! Output, real ( kind = 8 ) X(N), a vector of Chebyshev spaced data. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b real ( kind = 8 ) c integer ( kind = 4 ) i real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) theta real ( kind = 8 ) x(n) if ( n == 1 ) then x(1) = ( a + b ) / 2.0D+00 else do i = 1, n theta = real ( 2 * ( n - i ) + 1, kind = 8 ) * r8_pi & / real ( 2 * n, kind = 8 ) c = cos ( theta ) if ( mod ( n, 2 ) == 1 ) then if ( 2 * i - 1 == n ) then c = 0.0D+00 end if end if x(i) = ( ( 1.0D+00 - c ) * a & + ( 1.0D+00 + c ) * b ) & / 2.0D+00 end do end if return end subroutine r8vec_cheby1space ( n, a, b, x ) !*****************************************************************************80 ! !! R8VEC_CHEBY1SPACE creates Type 1 Chebyshev-spaced values in [A,B]. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 September 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A, B, the first and last entries. ! ! Output, real ( kind = 8 ) X(N), a vector of Type 1 Chebyshev spaced data. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b real ( kind = 8 ) c integer ( kind = 4 ) i real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) theta real ( kind = 8 ) x(n) if ( n == 1 ) then x(1) = ( a + b ) / 2.0D+00 else do i = 1, n theta = real ( n - i, kind = 8 ) * r8_pi / real ( n - 1, kind = 8 ) c = cos ( theta ) if ( mod ( n, 2 ) == 1 ) then if ( 2 * i - 1 == n ) then c = 0.0D+00 end if end if x(i) = ( ( 1.0D+00 - c ) * a & + ( 1.0D+00 + c ) * b ) & / 2.0D+00 end do end if return end subroutine r8vec_cheby2space ( n, a, b, x ) !*****************************************************************************80 ! !! R8VEC_CHEBY2SPACE creates Type 2 Chebyshev-spaced values in [A,B]. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 July 2017 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A, B, the first and last entries. ! ! Output, real ( kind = 8 ) X(N), a vector of Type 2 Chebyshev spaced data. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b real ( kind = 8 ) c integer ( kind = 4 ) i real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) theta real ( kind = 8 ) x(n) do i = 1, n theta = real ( n - i, kind = 8 ) * r8_pi / real ( n + 1, kind = 8 ) c = cos ( theta ) x(i) = ( ( 1.0D+00 - c ) * a & + ( 1.0D+00 + c ) * b ) & / 2.0D+00 end do return end subroutine r8vec_compare ( n, a1, a2, isgn ) !*****************************************************************************80 ! !! R8VEC_COMPARE compares two R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The lexicographic ordering is used. ! ! Example: ! ! Input: ! ! A1 = ( 2.0, 6.0, 2.0 ) ! A2 = ( 2.0, 8.0, 12.0 ) ! ! Output: ! ! ISGN = -1 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 February 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vectors. ! ! Input, real ( kind = 8 ) A1(N), A2(N), the vectors to be compared. ! ! Output, integer ( kind = 4 ) ISGN, the results of the comparison: ! -1, A1 < A2, ! 0, A1 = A2, ! +1, A1 > A2. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) integer ( kind = 4 ) isgn integer ( kind = 4 ) k isgn = 0 k = 1 do while ( k <= n ) if ( a1(k) < a2(k) ) then isgn = -1 return else if ( a2(k) < a1(k) ) then isgn = + 1 return end if k = k + 1 end do return end subroutine r8vec_concatenate ( n1, a, n2, b, c ) !*****************************************************************************80 ! !! R8VEC_CONCATENATE concatenates two R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8 values. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 22 November 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N1, the number of entries in the first vector. ! ! Input, real ( kind = 8 ) A(N1), the first vector. ! ! Input, integer ( kind = 4 ) N2, the number of entries in the second vector. ! ! Input, real ( kind = 8 ) B(N2), the second vector. ! ! Output, real ( kind = 8 ) C(N1+N2), the concatenation of A and B. ! implicit none integer ( kind = 4 ) n1 integer ( kind = 4 ) n2 real ( kind = 8 ) a(n1) real ( kind = 8 ) b(n2) real ( kind = 8 ) c(n1+n2) c( 1:n1) = a(1:n1) c(n1+1:n1+n2) = b(1:n2) return end subroutine r8vec_convolution ( m, x, n, y, z ) !*****************************************************************************80 ! !! R8VEC_CONVOLUTION returns the convolution of two R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The I-th entry of the convolution can be formed by summing the products ! that lie along the I-th diagonal of the following table: ! ! Y3 | 3 4 5 6 7 ! Y2 | 2 3 4 5 6 ! Y1 | 1 2 3 4 5 ! +------------------ ! X1 X2 X3 X4 X5 ! ! which will result in: ! ! Z = ( X1 * Y1, ! X1 * Y2 + X2 * Y1, ! X1 * Y3 + X2 * Y2 + X3 * Y1, ! X2 * Y3 + X3 * Y2 + X4 * Y1, ! X3 * Y3 + X4 * Y2 + X5 * Y1, ! X4 * Y3 + X5 * Y2, ! X5 * Y3 ) ! ! Example: ! ! Input: ! ! X = (/ 1, 2, 3, 4 /) ! Y = (/ -1, 5, 3 /) ! ! Output: ! ! Z = (/ -1, 3, 10, 17, 29, 12 /) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 May 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the dimension of X. ! ! Input, real ( kind = 8 ) X(M), the first vector to be convolved. ! ! Input, integer ( kind = 4 ) N, the dimension of Y. ! ! Input, real ( kind = 8 ) Y(N), the second vector to be convolved. ! ! Output, real ( kind = 8 ) Z(M+N-1), the convolution of X and Y. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n integer ( kind = 4 ) j real ( kind = 8 ) x(m) real ( kind = 8 ) y(n) real ( kind = 8 ) z(m+n-1) z(1:m+n-1) = 0.0D+00 do j = 1, n z(j:j+m-1) = z(j:j+m-1) + x(1:m) * y(j) end do return end subroutine r8vec_convolution_circ ( n, x, y, z ) !*****************************************************************************80 ! !! R8VEC_CONVOLUTION_CIRC: discrete circular convolution of two R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The formula used is: ! ! z(1+m) = xCCy(m) = sum ( 0 <= k <= n-1 ) x(1+k) * y(1+m-k) ! ! Here, if the index of Y becomes nonpositive, it is "wrapped around" ! by having N added to it. ! ! The circular convolution is equivalent to multiplication of Y by a ! circulant matrix formed from the vector X. ! ! Example: ! ! Input: ! ! X = (/ 1, 2, 3, 4 /) ! Y = (/ 1, 2, 4, 8 /) ! ! Output: ! ! Circulant form: ! ! Z = ( 1 4 3 2 ) ( 1 ) ! ( 2 1 4 3 ) ( 2 ) ! ( 3 2 1 4 ) * ( 4 ) ! ( 4 3 2 1 ) ( 8 ) ! ! The formula: ! ! Z = (/ 1*1 + 2*8 + 3*4 + 4*2, ! 1*2 + 2*1 + 3*8 + 4*4, ! 1*4 + 2*2 + 3*1 + 4*8, ! 1*8 + 2*4 + 3*2 + 4*1 /) ! ! Result: ! ! Z = (/ 37, 44, 43, 26 /) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 September 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of the vectors. ! ! Input, real ( kind = 8 ) X(N), Y(N), the vectors to be convolved. ! ! Output, real ( kind = 8 ) Z(N), the circular convolution of X and Y. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) m real ( kind = 8 ) x(n) real ( kind = 8 ) y(n) real ( kind = 8 ) z(n) do m = 1, n z(m) = dot_product ( x(1:m), y(m:1:-1) ) & + dot_product ( x(m+1:n), y(n:m+1:-1) ) end do return end subroutine r8vec_copy ( n, a1, a2 ) !*****************************************************************************80 ! !! R8VEC_COPY copies an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 September 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the length of the vectors. ! ! Input, real ( kind = 8 ) A1(N), the vector to be copied. ! ! Output, real ( kind = 8 ) A2(N), a copy of A1. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) a2(1:n) = a1(1:n) return end subroutine r8vec_correlation ( n, x, y, r ) !*****************************************************************************80 ! !! r8vec_correlation returns the correlation of two R8VEC's. ! ! Discussion: ! ! The correlation coefficient is also known as Pearson's r coefficient. ! ! It must be the case that -1 <= r <= +1. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 August 2019 ! ! Author: ! ! John Burkardt ! ! Input: ! ! integer ( kind = 4 ) N, the dimension of the vectors. ! ! real ( kind = 8 ) X(N), Y(N), the vectors. ! ! Output: ! ! real ( kind = 8 ) r, the correlation of X and Y. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) r real ( kind = 8 ) x(n) real ( kind = 8 ) x_centered(n) real ( kind = 8 ) x_mean real ( kind = 8 ) x_std real ( kind = 8 ) y(n) real ( kind = 8 ) y_centered(n) real ( kind = 8 ) y_mean real ( kind = 8 ) y_std if ( n <= 1 ) then r = 0.0D+00 else call r8vec_mean ( n, x, x_mean ) call r8vec_std_sample ( n, x, x_std ) x_centered = x - x_mean call r8vec_mean ( n, y, y_mean ) call r8vec_std_sample ( n, y, y_std ) y_centered = y - y_mean r = dot_product ( x_centered, y_centered ) / x_std / y_std / real ( n - 1, kind = 8 ) end if return end function r8vec_covariance ( n, x, y ) !*****************************************************************************80 ! !! R8VEC_COVARIANCE computes the covariance of two vectors. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 April 2013 ! ! Author: ! ! John Burkardt. ! ! Parameters: ! ! Input, real ( kind = 8 ) X(N), Y(N), the two vectors. ! ! Input, integer ( kind = 4 ) N, the dimension of the two vectors. ! ! Output, real ( kind = 8 ) R4VEC_COVARIANCE, the covariance of the vectors. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i real ( kind = 8 ) r8vec_covariance real ( kind = 8 ) value real ( kind = 8 ) x(n) real ( kind = 8 ) x_average real ( kind = 8 ) y(n) real ( kind = 8 ) y_average x_average = sum ( x(1:n) ) / real ( n, kind = 8 ) y_average = sum ( y(1:n) ) / real ( n, kind = 8 ) value = 0.0D+00 do i = 1, n value = value + ( x(i) - x_average ) * ( y(i) - y_average ) end do r8vec_covariance = value / real ( n - 1, kind = 8 ) return end function r8vec_cross_product_2d ( v1, v2 ) !*****************************************************************************80 ! !! R8VEC_CROSS_PRODUCT_2D finds the cross product of a pair of vectors in 2D. ! ! Discussion: ! ! Strictly speaking, the vectors V1 and V2 should be considered ! to lie in a 3D space, both having Z coordinate zero. The cross ! product value V3 then represents the standard cross product vector ! (0,0,V3). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 August 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) V1(2), V2(2), the vectors. ! ! Output, real ( kind = 8 ) R8VEC_CROSS_PRODUCT_2D, the cross product. ! implicit none real ( kind = 8 ) r8vec_cross_product_2d real ( kind = 8 ) v1(2) real ( kind = 8 ) v2(2) r8vec_cross_product_2d = v1(1) * v2(2) - v1(2) * v2(1) return end function r8vec_cross_product_affine_2d ( v0, v1, v2 ) !*****************************************************************************80 ! !! R8VEC_CROSS_PRODUCT_AFFINE_2D finds the affine cross product in 2D. ! ! Discussion: ! ! Strictly speaking, the vectors V1 and V2 should be considered ! to lie in a 3D space, both having Z coordinate zero. The cross ! product value V3 then represents the standard cross product vector ! (0,0,V3). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) V0(2), the base vector. ! ! Input, real ( kind = 8 ) V1(2), V2(2), the vectors. ! ! Output, real ( kind = 8 ) R8VEC_CROSS_PRODUCT_AFFINE_2D, ! the cross product (V1-V0) x (V2-V0). ! implicit none real ( kind = 8 ) r8vec_cross_product_affine_2d real ( kind = 8 ) v0(2) real ( kind = 8 ) v1(2) real ( kind = 8 ) v2(2) r8vec_cross_product_affine_2d = & ( v1(1) - v0(1) ) * ( v2(2) - v0(2) ) & - ( v2(1) - v0(1) ) * ( v1(2) - v0(2) ) return end subroutine r8vec_cross_product_3d ( v1, v2, v3 ) !*****************************************************************************80 ! !! R8VEC_CROSS_PRODUCT_3D computes the cross product of two R8VEC's in 3D. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The cross product in 3D can be regarded as the determinant of the ! symbolic matrix: ! ! | i j k | ! det | x1 y1 z1 | ! | x2 y2 z2 | ! ! = ( y1 * z2 - z1 * y2 ) * i ! + ( z1 * x2 - x1 * z2 ) * j ! + ( x1 * y2 - y1 * x2 ) * k ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 August 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) V1(3), V2(3), the two vectors. ! ! Output, real ( kind = 8 ) V3(3), the cross product vector. ! implicit none real ( kind = 8 ) v1(3) real ( kind = 8 ) v2(3) real ( kind = 8 ) v3(3) v3(1) = v1(2) * v2(3) - v1(3) * v2(2) v3(2) = v1(3) * v2(1) - v1(1) * v2(3) v3(3) = v1(1) * v2(2) - v1(2) * v2(1) return end subroutine r8vec_cross_product_affine_3d ( v0, v1, v2, v3 ) !*****************************************************************************80 ! !! R8VEC_CROSS_PRODUCT_AFFINE_3D computes the affine cross product in 3D. ! ! Discussion: ! ! The cross product in 3D can be regarded as the determinant of the ! symbolic matrix: ! ! | i j k | ! det | x1 y1 z1 | ! | x2 y2 z2 | ! ! = ( y1 * z2 - z1 * y2 ) * i ! + ( z1 * x2 - x1 * z2 ) * j ! + ( x1 * y2 - y1 * x2 ) * k ! ! Here, we use V0 as the base of an affine system so we compute ! the cross product of (V1-V0) and (V2-V0). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) V0(3), the base vector. ! ! Input, real ( kind = 8 ) V1(3), V2(3), the two vectors. ! ! Output, real ( kind = 8 ) V3(3), the cross product vector ! ( V1-V0) x (V2-V0). ! implicit none real ( kind = 8 ) v0(3) real ( kind = 8 ) v1(3) real ( kind = 8 ) v2(3) real ( kind = 8 ) v3(3) v3(1) = ( v1(2) - v0(2) ) * ( v2(3) - v0(3) ) & - ( v2(2) - v0(2) ) * ( v1(3) - v0(3) ) v3(2) = ( v1(3) - v0(3) ) * ( v2(1) - v0(1) ) & - ( v2(3) - v0(3) ) * ( v1(1) - v0(1) ) v3(3) = ( v1(1) - v0(1) ) * ( v2(2) - v0(2) ) & - ( v2(1) - v0(1) ) * ( v1(2) - v0(2) ) return end subroutine r8vec_cum ( n, a, a_cum ) !*****************************************************************************80 ! !! R8VEC_CUM computes the cumulutive sums of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Input: ! ! A = (/ 1.0, 2.0, 3.0, 4.0 /) ! ! Output: ! ! A_CUM = (/ 1.0, 3.0, 6.0, 10.0 /) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 May 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A(N), the vector to be summed. ! ! Output, real ( kind = 8 ) A_CUM(1:N), the cumulative sums. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) a_cum(n) integer ( kind = 4 ) i a_cum(1) = a(1) do i = 2, n a_cum(i) = a_cum(i-1) + a(i) end do return end subroutine r8vec_cum0 ( n, a, a_cum ) !*****************************************************************************80 ! !! R8VEC_CUM0 computes the cumulutive sums of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Input: ! ! A = (/ 1.0, 2.0, 3.0, 4.0 /) ! ! Output: ! ! A_CUM = (/ 0.0, 1.0, 3.0, 6.0, 10.0 /) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 May 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A(N), the vector to be summed. ! ! Output, real ( kind = 8 ) A_CUM(0:N), the cumulative sums. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) a_cum(0:n) integer ( kind = 4 ) i a_cum(0) = 0.0D+00 do i = 1, n a_cum(i) = a_cum(i-1) + a(i) end do return end subroutine r8vec_dif ( n, h, cof ) !*****************************************************************************80 ! !! R8VEC_DIF computes coefficients for estimating the N-th derivative. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The routine computes the N+1 coefficients for a centered finite difference ! estimate of the N-th derivative of a function. ! ! The estimate has the form ! ! FDIF(N,X) = Sum (I = 0 to N) COF(I) * F ( X(I) ) ! ! To understand the computation of the coefficients, it is enough ! to realize that the first difference approximation is ! ! FDIF(1,X) = F(X+DX) - F(X-DX) ) / (2*DX) ! ! and that the second difference approximation can be regarded as ! the first difference approximation repeated: ! ! FDIF(2,X) = FDIF(1,X+DX) - FDIF(1,X-DX) / (2*DX) ! = F(X+2*DX) - 2 F(X) + F(X-2*DX) / (4*DX) ! ! and so on for higher order differences. ! ! Thus, the next thing to consider is the integer coefficients of ! the sampled values of F, which are clearly the Pascal coefficients, ! but with an alternating negative sign. In particular, if we ! consider row I of Pascal's triangle to have entries j = 0 through I, ! then P(I,J) = P(I-1,J-1) - P(I-1,J), where P(*,-1) is taken to be 0, ! and P(0,0) = 1. ! ! 1 ! -1 1 ! 1 -2 1 ! -1 3 -3 1 ! 1 -4 6 -4 1 ! -1 5 -10 10 -5 1 ! 1 -6 15 -20 15 -6 1 ! ! Next, note that the denominator of the approximation for the ! N-th derivative will be (2*DX)^N. ! ! And finally, consider the location of the N+1 sampling ! points for F: ! ! X-N*DX, X-(N-2)*DX, X-(N-4)*DX, ..., X+(N-4)*DX, X+(N-2*DX), X+N*DX. ! ! Thus, a formula for evaluating FDIF(N,X) is ! ! fdif = 0.0 ! do i = 0, n ! xi = x + (2*i-n) * h ! fdif = fdif + cof(i) * f(xi) ! end do ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 February 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the derivative to be ! approximated. N must be 0 or greater. ! ! Input, real ( kind = 8 ) H, the half spacing between points. ! H must be positive. ! ! Output, real ( kind = 8 ) COF(0:N), the coefficients needed to approximate ! the N-th derivative of a function F. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) cof(0:n) real ( kind = 8 ) h integer ( kind = 4 ) i integer ( kind = 4 ) j if ( n < 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_DIF - Fatal error!' write ( *, '(a,i8)' ) ' Derivative order N = ', n write ( *, '(a)' ) ' but N must be at least 0.' stop 1 end if if ( h <= 0.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_DIF - Fatal error!' write ( *, '(a,g14.6)' ) ' The half sampling spacing is H = ', h write ( *, '(a)' ) ' but H must be positive.' stop 1 end if do i = 0, n cof(i) = 1.0D+00 do j = i - 1, 1, -1 cof(j) = - cof(j) + cof(j-1) end do if ( 0 < i ) then cof(0) = - cof(0) end if end do cof(0:n) = cof(0:n) / ( 2.0D+00 * h ) ** n return end function r8vec_diff_dot_product ( n, u1, v1, u2, v2 ) !*****************************************************************************80 ! !! R8VEC_DIFF_DOT_PRODUCT: dot product of a pair of R8VEC differences. ! ! Discussion: ! ! An R8VEC is a vector of R8 values. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 31 March 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of the vectors. ! ! Input, real ( kind = 8 ) U1(N), V1(N), defines the vector U1-V1. ! ! Input, real ( kind = 8 ) U2(N), V2(N), defines the vector U2-V2. ! ! Output, real ( kind = 8 ) R8VEC_DIFF_DOT_PRODUCT, the dot product ! of (U1-V1)*(U2-V2). ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i real ( kind = 8 ) r8vec_diff_dot_product real ( kind = 8 ) u1(n) real ( kind = 8 ) u2(n) real ( kind = 8 ) v1(n) real ( kind = 8 ) v2(n) real ( kind = 8 ) value value = 0.0D+00 do i = 1, n value = value + ( u1(i) - v1(i) ) * ( u2(i) - v2(i) ) end do r8vec_diff_dot_product = value return end function r8vec_diff_norm ( n, a, b ) !*****************************************************************************80 ! !! R8VEC_DIFF_NORM returns the L2 norm of the difference of R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The vector L2 norm is defined as: ! ! R8VEC_NORM_L2 = sqrt ( sum ( 1 <= I <= N ) A(I)^2 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 April 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), B(N), the vectors ! ! Output, real ( kind = 8 ) R8VEC_DIFF_NORM, the L2 norm of A - B. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) b(n) real ( kind = 8 ) r8vec_diff_norm r8vec_diff_norm = sqrt ( sum ( ( a(1:n) - b(1:n) ) ** 2 ) ) return end function r8vec_diff_norm_l1 ( n, a, b ) !*****************************************************************************80 ! !! R8VEC_DIFF_NORM_L1 returns the L1 norm of the difference of R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The vector L1 norm is defined as: ! ! R8VEC_NORM_L1 = sum ( 1 <= I <= N ) abs ( A(I) ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 April 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), B(N), the vectors. ! ! Output, real ( kind = 8 ) R8VEC_DIFF_NORM_L1, the L1 norm of A - B. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) b(n) real ( kind = 8 ) r8vec_diff_norm_l1 r8vec_diff_norm_l1 = sum ( abs ( a(1:n) - b(1:n) ) ) return end function r8vec_diff_norm_l2 ( n, a, b ) !*****************************************************************************80 ! !! R8VEC_DIFF_NORM_L2 returns the L2 norm of the difference of R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The vector L2 norm is defined as: ! ! R8VEC_NORM_L2 = sqrt ( sum ( 1 <= I <= N ) A(I)^2 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 April 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), B(N), the vectors ! ! Output, real ( kind = 8 ) R8VEC_DIFF_NORM_L2, the L2 norm of A - B. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) b(n) real ( kind = 8 ) r8vec_diff_norm_l2 r8vec_diff_norm_l2 = sqrt ( sum ( ( a(1:n) - b(1:n) ) ** 2 ) ) return end function r8vec_diff_norm_li ( n, a, b ) !*****************************************************************************80 ! !! R8VEC_DIFF_NORM_LI returns the L-oo norm of the difference of R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The vector L-oo norm is defined as: ! ! R8VEC_NORM_LI = max ( 1 <= I <= N ) abs ( A(I) ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 April 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), B(N), the vectors ! ! Output, real ( kind = 8 ) R8VEC_DIFF_NORM_LI, the L-oo norm of A - B. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) b(n) real ( kind = 8 ) r8vec_diff_norm_li r8vec_diff_norm_li = maxval ( abs ( a(1:n) - b(1:n) ) ) return end function r8vec_diff_norm_squared ( n, a, b ) !*****************************************************************************80 ! !! R8VEC_DIFF_NORM_SQUARED: square of the L2 norm of the difference of R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! R8VEC_DIFF_NORM_SQUARED = sum ( 1 <= I <= N ) ( A(I) - B(I) )^2 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 March 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), B(N), the vectors ! ! Output, real ( kind = 8 ) R8VEC_DIFF_NORM_SQUARED, the square of ! the L2 norm of A - B. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) b(n) real ( kind = 8 ) r8vec_diff_norm_squared r8vec_diff_norm_squared = sum ( ( a(1:n) - b(1:n) ) ** 2 ) return end subroutine r8vec_direct_product ( factor_index, factor_order, factor_value, & factor_num, point_num, x ) !*****************************************************************************80 ! !! R8VEC_DIRECT_PRODUCT creates a direct product of R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! To explain what is going on here, suppose we had to construct ! a multidimensional quadrature rule as the product of K rules ! for 1D quadrature. ! ! The product rule will be represented as a list of points and weights. ! ! The J-th item in the product rule will be associated with ! item J1 of 1D rule 1, ! item J2 of 1D rule 2, ! ..., ! item JK of 1D rule K. ! ! In particular, ! X(J) = ( X(1,J1), X(2,J2), ..., X(K,JK)) ! and ! W(J) = W(1,J1) * W(2,J2) * ... * W(K,JK) ! ! So we can construct the quadrature rule if we can properly ! distribute the information in the 1D quadrature rules. ! ! This routine carries out that task for the abscissas X. ! ! Another way to do this would be to compute, one by one, the ! set of all possible indices (J1,J2,...,JK), and then index ! the appropriate information. An advantage of the method shown ! here is that you can process the K-th set of information and ! then discard it. ! ! Example: ! ! Rule 1: ! Order = 4 ! X(1:4) = ( 1, 2, 3, 4 ) ! ! Rule 2: ! Order = 3 ! X(1:3) = ( 10, 20, 30 ) ! ! Rule 3: ! Order = 2 ! X(1:2) = ( 100, 200 ) ! ! Product Rule: ! Order = 24 ! X(1:24) = ! ( 1, 10, 100 ) ! ( 2, 10, 100 ) ! ( 3, 10, 100 ) ! ( 4, 10, 100 ) ! ( 1, 20, 100 ) ! ( 2, 20, 100 ) ! ( 3, 20, 100 ) ! ( 4, 20, 100 ) ! ( 1, 30, 100 ) ! ( 2, 30, 100 ) ! ( 3, 30, 100 ) ! ( 4, 30, 100 ) ! ( 1, 10, 200 ) ! ( 2, 10, 200 ) ! ( 3, 10, 200 ) ! ( 4, 10, 200 ) ! ( 1, 20, 200 ) ! ( 2, 20, 200 ) ! ( 3, 20, 200 ) ! ( 4, 20, 200 ) ! ( 1, 30, 200 ) ! ( 2, 30, 200 ) ! ( 3, 30, 200 ) ! ( 4, 30, 200 ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 April 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) FACTOR_INDEX, the index of the factor being ! processed. The first factor processed must be factor 1! ! ! Input, integer ( kind = 4 ) FACTOR_ORDER, the order of the factor. ! ! Input, real ( kind = 8 ) FACTOR_VALUE(FACTOR_ORDER), the factor values ! for factor FACTOR_INDEX. ! ! Input, integer ( kind = 4 ) FACTOR_NUM, the number of factors. ! ! Input, integer ( kind = 4 ) POINT_NUM, the number of elements in the ! direct product. ! ! Input/output, real ( kind = 8 ) X(FACTOR_NUM,POINT_NUM), the elements of ! the direct product, which are built up gradually. ! ! Local Parameters: ! ! Local, integer ( kind = 4 ) START, the first location of a block of ! values to set. ! ! Local, integer ( kind = 4 ) CONTIG, the number of consecutive values ! to set. ! ! Local, integer ( kind = 4 ) SKIP, the distance from the current value ! of START to the next location of a block of values to set. ! ! Local, integer ( kind = 4 ) REP, the number of blocks of values to set. ! implicit none integer ( kind = 4 ) factor_num integer ( kind = 4 ) factor_order integer ( kind = 4 ) point_num integer ( kind = 4 ), save :: contig integer ( kind = 4 ) factor_index real ( kind = 8 ) factor_value(factor_order) integer ( kind = 4 ) j integer ( kind = 4 ) k integer ( kind = 4 ), save :: rep integer ( kind = 4 ), save :: skip integer ( kind = 4 ) start real ( kind = 8 ) x(factor_num,point_num) if ( factor_index == 1 ) then contig = 1 skip = 1 rep = point_num x(1:factor_num,1:point_num) = 0.0D+00 end if rep = rep / factor_order skip = skip * factor_order do j = 1, factor_order start = 1 + ( j - 1 ) * contig do k = 1, rep x(factor_index,start:start+contig-1) = factor_value(j) start = start + skip end do end do contig = contig * factor_order return end subroutine r8vec_direct_product2 ( factor_index, factor_order, factor_value, & factor_num, point_num, w ) !*****************************************************************************80 ! !! R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! To explain what is going on here, suppose we had to construct ! a multidimensional quadrature rule as the product of K rules ! for 1D quadrature. ! ! The product rule will be represented as a list of points and weights. ! ! The J-th item in the product rule will be associated with ! item J1 of 1D rule 1, ! item J2 of 1D rule 2, ! ..., ! item JK of 1D rule K. ! ! In particular, ! X(J) = ( X(1,J1), X(2,J2), ..., X(K,JK)) ! and ! W(J) = W(1,J1) * W(2,J2) * ... * W(K,JK) ! ! So we can construct the quadrature rule if we can properly ! distribute the information in the 1D quadrature rules. ! ! This routine carries out the task involving the weights W. ! ! Another way to do this would be to compute, one by one, the ! set of all possible indices (J1,J2,...,JK), and then index ! the appropriate information. An advantage of the method shown ! here is that you can process the K-th set of information and ! then discard it. ! ! Example: ! ! Rule 1: ! Order = 4 ! W(1:4) = ( 2, 3, 5, 7 ) ! ! Rule 2: ! Order = 3 ! W(1:3) = ( 11, 13, 17 ) ! ! Rule 3: ! Order = 2 ! W(1:2) = ( 19, 23 ) ! ! Product Rule: ! Order = 24 ! W(1:24) = ! ( 2 * 11 * 19 ) ! ( 3 * 11 * 19 ) ! ( 4 * 11 * 19 ) ! ( 7 * 11 * 19 ) ! ( 2 * 13 * 19 ) ! ( 3 * 13 * 19 ) ! ( 5 * 13 * 19 ) ! ( 7 * 13 * 19 ) ! ( 2 * 17 * 19 ) ! ( 3 * 17 * 19 ) ! ( 5 * 17 * 19 ) ! ( 7 * 17 * 19 ) ! ( 2 * 11 * 23 ) ! ( 3 * 11 * 23 ) ! ( 5 * 11 * 23 ) ! ( 7 * 11 * 23 ) ! ( 2 * 13 * 23 ) ! ( 3 * 13 * 23 ) ! ( 5 * 13 * 23 ) ! ( 7 * 13 * 23 ) ! ( 2 * 17 * 23 ) ! ( 3 * 17 * 23 ) ! ( 5 * 17 * 23 ) ! ( 7 * 17 * 23 ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 April 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) FACTOR_INDEX, the index of the factor being ! processed. The first factor processed must be factor 1! ! ! Input, integer ( kind = 4 ) FACTOR_ORDER, the order of the factor. ! ! Input, real ( kind = 8 ) FACTOR_VALUE(FACTOR_ORDER), the factor values ! for factor FACTOR_INDEX. ! ! Input, integer ( kind = 4 ) FACTOR_NUM, the number of factors. ! ! Input, integer ( kind = 4 ) POINT_NUM, the number of elements in the ! direct product. ! ! Input/output, real ( kind = 8 ) W(POINT_NUM), the elements of the ! direct product, which are built up gradually. ! ! Local Parameters: ! ! Local, integer ( kind = 4 ) START, the first location of a block of values ! to set. ! ! Local, integer ( kind = 4 ) CONTIG, the number of consecutive values ! to set. ! ! Local, integer ( kind = 4 ) SKIP, the distance from the current value ! of START to the next location of a block of values to set. ! ! Local, integer ( kind = 4 ) REP, the number of blocks of values to set. ! implicit none integer ( kind = 4 ) factor_num integer ( kind = 4 ) factor_order integer ( kind = 4 ) point_num integer ( kind = 4 ), save :: contig integer ( kind = 4 ) factor_index real ( kind = 8 ) factor_value(factor_order) integer ( kind = 4 ) j integer ( kind = 4 ) k integer ( kind = 4 ), save :: rep integer ( kind = 4 ), save :: skip integer ( kind = 4 ) start real ( kind = 8 ) w(point_num) if ( factor_index == 1 ) then contig = 1 skip = 1 rep = point_num w(1:point_num) = 1.0D+00 end if rep = rep / factor_order skip = skip * factor_order do j = 1, factor_order start = 1 + ( j - 1 ) * contig do k = 1, rep w(start:start+contig-1) = w(start:start+contig-1) * factor_value(j) start = start + skip end do end do contig = contig * factor_order return end function r8vec_distance ( dim_num, v1, v2 ) !*****************************************************************************80 ! !! R8VEC_DISTANCE returns the Euclidean distance between two R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 August 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) DIM_NUM, the spatial dimension. ! ! Input, real ( kind = 8 ) V1(DIM_NUM), V2(DIM_NUM), the vectors. ! ! Output, real ( kind = 8 ) R8VEC_DISTANCE, the Euclidean distance ! between the vectors. ! implicit none integer ( kind = 4 ) dim_num real ( kind = 8 ) r8vec_distance real ( kind = 8 ) v1(dim_num) real ( kind = 8 ) v2(dim_num) r8vec_distance = sqrt ( sum ( ( v1(1:dim_num) - v2(1:dim_num) ) ** 2 ) ) return end function r8vec_dot_product ( n, v1, v2 ) !*****************************************************************************80 ! !! R8VEC_DOT_PRODUCT finds the dot product of a pair of R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! In FORTRAN90, the system routine DOT_PRODUCT should be called ! directly. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 May 2008 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of the vectors. ! ! Input, real ( kind = 8 ) V1(N), V2(N), the vectors. ! ! Output, real ( kind = 8 ) R8VEC_DOT_PRODUCT, the dot product. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) r8vec_dot_product real ( kind = 8 ) v1(n) real ( kind = 8 ) v2(n) r8vec_dot_product = dot_product ( v1(1:n), v2(1:n) ) return end function r8vec_dot_product_affine ( n, v0, v1, v2 ) !*****************************************************************************80 ! !! R8VEC_DOT_PRODUCT_AFFINE computes the affine dot product V1-V0 * V2-V0. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the spatial dimension. ! ! Input, real ( kind = 8 ) V0(N), the base vector. ! ! Input, real ( kind = 8 ) V1(N), V2(N), the vectors. ! ! Output, real ( kind = 8 ) R8VEC_DOT_PRODUCT_AFFINE, the dot product. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) r8vec_dot_product_affine real ( kind = 8 ) v0(n) real ( kind = 8 ) v1(n) real ( kind = 8 ) v2(n) r8vec_dot_product_affine = dot_product ( & v1(1:n) - v0(1:n), & v2(1:n) - v0(1:n) ) return end function r8vec_entropy ( n, x ) !*****************************************************************************80 ! !! R8VEC_ENTROPY computes the entropy of an R8VEC. ! ! Discussion: ! ! Typically, the entries represent probabilities, and must sum to 1. ! For this function, the only requirement is that the entries be nonnegative. ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 August 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries. ! ! Input, real ( kind = 8 ) X(N), the vector. ! Each entry must be nonnegative. ! ! Output, real ( kind = 8 ) R8VEC_ENTROPY, the entropy of the ! normalized vector. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i real ( kind = 8 ) r8_log_2 real ( kind = 8 ) r8vec_entropy real ( kind = 8 ) value real ( kind = 8 ) x(n) real ( kind = 8 ) x_sum real ( kind = 8 ) xi if ( any ( x(1:n) < 0.0D+00 ) ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_ENTROPY - Fatal error!' write ( *, '(a)' ) ' Some entries are negative.' stop 1 end if x_sum = sum ( x(1:n) ) if ( x_sum == 0.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_ENTROPY - Fatal error!' write ( *, '(a)' ) ' Entries sum to 0.' stop 1 end if value = 0.0D+00 do i = 1, n if ( 0.0D+00 < x(i) ) then xi = x(i) / x_sum value = value - r8_log_2 ( xi ) * xi end if end do r8vec_entropy = value return end function r8vec_eq ( n, a1, a2 ) !*****************************************************************************80 ! !! R8VEC_EQ is true if two R8VECs are equal. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vectors. ! ! Input, real ( kind = 8 ) A1(N), A2(N), two vectors to compare. ! ! Output, logical ( kind = 4 ) R8VEC_EQ, is TRUE if every pair of elements ! A1(I) and A2(I) are equal, and FALSE otherwise. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) logical ( kind = 4 ) r8vec_eq r8vec_eq = ( all ( a1(1:n) == a2(1:n) ) ) return end subroutine r8vec_even ( n, alo, ahi, a ) !*****************************************************************************80 ! !! R8VEC_EVEN returns an R8VEC of evenly spaced values. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! If N is 1, then the midpoint is returned. ! ! Otherwise, the two endpoints are returned, and N-2 evenly ! spaced points between them. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of values. ! ! Input, real ( kind = 8 ) ALO, AHI, the low and high values. ! ! Output, real ( kind = 8 ) A(N), N evenly spaced values. ! Normally, A(1) = ALO and A(N) = AHI. ! However, if N = 1, then A(1) = 0.5*(ALO+AHI). ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) ahi real ( kind = 8 ) alo integer ( kind = 4 ) i if ( n == 1 ) then a(1) = 0.5D+00 * ( alo + ahi ) else do i = 1, n a(i) = ( real ( n - i, kind = 8 ) * alo & + real ( i - 1, kind = 8 ) * ahi ) & / real ( n - 1, kind = 8 ) end do end if return end subroutine r8vec_even_select ( n, xlo, xhi, ival, xval ) !*****************************************************************************80 ! !! R8VEC_EVEN_SELECT returns the I-th of N evenly spaced values in [ XLO, XHI ]. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! XVAL = ( (N-IVAL) * XLO + (IVAL-1) * XHI ) / real ( N - 1 ) ! ! Unless N = 1, X(1) = XLO and X(N) = XHI. ! ! If N = 1, then X(1) = 0.5*(XLO+XHI). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of values. ! ! Input, real ( kind = 8 ) XLO, XHI, the low and high values. ! ! Input, integer ( kind = 4 ) IVAL, the index of the desired point. ! IVAL is normally between 1 and N, but may be any integer value. ! ! Output, real ( kind = 8 ) XVAL, the IVAL-th of N evenly spaced values ! between XLO and XHI. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) ival real ( kind = 8 ) xhi real ( kind = 8 ) xlo real ( kind = 8 ) xval if ( n == 1 ) then xval = 0.5D+00 * ( xlo + xhi ) else xval = ( real ( n - ival, kind = 8 ) * xlo & + real ( ival - 1, kind = 8 ) * xhi ) & / real ( n - 1, kind = 8 ) end if return end subroutine r8vec_even2 ( maxval, nfill, nold, xold, nval, xval ) !*****************************************************************************80 ! !! R8VEC_EVEN2 linearly interpolates new numbers into an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The number of values created between two old values can vary from ! one pair of values to the next. ! ! The interpolated values are evenly spaced. ! ! This routine is a generalization of R8VEC_EVEN. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 October 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) MAXVAL, the size of the XVAL array, as declared ! by the user. MAXVAL must be large enough to hold the NVAL values computed ! by this routine. In other words, MAXVAL must be at least equal to ! NOLD + SUM (1 <= I <= NOLD-1) NFILL(I). ! ! Input, integer ( kind = 4 ) NFILL(NOLD-1), the number of values ! to be interpolated between XOLD(I) and XOLD(I+1). ! NFILL(I) does not count the endpoints. Thus, if ! NFILL(I) is 1, there will be one new point generated ! between XOLD(I) and XOLD(I+1). ! NFILL(I) must be nonnegative. ! ! Input, integer ( kind = 4 ) NOLD, the number of values XOLD, ! between which extra values are to be interpolated. ! ! Input, real ( kind = 8 ) XOLD(NOLD), the original vector of numbers ! between which new values are to be interpolated. ! ! Output, integer ( kind = 4 ) NVAL, the number of values computed ! in the XVAL array. ! NVAL = NOLD + SUM ( 1 <= I <= NOLD-1 ) NFILL(I) ! ! Output, real ( kind = 8 ) XVAL(MAXVAL). On output, XVAL contains the ! NOLD values of XOLD, as well as the interpolated ! values, making a total of NVAL values. ! implicit none integer ( kind = 4 ) maxval integer ( kind = 4 ) nold integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) nadd integer ( kind = 4 ) nfill(nold-1) integer ( kind = 4 ) nval real ( kind = 8 ) xold(nold) real ( kind = 8 ) xval(maxval) nval = 1 do i = 1, nold - 1 if ( nfill(i) < 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_EVEN2 - Fatal error!' write ( *, '(a,i8)' ) ' NFILL(I) is negative for I = ', i write ( *, '(a,i8)' ) ' NFILL(I) = ', nfill(i) stop 1 end if if ( maxval < nval + nfill(i) + 1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_EVEN2 - Fatal error!' write ( *, '(a)' ) ' MAXVAL is not large enough. ' write ( *, '(a,i8)' ) ' MAXVAL = ', maxval write ( *, '(a)' ) ' which is exceeded by storage requirements' write ( *, '(a,i8)' ) ' for interpolating in interval ', i stop 1 end if nadd = nfill(i) + 2 do j = 1, nadd xval(nval+j-1) = ( real ( nadd - j, kind = 8 ) * xold(i) & + real ( j - 1, kind = 8 ) * xold(i+1) ) & / real ( nadd - 1, kind = 8 ) end do nval = nval + nfill(i) + 1 end do return end subroutine r8vec_even2_select ( n, xlo, xhi, ival, xval ) !*****************************************************************************80 ! !! R8VEC_EVEN2_SELECT returns the I-th of N evenly spaced midpoint values. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! This function returns the I-th of N evenly spaced midpoints of N ! equal subintervals of [XLO,XHI]. ! ! XVAL = ( ( 2 * N - 2 * IVAL + 1 ) * XLO ! + ( 2 * IVAL - 1 ) * XHI ) ! / ( 2 * N ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 July 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of values. ! ! Input, real ( kind = 8 ) XLO, XHI, the low and high values. ! ! Input, integer ( kind = 4 ) IVAL, the index of the desired point. ! IVAL is normally between 1 and N, but may be any integer value. ! ! Output, real ( kind = 8 ) XVAL, the IVAL-th of N evenly spaced midpoints ! between XLO and XHI. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) ival real ( kind = 8 ) xhi real ( kind = 8 ) xlo real ( kind = 8 ) xval xval = ( real ( 2 * n - 2 * ival + 1, kind = 8 ) * xlo & + real ( 2 * ival - 1, kind = 8 ) * xhi ) & / real ( 2 * n, kind = 8 ) return end subroutine r8vec_even3 ( nold, nval, xold, xval ) !*****************************************************************************80 ! !! R8VEC_EVEN3 evenly interpolates new data into an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! This routine accepts a short vector of numbers, and returns a longer ! vector of numbers, created by interpolating new values between ! the given values. ! ! Between any two original values, new values are evenly interpolated. ! ! Over the whole vector, the new numbers are interpolated in ! such a way as to try to minimize the largest distance interval size. ! ! The algorithm employed is not "perfect". ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 October 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) NOLD, the number of values XOLD, between ! which extra values are to be interpolated. ! ! Input, integer ( kind = 4 ) NVAL, the number of values to be computed ! in the XVAL array. NVAL should be at least NOLD. ! ! Input, real ( kind = 8 ) XOLD(NOLD), the original vector of numbers ! between which new values are to be interpolated. ! ! Output, real ( kind = 8 ) XVAL(NVAL). On output, XVAL contains the ! NOLD values of XOLD, as well as interpolated ! values, making a total of NVAL values. ! implicit none integer ( kind = 4 ) nval integer ( kind = 4 ) nold real ( kind = 8 ) density integer ( kind = 4 ) i integer ( kind = 4 ) ival integer ( kind = 4 ) j integer ( kind = 4 ) nmaybe integer ( kind = 4 ) npts integer ( kind = 4 ) ntemp integer ( kind = 4 ) ntot real ( kind = 8 ) xlen real ( kind = 8 ) xleni real ( kind = 8 ) xlentot real ( kind = 8 ) xold(nold) real ( kind = 8 ) xval(nval) xlen = 0.0D+00 do i = 1, nold - 1 xlen = xlen + abs ( xold(i+1) - xold(i) ) end do ntemp = nval - nold density = real ( ntemp, kind = 8 ) / xlen ival = 1 ntot = 0 xlentot = 0.0D+00 do i = 1, nold - 1 xleni = abs ( xold(i+1) - xold(i) ) npts = int ( density * xleni ) ntot = ntot + npts ! ! Determine if we have enough left-over density that it should ! be changed into a point. A better algorithm would agonize ! more over where that point should go. ! xlentot = xlentot + xleni nmaybe = nint ( xlentot * density ) if ( ntot < nmaybe ) then npts = npts + nmaybe - ntot ntot = nmaybe end if do j = 1, npts + 2 xval(ival+j-1) = ( real ( npts+2 - j, kind = 8 ) * xold(i) & + real ( j - 1, kind = 8 ) * xold(i+1) ) & / real ( npts+2 - 1, kind = 8 ) end do ival = ival + npts + 1 end do return end subroutine r8vec_expand_linear ( n, x, fat, xfat ) !*****************************************************************************80 ! !! R8VEC_EXPAND_LINEAR linearly interpolates new data into an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! This routine copies the old data, and inserts NFAT new values ! between each pair of old data values. This would be one way to ! determine places to evenly sample a curve, given the (unevenly ! spaced) points at which it was interpolated. ! ! Example: ! ! N = 3 ! NFAT = 2 ! ! X(1:N) = (/ 0.0, 6.0, 7.0 /) ! XFAT(1:2*3+1) = (/ 0.0, 2.0, 4.0, 6.0, 6.33, 6.66, 7.0 /) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 October 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of input data values. ! ! Input, real ( kind = 8 ) X(N), the original data. ! ! Input, integer ( kind = 4 ) FAT, the number of data values to interpolate ! between each pair of original data values. ! ! Output, real ( kind = 8 ) XFAT((N-1)*(FAT+1)+1), the "fattened" data. ! implicit none integer ( kind = 4 ) fat integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) x(n) real ( kind = 8 ) xfat((n-1)*(fat+1)+1) k = 0 do i = 1, n - 1 k = k + 1 xfat(k) = x(i) do j = 1, fat k = k + 1 xfat(k) = ( real ( fat - j + 1, kind = 8 ) * x(i) & + real ( j, kind = 8 ) * x(i+1) ) & / real ( fat + 1, kind = 8 ) end do end do k = k + 1 xfat(k) = x(n) return end subroutine r8vec_expand_linear2 ( n, x, before, fat, after, xfat ) !*****************************************************************************80 ! !! R8VEC_EXPAND_LINEAR2 linearly interpolates new data into an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! This routine starts with a vector of data. ! ! The intent is to "fatten" the data, that is, to insert more points ! between successive values of the original data. ! ! There will also be extra points placed BEFORE the first original ! value and AFTER that last original value. ! ! The "fattened" data is equally spaced between the original points. ! ! The BEFORE data uses the spacing of the first original interval, ! and the AFTER data uses the spacing of the last original interval. ! ! Example: ! ! N = 3 ! BEFORE = 3 ! FAT = 2 ! AFTER = 1 ! ! X = (/ 0.0, 6.0, 7.0 /) ! XFAT = (/ -6.0, -4.0, -2.0, 0.0, 2.0, 4.0, 6.0, 6.33, 6.66, 7.0, 7.66 /) ! 3 "BEFORE's" Old 2 "FATS" Old 2 "FATS" Old 1 "AFTER" ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 03 December 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of input data values. ! N must be at least 2. ! ! Input, real ( kind = 8 ) X(N), the original data. ! ! Input, integer ( kind = 4 ) BEFORE, the number of "before" values. ! ! Input, integer ( kind = 4 ) FAT, the number of data values to interpolate ! between each pair of original data values. ! ! Input, integer ( kind = 4 ) AFTER, the number of "after" values. ! ! Output, real ( kind = 8 ) XFAT(BEFORE+(N-1)*(FAT+1)+1+AFTER), the ! "fattened" data. ! implicit none integer ( kind = 4 ) after integer ( kind = 4 ) before integer ( kind = 4 ) fat integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) k real ( kind = 8 ) x(n) real ( kind = 8 ) xfat(before+(n-1)*(fat+1)+1+after) k = 0 ! ! Points BEFORE. ! do j = 1 - before + fat, fat k = k + 1 xfat(k) = ( real ( fat - j + 1, kind = 8 ) * ( x(1) - ( x(2) - x(1) ) ) & + real ( j, kind = 8 ) * x(1) ) & / real ( fat + 1, kind = 8 ) end do ! ! Original points and FAT points. ! do i = 1, n - 1 k = k + 1 xfat(k) = x(i) do j = 1, fat k = k + 1 xfat(k) = ( real ( fat - j + 1, kind = 8 ) * x(i) & + real ( j, kind = 8 ) * x(i+1) ) & / real ( fat + 1, kind = 8 ) end do end do k = k + 1 xfat(k) = x(n) ! ! Points AFTER. ! do j = 1, after k = k + 1 xfat(k) = ( real ( fat - j + 1, kind = 8 ) * x(n) & + real ( j, kind = 8 ) & * ( x(n) + ( x(n) - x(n-1) ) ) ) & / real ( fat + 1, kind = 8 ) end do return end subroutine r8vec_fill ( n, value, x ) !*****************************************************************************80 ! !! R8VEC_FILL sets all entries of an R8VEC to a given value. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 03 December 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the vector. ! ! Input, real ( kind = 8 ) VALUE, the value. ! ! Output, real ( kind = 8 ) X(N), the array. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) value real ( kind = 8 ) x(n) x(1:n) = value return end subroutine r8vec_first_index ( n, a, tol, first_index ) !*****************************************************************************80 ! !! R8VEC_FIRST_INDEX indexes the first occurrence of values in an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! For element A(I) of the vector, FIRST_INDEX(I) is the index in A of ! the first occurrence of the value A(I). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 August 2008 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Input, real ( kind = 8 ) TOL, a tolerance for equality. ! ! Output, integer ( kind = 4 ) FIRST_INDEX(N), the first occurrence index. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) first_index(n) integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) tol first_index(1:n) = -1 do i = 1, n if ( first_index(i) == -1 ) then first_index(i) = i do j = i + 1, n if ( abs ( a(i) - a(j) ) <= tol ) then first_index(j) = i end if end do end if end do return end subroutine r8vec_floor ( n, r8vec, floorvec ) !*****************************************************************************80 ! !! R8VEC_FLOOR rounds "down" (towards -oo) entries of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Example: ! ! R8 Value ! ! -1.1 -2 ! -1.0 -1 ! -0.9 -1 ! 0.0 0 ! 5.0 5 ! 5.1 5 ! 5.9 5 ! 6.0 6 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 April 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries. ! ! Input, real ( kind = 8 ) R8VEC(N), the values to be rounded down. ! ! Output, integer ( kind = 4 ) FLOORVEC(N), the rounded value. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) floorvec(n) integer ( kind = 4 ) i real ( kind = 8 ) r8vec(n) integer ( kind = 4 ) value do i = 1, n value = int ( r8vec(i) ) if ( r8vec(i) < real ( value, kind = 8 ) ) then value = value - 1 end if floorvec(i) = value end do return end subroutine r8vec_frac ( n, a, k, frac ) !*****************************************************************************80 ! !! R8VEC_FRAC searches for the K-th smallest entry in an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Hoare's algorithm is used. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 July 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input/output, real ( kind = 8 ) A(N). ! On input, A is the array to search. ! On output, the elements of A have been somewhat rearranged. ! ! Input, integer ( kind = 4 ) K, the fractile to be sought. If K = 1, the ! minimum entry is sought. If K = N, the maximum is sought. Other values ! of K search for the entry which is K-th in size. K must be at ! least 1, and no greater than N. ! ! Output, real ( kind = 8 ) FRAC, the value of the K-th fractile of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) frac integer ( kind = 4 ) i integer ( kind = 4 ) iryt integer ( kind = 4 ) j integer ( kind = 4 ) k integer ( kind = 4 ) left real ( kind = 8 ) temp real ( kind = 8 ) x if ( n <= 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_FRAC - Fatal error!' write ( *, '(a,i8)' ) ' Illegal nonpositive value of N = ', n stop 1 end if if ( k <= 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_FRAC - Fatal error!' write ( *, '(a,i8)' ) ' Illegal nonpositive value of K = ', k stop 1 end if if ( n < k ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_FRAC - Fatal error!' write ( *, '(a,i8)' ) ' Illegal N < K, K = ', k stop 1 end if left = 1 iryt = n do if ( iryt <= left ) then frac = a(k) exit end if x = a(k) i = left j = iryt do if ( j < i ) then if ( j < k ) then left = i end if if ( k < i ) then iryt = j end if exit end if ! ! Find I so that X <= A(I). ! do while ( a(i) < x ) i = i + 1 end do ! ! Find J so that A(J) <= X. ! do while ( x < a(j) ) j = j - 1 end do if ( i <= j ) then temp = a(i) a(i) = a(j) a(j) = temp i = i + 1 j = j - 1 end if end do end do return end subroutine r8vec_fraction ( n, x, fraction ) !*****************************************************************************80 ! !! R8VEC_FRACTION returns the fraction parts of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! If we regard a real number as ! ! R8 = SIGN * ( WHOLE + FRACTION ) ! ! where ! ! SIGN is +1 or -1, ! WHOLE is a nonnegative integer ! FRACTION is a nonnegative real number strictly less than 1, ! ! then this routine returns the value of FRACTION. ! ! Example: ! ! R8 R8_FRACTION ! ! 0.00 0.00 ! 1.01 0.01 ! 2.02 0.02 ! 19.73 0.73 ! -4.34 0.34 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 April 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of arguments. ! ! Input, real ( kind = 8 ) X(N), the arguments. ! ! Output, real ( kind = 8 ) FRACTION(N), the fraction parts. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) fraction(n) real ( kind = 8 ) x(n) fraction(1:n) = abs ( x(1:n) ) - real ( int ( abs ( x(1:n) ) ), kind = 8 ) return end function r8vec_gt ( n, a1, a2 ) !*****************************************************************************80 ! !! R8VEC_GT == ( A1 > A2 ) for R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The comparison is lexicographic. ! ! A1 > A2 <=> A1(1) > A2(1) or ! ( A1(1) == A2(1) and A1(2) > A2(2) ) or ! ... ! ( A1(1:N-1) == A2(1:N-1) and A1(N) > A2(N) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of the vectors. ! ! Input, real ( kind = 8 ) A1(N), A2(N), the vectors to be compared. ! ! Output, logical ( kind = 4 ) R8VEC_GT, is TRUE if and only if A1 > A2. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) integer ( kind = 4 ) i logical ( kind = 4 ) r8vec_gt r8vec_gt = .false. do i = 1, n if ( a2(i) < a1(i) ) then r8vec_gt = .true. exit else if ( a1(i) < a2(i) ) then r8vec_gt = .false. exit end if end do return end subroutine r8vec_heap_a ( n, a ) !*****************************************************************************80 ! !! R8VEC_HEAP_A reorders an R8VEC into an ascending heap. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! An ascending heap is an array A with the property that, for every index J, ! A(J) <= A(2*J) and A(J) <= A(2*J+1), (as long as the indices ! 2*J and 2*J+1 are legal). ! ! A(1) ! / \ ! A(2) A(3) ! / \ / \ ! A(4) A(5) A(6) A(7) ! / \ / \ ! A(8) A(9) A(10) A(11) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 July 2003 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the input array. ! ! Input/output, real ( kind = 8 ) A(N). ! On input, an unsorted array. ! On output, the array has been reordered into a heap. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) ifree real ( kind = 8 ) key integer ( kind = 4 ) m ! ! Only nodes N/2 down to 1 can be "parent" nodes. ! do i = n / 2, 1, -1 ! ! Copy the value out of the parent node. ! Position IFREE is now "open". ! key = a(i) ifree = i do ! ! Positions 2*IFREE and 2*IFREE + 1 are the descendants of position ! IFREE. (One or both may not exist because they exceed N.) ! m = 2 * ifree ! ! Does the first position exist? ! if ( n < m ) then exit end if ! ! Does the second position exist? ! if ( m + 1 <= n ) then ! ! If both positions exist, take the smaller of the two values, ! and update M if necessary. ! if ( a(m+1) < a(m) ) then m = m + 1 end if end if ! ! If the small descendant is smaller than KEY, move it up, ! and update IFREE, the location of the free position, and ! consider the descendants of THIS position. ! if ( key <= a(m) ) then exit end if a(ifree) = a(m) ifree = m end do ! ! Once there is no more shifting to do, KEY moves into the free spot. ! a(ifree) = key end do return end subroutine r8vec_heap_d ( n, a ) !*****************************************************************************80 ! !! R8VEC_HEAP_D reorders an R8VEC into an descending heap. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! A descending heap is an array A with the property that, for every index J, ! A(J) >= A(2*J) and A(J) >= A(2*J+1), (as long as the indices ! 2*J and 2*J+1 are legal). ! ! A(1) ! / \ ! A(2) A(3) ! / \ / \ ! A(4) A(5) A(6) A(7) ! / \ / \ ! A(8) A(9) A(10) A(11) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 July 2003 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the input array. ! ! Input/output, real ( kind = 8 ) A(N). ! On input, an unsorted array. ! On output, the array has been reordered into a heap. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) ifree real ( kind = 8 ) key integer ( kind = 4 ) m ! ! Only nodes N/2 down to 1 can be "parent" nodes. ! do i = n / 2, 1, -1 ! ! Copy the value out of the parent node. ! Position IFREE is now "open". ! key = a(i) ifree = i do ! ! Positions 2*IFREE and 2*IFREE + 1 are the descendants of position ! IFREE. (One or both may not exist because they exceed N.) ! m = 2 * ifree ! ! Does the first position exist? ! if ( n < m ) then exit end if ! ! Does the second position exist? ! if ( m + 1 <= n ) then ! ! If both positions exist, take the larger of the two values, ! and update M if necessary. ! if ( a(m) < a(m+1) ) then m = m + 1 end if end if ! ! If the large descendant is larger than KEY, move it up, ! and update IFREE, the location of the free position, and ! consider the descendants of THIS position. ! if ( a(m) <= key ) then exit end if a(ifree) = a(m) ifree = m end do ! ! Once there is no more shifting to do, KEY moves into the free spot IFREE. ! a(ifree) = key end do return end subroutine r8vec_heap_d_extract ( n, a, value ) !*****************************************************************************80 ! !! R8VEC_HEAP_D_EXTRACT: extract maximum from a heap descending sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! In other words, the routine finds the maximum value in the ! heap, returns that value to the user, deletes that value from ! the heap, and restores the heap to its proper form. ! ! This is one of three functions needed to model a priority queue. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 August 2010 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Thomas Cormen, Charles Leiserson, Ronald Rivest, ! Introduction to Algorithms, ! MIT Press, 2001, ! ISBN: 0262032937, ! LC: QA76.C662. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N, the number of items in the heap. ! ! Input/output, real ( kind = 8 ) A(N), the heap. ! ! Output, real ( kind = 8 ) VALUE, the item of maximum value, which has ! been removed from the heap. ! implicit none real ( kind = 8 ) a(*) integer ( kind = 4 ) n real ( kind = 8 ) value if ( n < 1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_HEAP_D_EXTRACT - Fatal error!' write ( *, '(a)' ) ' The heap is empty.' stop 1 end if ! ! Get the maximum value. ! value = a(1) if ( n == 1 ) then n = 0 return end if ! ! Shift the last value down. ! a(1) = a(n) ! ! Restore the heap structure. ! n = n - 1 call r8vec_sort_heap_d ( n, a ) return end subroutine r8vec_heap_d_insert ( n, a, value ) !*****************************************************************************80 ! !! R8VEC_HEAP_D_INSERT inserts a value into a heap descending sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! This is one of three functions needed to model a priority queue. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 August 2010 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Thomas Cormen, Charles Leiserson, Ronald Rivest, ! Introduction to Algorithms, ! MIT Press, 2001, ! ISBN: 0262032937, ! LC: QA76.C662. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N, the number of items in the heap. ! ! Input/output, real ( kind = 8 ) A(N), the heap. ! ! Input, real ( kind = 8 ) VALUE, the value to be inserted. ! implicit none real ( kind = 8 ) a(*) integer ( kind = 4 ) i integer ( kind = 4 ) n integer ( kind = 4 ) parent real ( kind = 8 ) value n = n + 1 i = n do while ( 1 < i ) parent = i / 2 if ( value <= a(parent) ) then exit end if a(i) = a(parent) i = parent end do a(i) = value return end subroutine r8vec_heap_d_max ( n, a, value ) !*****************************************************************************80 ! !! R8VEC_HEAP_D_MAX returns the maximum value in a heap descending sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! This is one of three functions needed to model a priority queue. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 August 2010 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Thomas Cormen, Charles Leiserson, Ronald Rivest, ! Introduction to Algorithms, ! MIT Press, 2001, ! ISBN: 0262032937, ! LC: QA76.C662. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items in the heap. ! ! Input, real ( kind = 8 ) A(N), the heap. ! ! Output, real ( kind = 8 ) VALUE, the maximum value in the heap. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) value value = a(1) return end subroutine r8vec_histogram ( n, a, a_lo, a_hi, histo_num, histo_gram ) !*****************************************************************************80 ! !! R8VEC_HISTOGRAM histograms an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Values between A_LO and A_HI will be histogrammed into the bins ! 1 through HISTO_NUM. Values below A_LO are counted in bin 0, ! and values greater than A_HI are counted in bin HISTO_NUM+1. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 September 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input, real ( kind = 8 ) A(N), the array to examine. ! ! Input, real ( kind = 8 ) A_LO, A_HI, the lowest and highest ! values to be histogrammed. These values will also define the bins. ! ! Input, integer ( kind = 4 ) HISTO_NUM, the number of bins to use. ! ! Output, integer ( kind = 4 ) HISTO_GRAM(0:HISTO_NUM+1), contains the ! number of entries of A in each bin. ! implicit none integer ( kind = 4 ) histo_num integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) a_hi real ( kind = 8 ) a_lo real ( kind = 8 ) delta integer ( kind = 4 ) histo_gram(0:histo_num+1) integer ( kind = 4 ) i integer ( kind = 4 ) j histo_gram(0:histo_num+1) = 0 delta = ( a_hi - a_lo ) / real ( 2 * histo_num, kind = 8 ) do i = 1, n if ( a(i) < a_lo ) then histo_gram(0) = histo_gram(0) + 1 else if ( a(i) <= a_hi ) then j = nint ( & ( ( a_hi - delta - a(i) ) & * real ( 1, kind = 8 ) & + ( - delta + a(i) - a_lo ) & * real ( histo_num, kind = 8 ) ) & / ( a_hi - 2.0D+00 * delta - a_lo ) ) histo_gram(j) = histo_gram(j) + 1 else if ( a_hi < a(i) ) then histo_gram(histo_num+1) = histo_gram(histo_num+1) + 1 end if end do return end subroutine r8vec_house_column ( n, a_vec, k, v ) !*****************************************************************************80 ! !! R8VEC_HOUSE_COLUMN defines a Householder premultiplier that "packs" a column. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The routine returns a vector V that defines a Householder ! premultiplier matrix H(V) that zeros out the subdiagonal entries of ! column K of the matrix A. ! ! H(V) = I - 2 * v * v' ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 February 2015 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix A. ! ! Input, real ( kind = 8 ) A_VEC(N), a row or column of the matrix A. ! ! Input, integer ( kind = 4 ) K, the "special" entry in A_VEC. ! The Householder matrix will zero out the entries after this. ! ! Output, real ( kind = 8 ) V(N), a vector of unit L2 norm which defines an ! orthogonal Householder premultiplier matrix H with the property ! that the K-th column of H*A is zero below the diagonal. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a_vec(n) integer ( kind = 4 ) k real ( kind = 8 ) s real ( kind = 8 ) v(n) v(1:n) = 0.0D+00 if ( k < 1 .or. n <= k ) then return end if s = sqrt ( dot_product ( a_vec(k:n), a_vec(k:n) ) ) if ( s == 0.0D+00 ) then return end if v(k) = a_vec(k) + sign ( s, a_vec(k) ) v(k+1:n) = a_vec(k+1:n) ! ! Normalize V. ! s = sqrt ( dot_product ( v(k:n), v(k:n) ) ) v(k:n) = v(k:n) / s return end function r8vec_i4vec_dot_product ( n, r8vec, i4vec ) !*****************************************************************************80 ! !! R8VEC_I4VEC_DOT_PRODUCT finds the dot product of an R8VEC and an I4VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! An I4VEC is a vector of I4's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 June 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of the vectors. ! ! Input, real ( kind = 8 ) R8VEC(N), the first vector. ! ! Input, integer ( kind = 4 ) I4VEC(N), the second vector. ! ! Output, real ( kind = 8 ) R8VEC_I4VEC_DOT_PRODUCT, the dot product. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i4vec(n) real ( kind = 8 ) r8vec(n) real ( kind = 8 ) r8vec_i4vec_dot_product r8vec_i4vec_dot_product = dot_product ( r8vec(1:n), & real ( i4vec(1:n), kind = 8 ) ) return end subroutine r8vec_identity_row ( n, i, a ) !*****************************************************************************80 ! !! R8VEC_IDENTITY_ROW returns the I-th row of the identity matrix. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 28 March 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input, integer ( kind = 4 ) I, the entry to be set to 1. ! ! Output, real ( kind = 8 ) A(N), the array. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i a(1:n) = 0.0D+00 if ( 1 <= i .and. i <= n ) then a(i) = 1.0D+00 end if return end subroutine r8vec_index_delete_all ( n, x, indx, xval ) !*****************************************************************************80 ! !! R8VEC_INDEX_DELETE_ALL deletes a value from an indexed sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Note that the value of N is adjusted because of the deletions! ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 October 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N, the size of the current list. ! ! Input/output, real ( kind = 8 ) X(N), the list. ! ! Input/output, integer ( kind = 4 ) INDX(N), the sort index of the list. ! ! Input, real ( kind = 8 ) XVAL, the value to be sought. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) equal integer ( kind = 4 ) equal1 integer ( kind = 4 ) equal2 integer ( kind = 4 ) get integer ( kind = 4 ) i integer ( kind = 4 ) indx(*) integer ( kind = 4 ) less integer ( kind = 4 ) more integer ( kind = 4 ) put real ( kind = 8 ) x(*) real ( kind = 8 ) xval if ( n < 1 ) then n = 0 return end if call r8vec_index_search ( n, x, indx, xval, less, equal, more ) if ( equal == 0 ) then return end if equal1 = equal do if ( equal1 <= 1 ) then exit end if if ( x(indx(equal1-1)) /= xval ) then exit end if equal1 = equal1 - 1 end do equal2 = equal do if ( n <= equal2 ) then exit end if if ( x(indx(equal2+1)) /= xval ) then exit end if equal2 = equal2 + 1 end do ! ! Discard certain X values. ! put = 0 do get = 1, n if ( x(get) /= xval ) then put = put + 1 x(put) = x(get) end if end do x(put+1:n) = 0.0D+00 ! ! Adjust the INDX values. ! do equal = equal1, equal2 do i = 1, n if ( indx(equal) < indx(i) ) then indx(i) = indx(i) - 1 end if end do end do ! ! Discard certain INDX values. ! indx(equal1:n+equal1-equal2-1) = indx(equal2+1:n) indx(n+equal1-equal2:n) = 0 ! ! Adjust N. ! n = put return end subroutine r8vec_index_delete_dupes ( n, x, indx, n2, x2, indx2 ) !*****************************************************************************80 ! !! R8VEC_INDEX_DELETE_DUPES deletes duplicates from an indexed sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The output quantities N2, X2, and INDX2 are computed from the ! input quantities by sorting, and eliminating duplicates. ! ! The output arrays should be dimensioned of size N, unless the user ! knows in advance what the value of N2 will be. ! ! The output arrays may be identified with the input arrays. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 October 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the input list. ! ! Input, real ( kind = 8 ) X(N), the list. ! ! Input, integer ( kind = 4 ) INDX(N), the sort index of the list. ! ! Output, integer ( kind = 4 ) N2, the number of unique entries in X. ! ! Output, real ( kind = 8 ) X2(N2), a copy of the list which has ! been sorted, and made unique. ! ! Output, integer ( kind = 4 ) INDX2(N2), the sort index of the new list. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) indx(n) integer ( kind = 4 ) indx2(n) integer ( kind = 4 ) n2 integer ( kind = 4 ) n3 real ( kind = 8 ) x(n) real ( kind = 8 ) x2(n) real ( kind = 8 ) x3(n) i = 0 n3 = 0 do i = i + 1 if ( n < i ) then exit end if if ( 1 < i ) then if ( x(indx(i)) == x3(n3) ) then cycle end if end if n3 = n3 + 1 x3(n3) = x(indx(i)) end do ! ! Copy data into output arrays. ! n2 = n3 x2(1:n2) = x3(1:n3) call i4vec_indicator1 ( n2, indx2 ) return end subroutine r8vec_index_delete_one ( n, x, indx, xval, n2, x2, indx2 ) !*****************************************************************************80 ! !! R8VEC_INDEX_DELETE_ONE deletes one copy of a value from indexed sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! If the value occurs in the list more than once, only one copy is deleted. ! ! Note that the value of N is adjusted because of the deletions. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 October 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the current list. ! ! Input, real ( kind = 8 ) X(N), the list. ! ! Input, integer ( kind = 4 ) INDX(N), the sort index of the list. ! ! Input, real ( kind = 8 ) XVAL, the value to be sought. ! ! Output, integer ( kind = 4 ) N2, the size of the current list. ! ! Output, real ( kind = 8 ) X2(N2), the list. ! ! Output, integer ( kind = 4 ) INDX2(N2), the sort index of the list. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) equal integer ( kind = 4 ) i integer ( kind = 4 ) indx(n) integer ( kind = 4 ) indx2(n) integer ( kind = 4 ) j integer ( kind = 4 ) less integer ( kind = 4 ) more integer ( kind = 4 ) n2 real ( kind = 8 ) x(n) real ( kind = 8 ) x2(n) real ( kind = 8 ) xval if ( n < 1 ) then n2 = 0 return end if n2 = n indx2(1:n2) = indx(1:n2) x2(1:n2) = x(1:n2) call r8vec_index_search ( n2, x2, indx2, xval, less, equal, more ) if ( equal /= 0 ) then j = indx2(equal) x2(j:n2-1) = x2(j+1:n2) indx2(equal:n2-1) = indx2(equal+1:n2) do i = 1, n2-1 if ( j < indx2(i) ) then indx2(i) = indx2(i) - 1 end if end do n2 = n2 - 1 end if return end subroutine r8vec_index_insert ( n, x, indx, xval ) !*****************************************************************************80 ! !! R8VEC_INDEX_INSERT inserts a value in an indexed sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 October 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N, the size of the current list. ! ! Input/output, real ( kind = 8 ) X(N), the list. ! ! Input/output, integer ( kind = 4 ) INDX(N), the sort index of the list. ! ! Input, real ( kind = 8 ) XVAL, the value to be sought. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) equal integer ( kind = 4 ) indx(*) integer ( kind = 4 ) less integer ( kind = 4 ) more real ( kind = 8 ) x(*) real ( kind = 8 ) xval if ( n <= 0 ) then n = 1 x(1) = xval indx(1) = 1 return end if call r8vec_index_search ( n, x, indx, xval, less, equal, more ) x(n+1) = xval indx(n+1:more+1:-1) = indx(n:more:-1) indx(more) = n + 1 n = n + 1 return end subroutine r8vec_index_insert_unique ( n, x, indx, xval ) !*****************************************************************************80 ! !! R8VEC_INDEX_INSERT_UNIQUE inserts a unique value in an indexed sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! If the value does not occur in the list, it is included in the list, ! and N, X and INDX are updated. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 October 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N, the size of the current list. ! ! Input/output, real ( kind = 8 ) X(N), the list. ! ! Input/output, integer ( kind = 4 ) INDX(N), the sort index of the list. ! ! Input, real ( kind = 8 ) XVAL, the value to be sought. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) equal integer ( kind = 4 ) indx(*) integer ( kind = 4 ) less integer ( kind = 4 ) more real ( kind = 8 ) x(*) real ( kind = 8 ) xval if ( n <= 0 ) then n = 1 x(1) = xval indx(1) = 1 return end if ! ! Does XVAL already occur in X? ! call r8vec_index_search ( n, x, indx, xval, less, equal, more ) if ( equal == 0 ) then x(n+1) = xval indx(n+1:more+1:-1) = indx(n:more:-1) indx(more) = n + 1 n = n + 1 end if return end subroutine r8vec_index_order ( n, x, indx ) !*****************************************************************************80 ! !! R8VEC_INDEX_ORDER sorts an R8VEC using an index vector. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The index vector itself is not modified. Therefore, the pair ! (X,INDX) no longer represents an index sorted vector. If this ! relationship is to be preserved, then simply set INDX(1:N)=(1:N). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 October 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the current list. ! ! Input/output, real ( kind = 8 ) X(N), the list. On output, the list ! has been sorted. ! ! Input, integer ( kind = 4 ) INDX(N), the sort index of the list. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) indx(n) real ( kind = 8 ) x(n) real ( kind = 8 ) y(n) y(1:n) = x(indx(1:n)) x(1:n) = y(1:n) return end subroutine r8vec_index_search ( n, x, indx, xval, less, equal, more ) !*****************************************************************************80 ! !! R8VEC_INDEX_SEARCH searches for a value in an indexed sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 October 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the current list. ! ! Input, real ( kind = 8 ) X(N), the list. ! ! Input, integer ( kind = 4 ) INDX(N), the sort index of the list. ! ! Input, real ( kind = 8 ) XVAL, the value to be sought. ! ! Output, integer ( kind = 4 ) LESS, EQUAL, MORE, the indexes in INDX of the ! entries of X that are just less than, equal to, and just greater ! than XVAL. If XVAL does not occur in X, then EQUAL is zero. ! If XVAL is the minimum entry of X, then LESS is 0. If XVAL ! is the greatest entry of X, then MORE is N+1. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) equal integer ( kind = 4 ) hi integer ( kind = 4 ) indx(n) integer ( kind = 4 ) less integer ( kind = 4 ) lo integer ( kind = 4 ) mid integer ( kind = 4 ) more real ( kind = 8 ) x(n) real ( kind = 8 ) xhi real ( kind = 8 ) xlo real ( kind = 8 ) xmid real ( kind = 8 ) xval if ( n <= 0 ) then less = 0 equal = 0 more = 0 return end if lo = 1 hi = n xlo = x(indx(lo)) xhi = x(indx(hi)) if ( xval < xlo ) then less = 0 equal = 0 more = 1 return else if ( xval == xlo ) then less = 0 equal = 1 more = 2 return end if if ( xhi < xval ) then less = n equal = 0 more = n + 1 return else if ( xval == xhi ) then less = n - 1 equal = n more = n + 1 return end if do if ( lo + 1 == hi ) then less = lo equal = 0 more = hi return end if mid = ( lo + hi ) / 2 xmid = x(indx(mid)) if ( xval == xmid ) then equal = mid less = equal - 1 more = equal + 1 return else if ( xval < xmid ) then hi = mid else if ( xmid < xval ) then lo = mid end if end do return end subroutine r8vec_index_sort_unique ( n, x, indx, n2 ) !*****************************************************************************80 ! !! R8VEC_INDEX_SORT_UNIQUE creates a sorted unique index for an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 October 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the current list. ! ! Input/output, real ( kind = 8 ) X(N), the list. On output, X contains only ! unique elements. ! ! Output, integer ( kind = 4 ) INDX(N), the sort index of the list. ! ! Output, integer ( kind = 4 ) N2, the number of unique elements in X. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) indx(n) integer ( kind = 4 ) n2 real ( kind = 8 ) x(n) real ( kind = 8 ) y(n) n2 = 0 do i = 1, n call r8vec_index_insert_unique ( n2, y, indx, x(i) ) end do x(1:n2) = y(1:n2) x(n2+1:n) = 0.0D+00 indx(n2+1:n) = 0 return end subroutine r8vec_index_sorted_range ( n, r, indx, r_lo, r_hi, i_lo, i_hi ) !*****************************************************************************80 ! !! R8VEC_INDEX_SORTED_RANGE: search index sorted vector for elements in a range. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 September 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items in the vector. ! ! Input, real ( kind = 8 ) R(N), the index sorted vector. ! ! Input, integer ( kind = 4 ) INDX(N), the vector used to sort R. ! The vector R(INDX(*)) is sorted. ! ! Input, real ( kind = 8 ) R_LO, R_HI, the limits of the range. ! ! Output, integer ( kind = 4 ) I_LO, I_HI, the range of indices ! so that I_LO <= I <= I_HI => R_LO <= R(INDX(I)) <= R_HI. If no ! values in R lie in the range, then I_HI < I_LO will be returned. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i_hi integer ( kind = 4 ) i_lo integer ( kind = 4 ) i1 integer ( kind = 4 ) i2 integer ( kind = 4 ) indx(n) integer ( kind = 4 ) j1 integer ( kind = 4 ) j2 real ( kind = 8 ) r(n) real ( kind = 8 ) r_hi real ( kind = 8 ) r_lo ! ! Cases we can handle immediately. ! if ( r(indx(n)) < r_lo ) then i_lo = n + 1 i_hi = n return end if if ( r_hi < r(indx(1)) ) then i_lo = 1 i_hi = 0 return end if ! ! Are there are least two intervals? ! if ( n == 1 ) then if ( r_lo <= r(indx(1)) .and. r(indx(1)) <= r_hi ) then i_lo = 1 i_hi = 1 else i_lo = 0 i_hi = -1 end if return end if ! ! Bracket R_LO. ! if ( r_lo <= r(indx(1)) ) then i_lo = 1 else ! ! R_LO is in one of the intervals spanned by R(INDX(J1)) to R(INDX(J2)). ! Examine the intermediate interval [R(INDX(I1)), R(INDX(I1+1))]. ! Does R_LO lie here, or below or above? ! j1 = 1 j2 = n i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 do if ( r_lo < r(indx(i1)) ) then j2 = i1 i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 else if ( r(indx(i2)) < r_lo ) then j1 = i2 i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 else i_lo = i1 exit end if end do end if ! ! Bracket R_HI. ! if ( r(indx(n)) <= r_hi ) then i_hi = n else j1 = i_lo j2 = n i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 do if ( r_hi < r(indx(i1)) ) then j2 = i1 i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 else if ( r(indx(i2)) < r_hi ) then j1 = i2 i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 else i_hi = i2 exit end if end do end if ! ! We expect to have computed the largest I_LO and smallest I_HI such that ! R(INDX(I_LO)) <= R_LO <= R_HI <= R(INDX(I_HI)) ! but what we want is actually ! R_LO <= R(INDX(I_LO)) <= R(INDX(I_HI)) <= R_HI ! which we can usually get simply by incrementing I_LO and decrementing I_HI. ! if ( r(indx(i_lo)) < r_lo ) then i_lo = i_lo + 1 if ( n < i_lo ) then i_hi = i_lo - 1 end if end if if ( r_hi < r(indx(i_hi)) ) then i_hi = i_hi - 1 if ( i_hi < 1 ) then i_lo = i_hi + 1 end if end if return end subroutine r8vec_indexed_heap_d ( n, a, indx ) !*****************************************************************************80 ! !! R8VEC_INDEXED_HEAP_D creates a descending heap from an indexed R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! An indexed R8VEC is an R8VEC of data values, and an R8VEC of N indices, ! each referencing an entry of the data vector. ! ! The function adjusts the index vector INDX so that, for 1 <= J <= N/2, ! we have: ! A(INDX(2*J)) <= A(INDX(J)) ! and ! A(INDX(2*J+1)) <= A(INDX(J)) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 August 2010 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the index array. ! ! Input, real ( kind = 8 ) A(*), the data vector. ! ! Input/output, integer ( kind = 4 ) INDX(N), the index array. ! Each entry of INDX must be a valid index for the array A. ! On output, the indices have been reordered into a descending heap. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(*) integer ( kind = 4 ) i integer ( kind = 4 ) ifree integer ( kind = 4 ) indx(n) integer ( kind = 4 ) key integer ( kind = 4 ) m ! ! Only nodes N/2 down to 1 can be "parent" nodes. ! do i = n / 2, 1, -1 ! ! Copy the value out of the parent node. ! Position IFREE is now "open". ! key = indx(i) ifree = i do ! ! Positions 2*IFREE and 2*IFREE + 1 are the descendants of position ! IFREE. (One or both may not exist because they exceed N.) ! m = 2 * ifree ! ! Does the first position exist? ! if ( n < m ) then exit end if ! ! Does the second position exist? ! if ( m + 1 <= n ) then ! ! If both positions exist, take the larger of the two values, ! and update M if necessary. ! if ( a(indx(m)) < a(indx(m+1)) ) then m = m + 1 end if end if ! ! If the large descendant is larger than KEY, move it up, ! and update IFREE, the location of the free position, and ! consider the descendants of THIS position. ! if ( a(indx(m)) <= a(key) ) then exit end if indx(ifree) = indx(m) ifree = m end do ! ! Once there is no more shifting to do, KEY moves into the free spot IFREE. ! indx(ifree) = key end do return end subroutine r8vec_indexed_heap_d_extract ( n, a, indx, indx_extract ) !*****************************************************************************80 ! !! R8VEC_INDEXED_HEAP_D_EXTRACT: extract from heap descending indexed R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! An indexed R8VEC is an R8VEC of data values, and an R8VEC of N indices, ! each referencing an entry of the data vector. ! ! The routine finds the maximum value in the heap, returns that value to the ! user, deletes that value from the heap, and restores the heap to its ! proper form. ! ! Note that the argument N must be a variable, which will be decremented ! before return, and that INDX will hold one less value on output than it ! held on input. ! ! This is one of three functions needed to model a priority queue. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 August 2010 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Thomas Cormen, Charles Leiserson, Ronald Rivest, ! Introduction to Algorithms, ! MIT Press, 2001, ! ISBN: 0262032937, ! LC: QA76.C662. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N, the number of items in the ! index vector. ! ! Input, real ( kind = 8 ) A(*), the data vector. ! ! Input/output, integer ( kind = 4 ) INDX(N), the index vector. ! ! Output, integer ( kind = 4 ) INDX_EXTRACT, the index in A of the item of ! maximum value, which has now been removed from the heap. ! implicit none real ( kind = 8 ) a(*) integer ( kind = 4 ) indx(*) integer ( kind = 4 ) indx_extract integer ( kind = 4 ) n if ( n < 1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_INDEXED_HEAP_D_EXTRACT - Fatal error!' write ( *, '(a)' ) ' The heap is empty.' stop 1 end if ! ! Get the index of the maximum value. ! indx_extract = indx(1) if ( n == 1 ) then n = 0 return end if ! ! Shift the last index down. ! indx(1) = indx(n) ! ! Restore the heap structure. ! n = n - 1 call r8vec_indexed_heap_d ( n, a, indx ) return end subroutine r8vec_indexed_heap_d_insert ( n, a, indx, indx_insert ) !*****************************************************************************80 ! !! R8VEC_INDEXED_HEAP_D_INSERT: insert value into heap descending indexed R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! An indexed R8VEC is an R8VEC of data values, and an R8VEC of N indices, ! each referencing an entry of the data vector. ! ! Note that the argument N must be a variable, and will be incremented before ! return, and that INDX must be able to hold one more entry on output than ! it held on input. ! ! This is one of three functions needed to model a priority queue. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 August 2010 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Thomas Cormen, Charles Leiserson, Ronald Rivest, ! Introduction to Algorithms, ! MIT Press, 2001, ! ISBN: 0262032937, ! LC: QA76.C662. ! ! Parameters: ! ! Input/output, integer ( kind = 4 ) N, the number of items in the ! index vector. ! ! Input, real ( kind = 8 ) A(*), the data vector. ! ! Input/output, integer ( kind = 4 ) INDX(N), the index vector. ! ! Input, integer ( kind = 4 ) INDX_INSERT, the index in A of the value ! to be inserted into the heap. ! implicit none real ( kind = 8 ) a(*) integer ( kind = 4 ) i integer ( kind = 4 ) indx(*) integer ( kind = 4 ) indx_insert integer ( kind = 4 ) n integer ( kind = 4 ) parent n = n + 1 i = n do while ( 1 < i ) parent = i / 2 if ( a(indx_insert) <= a(indx(parent)) ) then exit end if indx(i) = indx(parent) i = parent end do indx(i) = indx_insert return end subroutine r8vec_indexed_heap_d_max ( n, a, indx, indx_max ) !*****************************************************************************80 ! !! R8VEC_INDEXED_HEAP_D_MAX: maximum value in heap descending indexed R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! An indexed R8VEC is an R8VEC of data values, and an R8VEC of N indices, ! each referencing an entry of the data vector. ! ! This is one of three functions needed to model a priority queue. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 August 2010 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Thomas Cormen, Charles Leiserson, Ronald Rivest, ! Introduction to Algorithms, ! MIT Press, 2001, ! ISBN: 0262032937, ! LC: QA76.C662. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items in the index vector. ! ! Input, real ( kind = 8 ) A(*), the data vector. ! ! Input, integer ( kind = 4 ) INDX(N), the index vector. ! ! Output, integer ( kind = 4 ) INDX_MAX, the index in A of the maximum value ! in the heap. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(*) integer ( kind = 4 ) indx(n) integer ( kind = 4 ) indx_max indx_max = indx(1) return end subroutine r8vec_indicator0 ( n, a ) !*****************************************************************************80 ! !! R8VEC_INDICATOR0 sets an R8VEC to the indicator vector (0,1,2,...). ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 September 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Output, real ( kind = 8 ) A(N), the array. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i do i = 1, n a(i) = real ( i - 1, kind = 8 ) end do return end subroutine r8vec_indicator1 ( n, a ) !*****************************************************************************80 ! !! R8VEC_INDICATOR1 sets an R8VEC to the indicator vector (1,2,3,...). ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 September 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Output, real ( kind = 8 ) A(N), the array. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i do i = 1, n a(i) = real ( i, kind = 8 ) end do return end subroutine r8vec_insert ( n, a, pos, value ) !*****************************************************************************80 ! !! R8VEC_INSERT inserts a value into an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 February 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of the array on input. ! ! Input/output, real ( kind = 8 ) A(N+1), the array. On input, A is ! assumed to contain only N entries, while on output, A actually ! contains N+1 entries. ! ! Input, integer ( kind = 4 ) POS, the position to be assigned the new entry. ! 1 <= POS <= N+1. ! ! Input, real ( kind = 8 ) VALUE, the value to be inserted. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n+1) integer ( kind = 4 ) i integer ( kind = 4 ) pos real ( kind = 8 ) value if ( pos < 1 .or. n + 1 < pos ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_INSERT - Fatal error!' write ( *, '(a,i8)' ) ' Illegal insertion position = ', pos stop 1 else do i = n + 1, pos + 1, -1 a(i) = a(i-1) end do a(pos) = value end if return end function r8vec_is_ascending ( n, x ) !*****************************************************************************80 ! !! R8VEC_IS_ASCENDING determines if an R8VEC is (weakly) ascending. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! For example, if: ! ! X = ( -8.1, 1.3, 2.2, 3.4, 7.5, 7.5, 9.8 ) ! ! then ! ! R8VEC_IS_ASCENDING = TRUE ! ! The sequence is not required to be strictly ascending. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 April 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the array. ! ! Input, real ( kind = 8 ) X(N), the array to be examined. ! ! Output, logical ( kind = 4 ) R8VEC_IS_ASCENDING, is TRUE if the ! entries ascend. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i logical ( kind = 4 ) r8vec_is_ascending real ( kind = 8 ) x(n) logical ( kind = 4 ) value value = .true. do i = 1, n - 1 if ( x(i+1) < x(i) ) then value = .false. exit end if end do r8vec_is_ascending = value return end function r8vec_is_ascending_strictly ( n, x ) !*****************************************************************************80 ! !! R8VEC_IS_ASCENDING_STRICTLY determines if an R8VEC is strictly ascending. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Notice the effect of entry number 6 in the following results: ! ! X = ( -8.1, 1.3, 2.2, 3.4, 7.5, 7.4, 9.8 ) ! Y = ( -8.1, 1.3, 2.2, 3.4, 7.5, 7.5, 9.8 ) ! Z = ( -8.1, 1.3, 2.2, 3.4, 7.5, 7.6, 9.8 ) ! ! R8VEC_IS_ASCENDING_STRICTLY ( X ) = FALSE ! R8VEC_IS_ASCENDING_STRICTLY ( Y ) = FALSE ! R8VEC_IS_ASCENDING_STRICTLY ( Z ) = TRUE ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 03 December 2007 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the array. ! ! Input, real ( kind = 8 ) X(N), the array to be examined. ! ! Output, logical ( kind = 4 ) R8VEC_IS_ASCENDING_STRICTLY, is TRUE if the ! entries strictly ascend. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i logical ( kind = 4 ) r8vec_is_ascending_strictly real ( kind = 8 ) x(n) logical ( kind = 4 ) value value = .true. do i = 1, n - 1 if ( x(i+1) <= x(i) ) then value = .false. exit end if end do r8vec_is_ascending_strictly = value return end function r8vec_is_binary ( n, x ) !*****************************************************************************80 ! !! R8VEC_IS_BINARY is true if an R8VEC only contains 0 and 1 entries. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 March 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of the vectors. ! ! Input, real ( kind = 8 ) X(N), the vector to be checked. ! ! Output, logical R8VEC_IS_BINARY, is true if all entries are 0 or 1. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i logical r8vec_is_binary logical value real ( kind = 8 ) x(n) value = .true. do i = 1, n if ( x(i) /= 0.0D+00 .and. x(i) /= 1.0D+00 ) then value = .false. exit end if end do r8vec_is_binary = value return end function r8vec_is_distinct ( n, a ) !*****************************************************************************80 ! !! R8VEC_IS_DISTINCT is true if the entries in an R8VEC are distinct. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 31 March 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A(N), the vector to be checked. ! ! Output, logical ( kind = 4 ) R8VEC_IS_DISTINCT is TRUE if the entries ! are distinct. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) j logical ( kind = 4 ) r8vec_is_distinct logical ( kind = 4 ) value value = .true. do i = 2, n do j = 1, i - 1 if ( a(i) == a(j) ) then value = .false. exit end if end do end do r8vec_is_distinct = value return end function r8vec_is_in_01 ( n, a ) !*****************************************************************************80 ! !! R8VEC_IS_IN_01 is TRUE if the entries of an R8VEC are in the range [0,1]. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 October 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, logical ( kind = 4 ) R8VEC_IS_IN_01, is TRUE if every entry is ! between 0 and 1. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) logical ( kind = 4 ) r8vec_is_in_01 logical ( kind = 4 ) value if ( any ( a(1:n) < 0.0D+00 .or. 1.0D+00 < a(1:n) ) ) then value = .false. else value = .true. end if r8vec_is_in_01 = value; return end function r8vec_is_in_ab ( n, x, a, b ) !*****************************************************************************80 ! !! R8VEC_IS_IN_AB is TRUE if the entries of an R8VEC are in the range [A,B]. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 April 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in X. ! ! Input, real ( kind = 8 ) X(N), the vector. ! ! Input, real ( kind = 8 ) A, B, the limits of the range. ! ! Output, logical ( kind = 4 ) R8VEC_IS_IN_AB, is TRUE if every entry is ! between A and B. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b logical ( kind = 4 ) r8vec_is_in_ab real ( kind = 8 ) x(n) if ( any ( x(1:n) < a .or. b < x(1:n) ) ) then r8vec_is_in_ab = .false. else r8vec_is_in_ab = .true. end if return end function r8vec_is_insignificant ( n, r, s ) !*****************************************************************************80 ! !! R8VEC_IS_INSIGNIFICANT determines if an R8VEC is insignificant. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 November 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of the vectors. ! ! Input, real ( kind = 8 ) R(N), the vector to be compared against. ! ! Input, real ( kind = 8 ) S(N), the vector to be compared. ! ! Output, logical ( kind = 4 ) R8VEC_IS_INSIGNIFICANT, is TRUE if S is ! insignificant compared to R. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i real ( kind = 8 ) r(n) logical ( kind = 4 ) r8vec_is_insignificant real ( kind = 8 ) s(n) real ( kind = 8 ) t real ( kind = 8 ) tol logical ( kind = 4 ) value value = .true. do i = 1, n t = r(i) + s(i) tol = epsilon ( r(i) ) * abs ( r(i) ) if ( tol < abs ( r(i) - t ) ) then value = .false. exit end if end do r8vec_is_insignificant = value return end function r8vec_is_integer ( n, a ) !*****************************************************************************80 ! !! R8VEC_IS_INTEGER is TRUE if the entries of an R8VEC are integers. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 October 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, logical ( kind = 4 ) R8VEC_IS_INTEGER, is TRUE if every entry ! is an integer. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) logical ( kind = 4 ) r8vec_is_integer r8vec_is_integer = all ( a(1:n) == aint ( a(1:n) ) ) return end function r8vec_is_negative ( n, a ) !*****************************************************************************80 ! !! R8VEC_IS_NEGATIVE: every element of an R8VEC is strictly negative. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 June 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A(N). ! ! Output, logical ( kind = 4 ) R8VEC_IS_NEGATIVE, is TRUE every entry ! is strictly negative. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) logical ( kind = 4 ) r8vec_is_negative r8vec_is_negative = ( all ( a(1:n) < 0.0D+00 ) ) return end function r8vec_is_negative_any ( n, a ) !*****************************************************************************80 ! !! R8VEC_IS_NEGATIVE_ANY: ( any A < 0 ) for R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 October 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, logical ( kind = 4 ) R8VEC_IS_NEGATIVE_ANY is TRUE if any entry ! is negative. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) logical ( kind = 4 ) r8vec_is_negative_any r8vec_is_negative_any = any ( a(1:n) < 0.0D+00 ) return end function r8vec_is_nonnegative ( n, a ) !*****************************************************************************80 ! !! R8VEC_IS_NONNEGATIVE is TRUE if all the entries of an R8VEC are nonnegative. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 March 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, logical ( kind = 4 ) R8VEC_IS_NONNEGATIVE, the value of ! the condition. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) logical ( kind = 4 ) r8vec_is_nonnegative r8vec_is_nonnegative = all ( 0.0D+00 <= a(1:n) ) return end function r8vec_is_nonpositive ( n, a ) !*****************************************************************************80 ! !! R8VEC_IS_NONPOSITIVE: ( all ( A <= 0 ) ) for R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 October 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries. ! ! Input, double ( kind = 8 ) A(N), the vector. ! ! Output, logical ( kind = 4 ) R8VEC_IS_NONPOSITIVE is TRUE if all entries ! are less than or equal to zero. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) logical ( kind = 4 ) r8vec_is_nonpositive r8vec_is_nonpositive = all ( a(1:n) <= 0.0D+00 ) return end function r8vec_is_nonzero_any ( n, a ) !*****************************************************************************80 ! !! R8VEC_IS_NONZERO_ANY: ( any A nonzero ) for R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 December 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, logical ( kind = 4 ) R8VEC_IS_NONZERO_ANY is TRUE if any entry ! is nonzero. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) logical ( kind = 4 ) r8vec_is_nonzero_any r8vec_is_nonzero_any = any ( a(1:n) /= 0.0D+00 ) return end function r8vec_is_one ( n, a ) !*****************************************************************************80 ! !! R8VEC_IS_ONE is TRUE if all the entries of an R8VEC are one. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 March 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, logical ( kind = 4 ) R8VEC_IS_ONE, the value of the condition. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) logical ( kind = 4 ) r8vec_is_one r8vec_is_one = all ( a(1:n) == 1.0D+00 ) return end function r8vec_is_positive ( n, a ) !*****************************************************************************80 ! !! R8VEC_IS_POSITIVE: every element of an R8VEC is strictly positive. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 June 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A(N). ! ! Output, logical ( kind = 4 ) R8VEC_IS_POSITIVE, is TRUE every entry ! is strictly positive. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) logical ( kind = 4 ) r8vec_is_positive r8vec_is_positive = ( all ( 0.0D+00 < a(1:n) ) ) return end function r8vec_is_zero ( n, a ) !*****************************************************************************80 ! !! R8VEC_IS_ZERO is TRUE if all the entries of an R8VEC are zero. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 March 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, logical ( kind = 4 ) R8VEC_IS_ZERO, the value of the condition. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) logical ( kind = 4 ) r8vec_is_zero r8vec_is_zero = all ( a(1:n) == 0.0D+00 ) return end subroutine r8vec_legendre ( n, x_first, x_last, x ) !*****************************************************************************80 ! !! R8VEC_LEGENDRE creates a vector of Legendre-spaced values. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 June 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) X_FIRST, X_LAST, the first and last entries. ! ! Output, real ( kind = 8 ) X(N), a vector of Legendre-spaced data. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) x(n) real ( kind = 8 ) x_first real ( kind = 8 ) x_last call legendre_zeros ( n, x ) x(1:n) = ( ( 1.0D+00 - x(1:n) ) * x_first & + ( 1.0D+00 + x(1:n) ) * x_last ) & / 2.0D+00 return end subroutine r8vec_linspace ( n, a, b, x ) !*****************************************************************************80 ! !! R8VEC_LINSPACE creates a vector of linearly spaced values. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! 4 points evenly spaced between 0 and 12 will yield 0, 4, 8, 12. ! ! In other words, the interval is divided into N-1 even subintervals, ! and the endpoints of intervals are used as the points. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 March 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A, B, the first and last entries. ! ! Output, real ( kind = 8 ) X(N), a vector of linearly spaced data. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b integer ( kind = 4 ) i real ( kind = 8 ) x(n) if ( n == 1 ) then x(1) = ( a + b ) / 2.0D+00 else do i = 1, n x(i) = ( real ( n - i, kind = 8 ) * a & + real ( i - 1, kind = 8 ) * b ) & / real ( n - 1, kind = 8 ) end do end if return end subroutine r8vec_linspace2 ( n, a, b, x ) !*****************************************************************************80 ! !! R8VEC_LINSPACE2 creates a vector of linearly spaced values. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! 5 points evenly spaced between 0 and 12 will yield 2, 4, 6, 8, 10. ! ! In other words, the interval is divided into N+1 even subintervals, ! and the endpoints of internal intervals are used as the points. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 September 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A, B, the first and last entries. ! ! Output, real ( kind = 8 ) X(N), a vector of linearly spaced data. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b integer ( kind = 4 ) i real ( kind = 8 ) x(n) do i = 1, n x(i) = ( real ( n - i + 1, kind = 8 ) * a & + real ( i, kind = 8 ) * b ) & / real ( n + 1, kind = 8 ) end do return end function r8vec_lt ( n, a1, a2 ) !*****************************************************************************80 ! !! R8VEC_LT == ( A1 < A2 ) for R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The comparison is lexicographic. ! ! A1 < A2 <=> A1(1) < A2(1) or ! ( A1(1) == A2(1) and A1(2) < A2(2) ) or ! ... ! ( A1(1:N-1) == A2(1:N-1) and A1(N) < A2(N) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of the vectors. ! ! Input, real ( kind = 8 ) A1(N), A2(N), the vectors to be compared. ! ! Output, logical ( kind = 4 ) R8VEC_LT, is TRUE if and only if A1 < A2. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) logical ( kind = 4 ) r8vec_lt integer ( kind = 4 ) i r8vec_lt = .false. do i = 1, n if ( a1(i) < a2(i) ) then r8vec_lt = .true. exit else if ( a2(i) < a1(i) ) then r8vec_lt = .false. exit end if end do return end subroutine r8vec_mask_print ( n, a, mask_num, mask, title ) !*****************************************************************************80 ! !! R8VEC_MASK_PRINT prints a masked R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 September 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of components of the vector. ! ! Input, real ( kind = 8 ) A(N), the vector to be printed. ! ! Input, integer ( kind = 4 ) MASK_NUM, the number of masked elements. ! ! Input, integer ( kind = 4 ) MASK(MASK_NUM), the indices of the vector ! to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) mask_num integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) mask(mask_num) character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) ' Masked vector printout:' write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' do i = 1, mask_num write ( *, '(2x,i8,a,1x,i8,2x,g14.6)' ) i, ':', mask(i), a(mask(i)) end do return end function r8vec_max ( n, a ) !*****************************************************************************80 ! !! R8VEC_MAX returns the maximum value in an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 January 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, real ( kind = 8 ) R8VEC_MAX, the value of the largest entry. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_max real ( kind = 8 ) value value = maxval ( a(1:n) ) r8vec_max = value return end subroutine r8vec_max_abs_index ( n, a, max_index ) !*****************************************************************************80 ! !! R8VEC_MAX_ABS_INDEX: index of the maximum absolute value in an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 August 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, integer ( kind = 4 ) MAX_INDEX, the index of the largest entry. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) max_index if ( n <= 0 ) then max_index = -1 else max_index = 1 do i = 2, n if ( abs ( a(max_index) ) < abs ( a(i) ) ) then max_index = i end if end do end if return end subroutine r8vec_max_index ( n, a, max_index ) !*****************************************************************************80 ! !! R8VEC_MAX_INDEX returns the index of the maximum value in an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 August 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, integer ( kind = 4 ) MAX_INDEX, the index of the largest entry. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) max_index if ( n <= 0 ) then max_index = -1 else max_index = 1 do i = 2, n if ( a(max_index) < a(i) ) then max_index = i end if end do end if return end subroutine r8vec_mean ( n, a, mean ) !*****************************************************************************80 ! !! R8VEC_MEAN returns the mean of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 February 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A(N), the vector whose mean is desired. ! ! Output, real ( kind = 8 ) MEAN, the mean of the vector entries. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) mean mean = sum ( a(1:n) ) / real ( n, kind = 8 ) return end subroutine r8vec_mean_geometric ( n, a, mean ) !*****************************************************************************80 ! !! R8VEC_MEAN_GEOMETRIC returns the geometric mean of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 April 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A(N), the vector whose mean is desired. ! All entries should be nonnegative. ! ! Output, real ( kind = 8 ) MEAN, the geometric mean of the vector entries. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) mean mean = exp ( sum ( log ( a(1:n) ) ) / real ( n, kind = 8 ) ) return end subroutine r8vec_mean_running ( n, v, a ) !*****************************************************************************80 ! !! R8VEC_MEAN_RUNNING computes the running means of an R8VEC. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 22 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items. ! ! Input, real ( kind = 8 ) V(N), the data. ! ! Output, real ( kind = 8 ) A(N+1), the running average. A(I) is the ! average value of the first I-1 values in V. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n+1) integer ( kind = 4 ) i real ( kind = 8 ) v(n) ! ! Sum. ! a(1) = 0.0D+00 do i = 2, n + 1 a(i) = a(i-1) + v(i-1) end do ! ! Average. ! do i = 2, n + 1 a(i) = a(i) / real ( i - 1, kind = 8 ) end do return end subroutine r8vec_mean_update ( nm1, mean_nm1, xn, n, mean_n ) !*****************************************************************************80 ! !! R8VEC_MEAN_UPDATE updates the mean of an R8VEC with one new value. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 November 2017 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) NM1, the number of entries in the old vector. ! ! Input, real ( kind = 8 ) MEAN_NM1, the mean of the old vector. ! ! Input, real ( kind = 8 ) XN, the new N-th entry of the vector. ! ! Output, integer ( kind = 4 ) N, the number of entries in the new vector. ! ! Output, real ( kind = 8 ) MEAN_N, the mean of the new vector. ! implicit none real ( kind = 8 ) mean_n real ( kind = 8 ) mean_nm1 integer ( kind = 4 ) n integer ( kind = 4 ) nm1 real ( kind = 8 ) xn if ( nm1 <= 0 ) then n = 1 mean_n = xn else n = nm1 + 1 mean_n = mean_nm1 + ( xn - mean_nm1 ) / real ( n, kind = 8 ) end if return end subroutine r8vec_median ( n, a, median ) !*****************************************************************************80 ! !! R8VEC_MEDIAN returns the median of an unsorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Hoare's algorithm is used. The values of the vector are ! rearranged by this routine. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 September 2000 ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input/output, real ( kind = 8 ) A(N), the array to search. On output, ! the order of the elements of A has been somewhat changed. ! ! Output, real ( kind = 8 ) MEDIAN, the value of the median of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) k real ( kind = 8 ) median k = ( n + 1 ) / 2 call r8vec_frac ( n, a, k, median ) return end subroutine r8vec_mesh_2d ( nx, ny, xvec, yvec, xmat, ymat ) !*****************************************************************************80 ! !! R8VEC_MESH_2D creates a 2D mesh from X and Y vectors. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! NX = 2 ! XVEC = ( 1, 2, 3 ) ! NY = 3 ! YVEC = ( 4, 5 ) ! ! XMAT = ( ! 1, 2, 3 ! 1, 2, 3 ) ! ! YMAT = ( ! 4, 4, 4 ! 5, 5, 5 ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 22 July 2013 ! ! Parameters: ! ! Input, integer ( kind = 4 ) NX, NY, the number of X and Y values. ! ! Input, real ( kind = 8 ) XVEC(NX), YVEC(NY), the X and Y coordinate ! values. ! ! Output, real ( kind = 8 ) XMAT(NX,NY), YMAT(NX,NY), the coordinate ! values of points on an NX by NY mesh. ! implicit none integer ( kind = 4 ) nx integer ( kind = 4 ) ny integer ( kind = 4 ) j real ( kind = 8 ) xmat(nx,ny) real ( kind = 8 ) xvec(nx) real ( kind = 8 ) ymat(nx,ny) real ( kind = 8 ) yvec(ny) do j = 1, ny xmat(1:nx,j) = xvec(1:nx) end do do j = 1, ny ymat(1:nx,j) = yvec(j) end do return end subroutine r8vec_midspace ( n, a, b, x ) !*****************************************************************************80 ! !! R8VEC_MIDSPACE creates a vector of linearly spaced values. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! This function divides the interval [a,b] into n subintervals, and then ! returns the midpoints of those subintervals. ! ! Example: ! ! N = 5, A = 10, B = 20 ! X = [ 11, 13, 15, 17, 19 ] ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 03 June 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A, B, the endpoints of the interval. ! ! Output, real ( kind = 8 ) X(N), a vector of linearly spaced data. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b integer ( kind = 4 ) i real ( kind = 8 ) x(n) do i = 1, n x(i) = ( real ( 2 * n - 2 * i + 1, kind = 8 ) * a & + real ( 2 * i - 1, kind = 8 ) * b ) & / real ( 2 * n, kind = 8 ) end do return end function r8vec_min ( n, a ) !*****************************************************************************80 ! !! R8VEC_MIN returns the minimum value of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 November 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, real ( kind = 8 ) R8VEC_MIN, the value of the smallest entry. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_min real ( kind = 8 ) value value = minval ( a(1:n) ) r8vec_min = value return end subroutine r8vec_min_index ( n, a, min_index ) !*****************************************************************************80 ! !! R8VEC_MIN_INDEX returns the index of the minimum value in an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 August 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, integer ( kind = 4 ) MIN_INDEX, the index of the smallest entry. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) min_index if ( n <= 0 ) then min_index = -1 else min_index = 1 do i = 2, n if ( a(i) < a(min_index) ) then min_index = i end if end do end if return end function r8vec_min_pos ( n, a ) !*****************************************************************************80 ! !! R8VEC_MIN_POS returns the minimum positive value of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 November 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, real ( kind = 8 ) R8VEC_MIN_POS, the smallest positive entry. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i real ( kind = 8 ) r8vec_min_pos real ( kind = 8 ) value value = huge ( value ) do i = 1, n if ( 0.0D+00 < a(i) ) then value = min ( value, a(i) ) end if end do r8vec_min_pos = value return end subroutine r8vec_mirror_next ( n, a, done ) !*****************************************************************************80 ! !! R8VEC_MIRROR_NEXT steps through all sign variations of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! In normal use, the user would set every element of A to be positive. ! The routine will take the input value of A, and output a copy in ! which the signs of one or more entries have been changed. Repeatedly ! calling the routine with the output from the previous call will generate ! every distinct "variation" of A; that is, all possible sign variations. ! ! When the output variable DONE is TRUE (or equal to 1), then the ! output value of A_NEW is the last in the series. ! ! Note that A may have some zero values. The routine will essentially ! ignore such entries; more exactly, it will not stupidly assume that -0 ! is a proper "variation" of 0! ! ! Also, it is possible to call this routine with the signs of A set ! in any way you like. The routine will operate properly, but it ! will nonethess terminate when it reaches the value of A in which ! every nonzero entry has negative sign. ! ! More efficient algorithms using the Gray code seem to require internal ! memory in the routine, which is not one of MATLAB's strong points, ! or the passing back and forth of a "memory array", or the use of ! global variables, or unnatural demands on the user. This form of ! the routine is about as clean as I can make it. ! ! Example: ! ! Input Output ! --------- -------------- ! A A_NEW DONE ! --------- -------- ---- ! 1 2 3 -1 2 3 false ! -1 2 3 1 -2 3 false ! 1 -2 3 -1 -2 3 false ! -1 -2 3 1 2 -3 false ! 1 2 -3 -1 2 -3 false ! -1 2 -3 1 -2 -3 false ! 1 -2 -3 -1 -2 -3 false ! -1 -2 -3 1 2 3 true ! ! 1 0 3 -1 0 3 false ! -1 0 3 1 0 -3 false ! 1 0 -3 -1 0 -3 false ! -1 0 -3 1 0 3 true ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 April 2005 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input/output, real ( kind = 8 ) A(N), a vector of real numbers. ! On output, the signs of some entries have been changed. ! ! Output, logical ( kind = 4 ) DONE, is TRUE if the input vector A was the ! last element in the series (every entry was nonpositive); the output ! vector is reset so that all entries are nonnegative, but presumably the ! ride is over! ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) logical ( kind = 4 ) done integer ( kind = 4 ) i integer ( kind = 4 ) positive ! ! Seek the first strictly positive entry of A. ! positive = 0 do i = 1, n if ( 0.0D+00 < a(i) ) then positive = i exit end if end do ! ! If there is no strictly positive entry of A, there is no successor. ! if ( positive == 0 ) then a(1:n) = - a(1:n) done = .true. return end if ! ! Otherwise, negate A up to the positive entry. ! a(1:positive) = - a(1:positive) done = .false. return end subroutine r8vec_mirror_ab_next ( m, a, b, x, done ) !*****************************************************************************80 ! !! R8VEC_MIRROR_AB_NEXT steps through "mirrored" versions of vector X. ! ! Discussion: ! ! X is an M component vector contained in a rectangle described by A and B, ! that is, for each index I, we have ! A(I) <= X(I) <= B(I). ! ! "Mirrored" versions of the vector X have one or more components ! reflected about the A or B limit. ! ! As long as each component of X is strictly between the limits A and B, ! this means there will be 3^M possible versions of the vector. ! ! If one component of X is equal to one limit or the other, this ! suppresses mirroring across that limit. If one component of ! X, A and B are equal, then no mirroring is done at all in that component. ! ! Example: ! ! A = 0, 0, 0 ! X = 1, 1, 1 ! B = 2, 2, 2 ! results in the following sequence of 3x3x3 values: ! ! 0 0 0 ! 0 0 1 ! 0 0 2 ! 0 1 0 ! 0 1 1 ! ....... ! 2 1 1 ! 2 1 2 ! 2 2 0 ! 2 2 1 ! 2 2 2 ! ! A = 0 1 0 ! X = 1 1 1 ! B = 2 2 2 ! results in the following sequence of 3x2x3 values: ! ! 0 1 0 ! 0 1 1 ! 0 1 2 ! 0 2 0 ! 0 2 1 ! 0 2 2 ! 1 1 0 ! 1 1 1 ! 1 1 2 ! 1 2 0 ! 1 2 1 ! 1 2 2 ! 2 1 0 ! 2 1 1 ! 2 1 2 ! 2 2 0 ! 2 2 1 ! 2 2 2 ! ! A = 0 1 0 ! X = 1 1 1 ! B = 2 1 2 ! results in the following sequence of 3x1x3 values: ! ! 0 1 0 ! 0 1 1 ! 0 1 2 ! 1 1 0 ! 1 1 1 ! 1 1 2 ! 2 1 0 ! 2 1 1 ! 2 1 2 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 August 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A(M), B(M), the lower and upper limits. ! ! Input/output, real ( kind = 8 ) X(M), a vector being manipulated. ! ! Input/output, logical DONE. On first call, DONE should be TRUE, and ! A(I) <= X(I) <= B(I) for each index I. On output, if DONE is TRUE, ! then the returned value is the last entry in the sequence. ! implicit none integer ( kind = 4 ) m real ( kind = 8 ) a(m) real ( kind = 8 ) b(m) logical done integer ( kind = 4 ) i real ( kind = 8 ) x(m) ! ! First call: ! if ( done ) then ! ! Ensure all A(I) <= X(I) <= B(I). ! do i = 1, m if ( x(i) < a(i) ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'R8VEC_MIRROR_AB_NEXT - Fatal error!' write ( *, '(a)' ) ' Not every A(I) <= X(I).' stop 1 end if if ( b(i) < x(i) ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'R8VEC_MIRROR_AB_NEXT - Fatal error!' write ( *, '(a)' ) ' Not every X(I) <= B(I).' stop 1 end if end do ! ! Set first element of sequence. ! x(1:m) = 2.0D+00 * a(1:m) - x(1:m) ! ! Unless A = B, our sequence is not done. ! done = .true. do i = 1, m if ( a(i) /= b(i) ) then done = .false. exit end if end do ! ! Subsequent calls. ! else ! ! Initialize index to last. ! loop ! if index < 1, set DONE = true and return. ! if the index-th value is below B, increment it and return; ! otherwise reset index-th value to A. ! decrement the index. ! i = m do while ( 1 <= i ) if ( x(i) < a(i) ) then x(i) = 2.0D+00 * a(i) - x(i) return else if ( x(i) < b(i) ) then x(i) = 2.0D+00 * b(i) - x(i) return else x(i) = x(i) - 2.0D+00 * ( b(i) - a(i) ) end if i = i - 1 end do done = .true. end if return end subroutine r8vec_mm_to_01 ( n, a ) !*****************************************************************************80 ! !! R8VEC_MM_TO_01 shifts and rescales an R8VEC to lie within [0,1]. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! On input, A contains the original data. On output, A has been shifted ! and scaled so that all entries lie between 0 and 1. ! ! Formula: ! ! A(I) := ( A(I) - AMIN ) / ( AMAX - AMIN ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of data values. ! ! Input/output, real ( kind = 8 ) A(N), the data to be rescaled. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) amax real ( kind = 8 ) amin amax = maxval ( a(1:n) ) amin = minval ( a(1:n) ) if ( amin == amax ) then a(1:n) = 0.5D+00 else a(1:n) = ( a(1:n) - amin ) / ( amax - amin ) end if return end subroutine r8vec_mm_to_cd ( n, a, bmin, bmax, b ) !*****************************************************************************80 ! !! R8VEC_MM_TO_CD shifts and rescales an R8VEC from one interval to another. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The mininum entry of A is mapped to BMIN, the maximum entry ! to BMAX, and values in between are mapped linearly. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of data values. ! ! Input, real ( kind = 8 ) A(N), the data to be remapped. ! ! Input, real ( kind = 8 ) BMIN, BMAX, the values to which min(A) and max(A) ! are to be assigned. ! ! Output, real ( kind = 8 ) B(N), the remapped data. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) amax real ( kind = 8 ) amin real ( kind = 8 ) b(n) real ( kind = 8 ) bmax real ( kind = 8 ) bmin if ( bmax == bmin ) then b(1:n) = bmin return end if amin = minval ( a(1:n) ) amax = maxval ( a(1:n) ) if ( amax == amin ) then b(1:n) = 0.5D+00 * ( bmax + bmin ) return end if b(1:n) = ( ( amax - a(1:n) ) * bmin & + ( a(1:n) - amin ) * bmax ) & / ( amax - amin ) return end subroutine r8vec_nint ( n, a ) !*****************************************************************************80 ! !! R8VEC_NINT rounds entries of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 March 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input/output, real ( kind = 8 ) A(N), the vector to be NINT'ed. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) a(1:n) = nint ( real ( a(1:n), kind = 8 ) ) return end function r8vec_norm ( n, a ) !*****************************************************************************80 ! !! R8VEC_NORM returns the L2 norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The vector L2 norm is defined as: ! ! R8VEC_NORM = sqrt ( sum ( 1 <= I <= N ) A(I)^2 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 August 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector whose L2 norm is desired. ! ! Output, real ( kind = 8 ) R8VEC_NORM, the L2 norm of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_norm r8vec_norm = sqrt ( sum ( a(1:n) ** 2 ) ) return end function r8vec_norm_affine ( n, v0, v1 ) !*****************************************************************************80 ! !! R8VEC_NORM_AFFINE returns the affine norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The affine vector L2 norm is defined as: ! ! R8VEC_NORM_AFFINE(V0,V1) ! = sqrt ( sum ( 1 <= I <= N ) ( V1(I) - V0(I) )^2 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the vectors. ! ! Input, real ( kind = 8 ) V0(N), the base vector. ! ! Input, real ( kind = 8 ) V1(N), the vector whose affine norm is desired. ! ! Output, real ( kind = 8 ) R8VEC_NORM_AFFINE, the L2 norm of V1-V0. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) r8vec_norm_affine real ( kind = 8 ) v0(n) real ( kind = 8 ) v1(n) r8vec_norm_affine = sqrt ( sum ( ( v0(1:n) - v1(1:n) ) ** 2 ) ) return end function r8vec_norm_l0 ( n, a ) !*****************************************************************************80 ! !! R8VEC_NORM_L0 returns the l0 "norm" of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The l0 "norm" simply counts the number of nonzero entries in the vector. ! It is not a true norm, but has some similarities to one. It is useful ! in the study of compressive sensing. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 June 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, real ( kind = 8 ) R8VEC_NORM_L0, the value of the norm. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i real ( kind = 8 ) r8vec_norm_l0 real ( kind = 8 ) value value = 0.0D+00 do i = 1, n if ( a(i) /= 0.0D+00 ) then value = value + 1.0D+00 end if end do r8vec_norm_l0 = value return end function r8vec_norm_l1 ( n, a ) !*****************************************************************************80 ! !! R8VEC_NORM_L1 returns the L1 norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The vector L1 norm is defined as: ! ! R8VEC_NORM_L1 = sum ( 1 <= I <= N ) abs ( A(I) ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 April 2002 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector whose L1 norm is desired. ! ! Output, real ( kind = 8 ) R8VEC_NORM_L1, the L1 norm of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_norm_l1 r8vec_norm_l1 = sum ( abs ( a(1:n) ) ) return end function r8vec_norm_l2 ( n, a ) !*****************************************************************************80 ! !! R8VEC_NORM_L2 returns the L2 norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The vector L2 norm is defined as: ! ! R8VEC_NORM_L2 = sqrt ( sum ( 1 <= I <= N ) A(I)^2 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 April 2002 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector whose L2 norm is desired. ! ! Output, real ( kind = 8 ) R8VEC_NORM_L2, the L2 norm of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_norm_l2 r8vec_norm_l2 = sqrt ( sum ( a(1:n) ** 2 ) ) return end function r8vec_norm_li ( n, a ) !*****************************************************************************80 ! !! R8VEC_NORM_LI returns the L-oo norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The vector L-oo norm is defined as: ! ! R8VEC_NORM_LI = max ( 1 <= I <= N ) abs ( A(I) ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 April 2002 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector whose L-oo norm is desired. ! ! Output, real ( kind = 8 ) R8VEC_NORM_LI, the L-oo norm of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_norm_li r8vec_norm_li = maxval ( abs ( a(1:n) ) ) return end function r8vec_norm_lp ( n, a, p ) !*****************************************************************************80 ! !! R8VEC_NORM_LP returns the LP norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The vector LP norm is defined as: ! ! R8VEC_NORM_LP = ( sum ( 1 <= I <= N ) ( abs ( A(I) ) )^P )^(1/P). ! ! Usually, the LP norms with ! 1 <= P <= oo ! are of interest. This routine allows ! 0 < P <= Huge ( P ). ! If P = Huge ( P ), then the L-oo norm is returned, which is ! simply the maximum of the absolute values of the vector components. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 April 2002 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector whose LP norm is desired. ! ! Input, real ( kind = 8 ) P, the index of the norm. ! ! Output, real ( kind = 8 ) R8VEC_NORM_LP, the LP norm of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) p real ( kind = 8 ) r8vec_norm_lp if ( p <= 0.0D+00 ) then r8vec_norm_lp = -1.0D+00 else if ( p == huge ( p ) ) then r8vec_norm_lp = maxval ( abs ( a(1:n) ) ) else if ( p == 1.0D+00 ) then r8vec_norm_lp = sum ( abs ( a(1:n) ) ) else if ( p == 2.0D+00 ) then r8vec_norm_lp = sqrt ( sum ( a(1:n) ** 2 ) ) else r8vec_norm_lp = ( sum ( ( abs ( a(1:n) ) ) ** p ) ) ** ( 1.0D+00 / p ) end if return end function r8vec_norm_rms ( n, a ) !*****************************************************************************80 ! !! R8VEC_NORM_RMS returns the RMS norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The vector RMS norm is defined as: ! ! R8VEC_NORM_RMS = sqrt ( sum ( 1 <= I <= N ) A(I)^2 / N ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 October 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, real ( kind = 8 ) R8VEC_NORM_RMS, the RMS norm of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_norm_rms r8vec_norm_rms = sqrt ( sum ( a(1:n) ** 2 ) / real ( n, kind = 8 ) ) return end function r8vec_norm_squared ( n, a ) !*****************************************************************************80 ! !! R8VEC_NORM_SQUARED returns the square of the L2 norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! R8VEC_NORM_SQUARED = sqrt ( sum ( 1 <= I <= N ) A(I)^2 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 March 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, real ( kind = 8 ) R8VEC_NORM_SQUARED, the squared L2 norm of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_norm_squared r8vec_norm_squared = sum ( a(1:n) ** 2 ) return end subroutine r8vec_normal_01 ( n, seed, x ) !*****************************************************************************80 ! !! R8VEC_NORMAL_01 returns a unit pseudonormal R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The standard normal probability distribution function (PDF) has ! mean 0 and standard deviation 1. ! ! This routine can generate a vector of values on one call. It ! has the feature that it should provide the same results ! in the same order no matter how we break up the task. ! ! Before calling this routine, the user may call RANDOM_SEED ! in order to set the seed of the random number generator. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 August 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of values desired. ! ! Input/output, integer ( kind = 4 ) SEED, a seed for the random ! number generator. ! ! Output, real ( kind = 8 ) X(N), a sample of the standard normal PDF. ! ! Local parameters: ! ! Local, real ( kind = 8 ) R(N+1), is used to store some uniform ! random values. Its dimension is N+1, but really it is only needed ! to be the smallest even number greater than or equal to N. ! ! Local, integer X_LO_INDEX, X_HI_INDEX, records the range of entries of ! X that we need to compute. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) m real ( kind = 8 ) r(n+1) real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) r8_uniform_01 integer ( kind = 4 ) seed real ( kind = 8 ) x(n) integer ( kind = 4 ) x_hi_index integer ( kind = 4 ) x_lo_index ! ! Record the range of X we need to fill in. ! x_lo_index = 1 x_hi_index = n ! ! Maybe we don't need any more values. ! if ( x_hi_index - x_lo_index + 1 == 1 ) then r(1) = r8_uniform_01 ( seed ) if ( r(1) == 0.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_NORMAL_01 - Fatal error!' write ( *, '(a)' ) ' R8_UNIFORM_01 returned a value of 0.' stop 1 end if r(2) = r8_uniform_01 ( seed ) x(x_hi_index) = & sqrt ( -2.0D+00 * log ( r(1) ) ) * cos ( 2.0D+00 * r8_pi * r(2) ) ! ! If we require an even number of values, that's easy. ! else if ( mod ( x_hi_index - x_lo_index + 1, 2 ) == 0 ) then m = ( x_hi_index - x_lo_index + 1 ) / 2 call r8vec_uniform_01 ( 2*m, seed, r ) x(x_lo_index:x_hi_index-1:2) = & sqrt ( -2.0D+00 * log ( r(1:2*m-1:2) ) ) & * cos ( 2.0D+00 * r8_pi * r(2:2*m:2) ) x(x_lo_index+1:x_hi_index:2) = & sqrt ( -2.0D+00 * log ( r(1:2*m-1:2) ) ) & * sin ( 2.0D+00 * r8_pi * r(2:2*m:2) ) ! ! If we require an odd number of values, we generate an even number, ! and handle the last pair specially, storing one in X(N), and ! saving the other for later. ! else x_hi_index = x_hi_index - 1 m = ( x_hi_index - x_lo_index + 1 ) / 2 + 1 call r8vec_uniform_01 ( 2 * m, seed, r ) x(x_lo_index:x_hi_index-1:2) = & sqrt ( -2.0D+00 * log ( r(1:2*m-3:2) ) ) & * cos ( 2.0D+00 * r8_pi * r(2:2*m-2:2) ) x(x_lo_index+1:x_hi_index:2) = & sqrt ( -2.0D+00 * log ( r(1:2*m-3:2) ) ) & * sin ( 2.0D+00 * r8_pi * r(2:2*m-2:2) ) x(n) = sqrt ( -2.0D+00 * log ( r(2*m-1) ) ) & * cos ( 2.0D+00 * r8_pi * r(2*m) ) end if return end subroutine r8vec_normal_ab ( n, a, b, seed, x ) !*****************************************************************************80 ! !! R8VEC_NORMAL_AB returns a scaled pseudonormal R8VEC. ! ! Discussion: ! ! The standard normal probability distribution function (PDF) has ! mean 0 and standard deviation 1. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 August 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of values desired. ! ! Input, real ( kind = 8 ) A, B, the mean and standard deviation. ! ! Input/output, integer ( kind = 4 ) SEED, a seed for the random ! number generator. ! ! Output, real ( kind = 8 ) X(N), a sample of the standard normal PDF. ! ! Local parameters: ! ! Local, real ( kind = 8 ) R(N+1), is used to store some uniform ! random values. Its dimension is N+1, but really it is only needed ! to be the smallest even number greater than or equal to N. ! ! Local, integer ( kind = 4 ) X_LO_INDEX, X_HI_INDEX, records the range ! of entries of X that we need to compute. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b integer ( kind = 4 ) m real ( kind = 8 ) r(n+1) real ( kind = 8 ), parameter :: r8_pi = 3.141592653589793D+00 real ( kind = 8 ) r8_uniform_01 integer ( kind = 4 ) seed real ( kind = 8 ) x(n) integer ( kind = 4 ) x_hi_index integer ( kind = 4 ) x_lo_index ! ! Record the range of X we need to fill in. ! x_lo_index = 1 x_hi_index = n ! ! If we need just one new value, do that here to avoid null arrays. ! if ( x_hi_index - x_lo_index + 1 == 1 ) then r(1) = r8_uniform_01 ( seed ) if ( r(1) == 0.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_NORMAL_AB - Fatal error!' write ( *, '(a)' ) ' R8_UNIFORM_01 returned a value of 0.' stop 1 end if r(2) = r8_uniform_01 ( seed ) x(x_hi_index) = & sqrt ( - 2.0D+00 * log ( r(1) ) ) * cos ( 2.0D+00 * r8_pi * r(2) ) ! ! If we require an even number of values, that's easy. ! else if ( mod ( x_hi_index - x_lo_index, 2 ) == 1 ) then m = ( x_hi_index - x_lo_index + 1 ) / 2 call r8vec_uniform_01 ( 2*m, seed, r ) x(x_lo_index:x_hi_index-1:2) = & sqrt ( - 2.0D+00 * log ( r(1:2*m-1:2) ) ) & * cos ( 2.0D+00 * r8_pi * r(2:2*m:2) ) x(x_lo_index+1:x_hi_index:2) = & sqrt ( - 2.0D+00 * log ( r(1:2*m-1:2) ) ) & * sin ( 2.0D+00 * r8_pi * r(2:2*m:2) ) ! ! If we require an odd number of values, we generate an even number, ! and handle the last pair specially, storing one in X(N), and ! saving the other for later. ! else x_hi_index = x_hi_index - 1 m = ( x_hi_index - x_lo_index + 1 ) / 2 + 1 call r8vec_uniform_01 ( 2*m, seed, r ) x(x_lo_index:x_hi_index-1:2) = & sqrt ( - 2.0D+00 * log ( r(1:2*m-3:2) ) ) & * cos ( 2.0D+00 * r8_pi * r(2:2*m-2:2) ) x(x_lo_index+1:x_hi_index:2) = & sqrt ( - 2.0D+00 * log ( r(1:2*m-3:2) ) ) & * sin ( 2.0D+00 * r8_pi * r(2:2*m-2:2) ) x(n) = sqrt ( - 2.0D+00 * log ( r(2*m-1) ) ) & * cos ( 2.0D+00 * r8_pi * r(2*m) ) end if x(1:n) = a + b * x(1:n) return end subroutine r8vec_normalize ( n, a ) !*****************************************************************************80 ! !! R8VEC_NORMALIZE normalizes an R8VEC in the Euclidean norm. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The euclidean norm is also sometimes called the l2 or ! least squares norm. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of the vector. ! ! Input/output, real ( kind = 8 ) A(N), the vector to be normalized. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) norm norm = sqrt ( sum ( a(1:n) ** 2 ) ) if ( norm /= 0.0D+00 ) then a(1:n) = a(1:n) / norm end if return end subroutine r8vec_normalize_l1 ( n, a ) !*****************************************************************************80 ! !! R8VEC_NORMALIZE_L1 normalizes an R8VEC to have unit sum. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 July 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input/output, real ( kind = 8 ) A(N), the vector to be normalized. ! On output, the entries of A should have unit sum. However, if ! the input vector has zero sum, the routine halts. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) a_sum a_sum = sum ( a(1:n) ) if ( a_sum == 0.0D+00 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_NORMALIZE_L1 - Fatal error!' write ( *, '(a)' ) ' The vector entries sum to 0.' stop 1 end if a(1:n) = a(1:n) / a_sum return end function r8vec_normsq ( n, v ) !*****************************************************************************80 ! !! R8VEC_NORMSQ returns the square of the L2 norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The square of the vector L2 norm is defined as: ! ! R8VEC_NORMSQ = sum ( 1 <= I <= N ) V(I)^2. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 28 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the vector dimension. ! ! Input, real ( kind = 8 ) V(N), the vector. ! ! Output, real ( kind = 8 ) R8VEC_NORMSQ, the squared L2 norm. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) r8vec_normsq real ( kind = 8 ) v(n) r8vec_normsq = sum ( v(1:n) ** 2 ) return end function r8vec_normsq_affine ( n, v0, v1 ) !*****************************************************************************80 ! !! R8VEC_NORMSQ_AFFINE returns the affine squared norm of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The affine squared vector L2 norm is defined as: ! ! R8VEC_NORMSQ_AFFINE(V0,V1) ! = sum ( 1 <= I <= N ) ( V1(I) - V0(I) )^2 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 28 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the vector dimension. ! ! Input, real ( kind = 8 ) V0(N), the base vector. ! ! Input, real ( kind = 8 ) V1(N), the vector. ! ! Output, real ( kind = 8 ) R8VEC_NORMSQ_AFFINE, the squared affine L2 norm. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) r8vec_normsq_affine real ( kind = 8 ) v0(n) real ( kind = 8 ) v1(n) r8vec_normsq_affine = sum ( ( v0(1:n) - v1(1:n) ) ** 2 ) return end subroutine r8vec_order_type ( n, a, order ) !*****************************************************************************80 ! !! R8VEC_ORDER_TYPE determines if R8VEC is (non)strictly ascending/descending. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 15 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of the array. ! ! Input, real ( kind = 8 ) A(N), the array to be checked. ! ! Output, integer ( kind = 4 ) ORDER, order indicator: ! -1, no discernable order; ! 0, all entries are equal; ! 1, ascending order; ! 2, strictly ascending order; ! 3, descending order; ! 4, strictly descending order. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) order ! ! Search for the first value not equal to A(1). ! i = 1 do i = i + 1 if ( n < i ) then order = 0 return end if if ( a(1) < a(i) ) then if ( i == 2 ) then order = 2 else order = 1 end if exit else if ( a(i) < a(1) ) then if ( i == 2 ) then order = 4 else order = 3 end if exit end if end do ! ! Now we have a "direction". Examine subsequent entries. ! do while ( i < n ) i = i + 1 if ( order == 1 ) then if ( a(i) < a(i-1) ) then order = -1 exit end if else if ( order == 2 ) then if ( a(i) < a(i-1) ) then order = -1 exit else if ( a(i) == a(i-1) ) then order = 1 end if else if ( order == 3 ) then if ( a(i-1) < a(i) ) then order = -1 exit end if else if ( order == 4 ) then if ( a(i-1) < a(i) ) then order = -1 exit else if ( a(i) == a(i-1) ) then order = 3 end if end if end do return end subroutine r8vec_part_quick_a ( n, a, l, r ) !*****************************************************************************80 ! !! R8VEC_PART_QUICK_A reorders an R8VEC as part of a quick sort. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The routine reorders the entries of A. Using A(1) as the key, ! all entries of A that are less than or equal to the key will ! precede the key which precedes all entries that are greater than the key. ! ! Example: ! ! Input: ! ! N = 8 ! ! A = ( 6, 7, 3, 1, 6, 8, 2, 9 ) ! ! Output: ! ! L = 3, R = 6 ! ! A = ( 3, 1, 2, 6, 6, 8, 9, 7 ) ! ------- ------- ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of A. ! ! Input/output, real ( kind = 8 ) A(N). On input, the array to be checked. ! On output, A has been reordered as described above. ! ! Output, integer ( kind = 4 ) L, R, the indices of A that define ! the three segments. Let KEY = the input value of A(1). Then ! I <= L A(I) < KEY; ! L < I < R A(I) = KEY; ! R <= I KEY < A(I). ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i real ( kind = 8 ) key integer ( kind = 4 ) l integer ( kind = 4 ) m integer ( kind = 4 ) r real ( kind = 8 ) temp if ( n < 1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_PART_QUICK_A - Fatal error!' write ( *, '(a)' ) ' N < 1.' stop 1 else if ( n == 1 ) then l = 0 r = 2 return end if key = a(1) m = 1 ! ! The elements of unknown size have indices between L+1 and R-1. ! l = 1 r = n + 1 do i = 2, n if ( key < a(l+1) ) then r = r - 1 temp = a(r) a(r) = a(l+1) a(l+1) = temp else if ( a(l+1) == key ) then m = m + 1 temp = a(m) a(m) = a(l+1) a(l+1) = temp l = l + 1 else if ( a(l+1) < key ) then l = l + 1 end if end do ! ! Now shift small elements to the left, and KEY elements to center. ! do i = 1, l - m a(i) = a(i+m) end do ! ! Out of bounds here, occasionally ! l = l - m a(l+1:l+m) = key return end subroutine r8vec_permute ( n, p, a ) !*****************************************************************************80 ! !! R8VEC_PERMUTE permutes an R8VEC in place. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! This routine permutes an array of real "objects", but the same ! logic can be used to permute an array of objects of any arithmetic ! type, or an array of objects of any complexity. The only temporary ! storage required is enough to store a single object. The number ! of data movements made is N + the number of cycles of order 2 or more, ! which is never more than N + N/2. ! ! P(I) = J means that the I-th element of the output array should be ! the J-th element of the input array. P must be a legal permutation ! of the integers from 1 to N, otherwise the algorithm will ! fail catastrophically. ! ! Example: ! ! Input: ! ! N = 5 ! P = ( 2, 4, 5, 1, 3 ) ! A = ( 1.0, 2.0, 3.0, 4.0, 5.0 ) ! ! Output: ! ! A = ( 2.0, 4.0, 5.0, 1.0, 3.0 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 July 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of objects. ! ! Input, integer ( kind = 4 ) P(N), the permutation. ! ! Input/output, real ( kind = 8 ) A(N), the array to be permuted. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) a_temp integer ( kind = 4 ) ierror integer ( kind = 4 ) iget integer ( kind = 4 ) iput integer ( kind = 4 ) istart integer ( kind = 4 ) p(n) integer ( kind = 4 ) perm1_check ierror = perm1_check ( n, p ) if ( ierror .ne. 0 ) then write ( *, '(a)' ) '' write ( *, '(a)' ) 'R8VEC_PERMUTE - Fatal error!' write ( *, '(a)' ) ' PERM1_CHECK returned error.' stop 1 end if ! ! Search for the next element of the permutation that has not been used. ! do istart = 1, n if ( p(istart) < 0 ) then cycle else if ( p(istart) == istart ) then p(istart) = - p(istart) cycle else a_temp = a(istart) iget = istart ! ! Copy the new value into the vacated entry. ! do iput = iget iget = p(iget) p(iput) = - p(iput) if ( iget < 1 .or. n < iget ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_PERMUTE - Fatal error!' write ( *, '(a)' ) ' A permutation index is out of range.' write ( *, '(a,i8,a,i8)' ) ' P(', iput, ') = ', iget stop 1 end if if ( iget == istart ) then a(iput) = a_temp exit end if a(iput) = a(iget) end do end if end do ! ! Restore the signs of the entries. ! p(1:n) = - p(1:n) return end subroutine r8vec_permute_cyclic ( n, k, a ) !*****************************************************************************80 ! !! R8VEC_PERMUTE_CYCLIC performs a cyclic permutation of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! For 0 <= K < N, this function cyclically permutes the input vector ! to have the form ! ! ( A(K+1), A(K+2), ..., A(N), A(1), ..., A(K) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 22 August 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of objects. ! ! Input, integer ( kind = 4 ) K, the increment used. ! ! Input/output, real ( kind = 8 ) A(N), the array to be permuted. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) b(n) integer ( kind = 4 ) i integer ( kind = 4 ) i4_wrap integer ( kind = 4 ) ipk integer ( kind = 4 ) k do i = 1, n ipk = i4_wrap ( i + k, 1, n ) b(i) = a(ipk) end do a(1:n) = b(1:n) return end subroutine r8vec_permute_uniform ( n, a, seed ) !*****************************************************************************80 ! !! R8VEC_PERMUTE_UNIFORM randomly permutes an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 April 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of objects. ! ! Input/output, real ( kind = 8 ) A(N), the array to be permuted. ! ! Input/output, integer ( kind = 4 ) SEED, a seed for the random ! number generator. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) p(n) integer ( kind = 4 ) seed call perm1_uniform ( n, seed, p ) call r8vec_permute ( n, p, a ) return end subroutine r8vec_polarize ( n, a, p, a_normal, a_parallel ) !*****************************************************************************80 ! !! R8VEC_POLARIZE decomposes an R8VEC into normal and parallel components. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The (nonzero) vector P defines a direction. ! ! The vector A can be written as the sum ! ! A = A_normal + A_parallel ! ! where A_parallel is a linear multiple of P, and A_normal ! is perpendicular to P. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 November 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the vector to be polarized. ! ! Input, real ( kind = 8 ) P(N), the polarizing direction. ! ! Output, real ( kind = 8 ) A_NORMAL(N), A_PARALLEL(N), the normal ! and parallel components of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) a_dot_p real ( kind = 8 ) a_normal(n) real ( kind = 8 ) a_parallel(n) real ( kind = 8 ) p(n) real ( kind = 8 ) p_norm p_norm = sqrt ( sum ( p(1:n) ** 2 ) ) if ( p_norm == 0.0D+00 ) then a_normal(1:n) = a(1:n) a_parallel(1:n) = 0.0D+00 return end if a_dot_p = dot_product ( a(1:n), p(1:n) ) / p_norm a_parallel(1:n) = a_dot_p * p(1:n) / p_norm a_normal(1:n) = a(1:n) - a_parallel(1:n) return end subroutine r8vec_print ( n, a, title ) !*****************************************************************************80 ! !! R8VEC_PRINT prints an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 22 August 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of components of the vector. ! ! Input, real ( kind = 8 ) A(N), the vector to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' do i = 1, n write ( *, '(2x,i8,a,1x,g16.8)' ) i, ':', a(i) end do return end subroutine r8vec_print_16 ( n, a, title ) !*****************************************************************************80 ! !! R8VEC_PRINT_16 prints an R8VEC to 16 decimal places. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 May 2014 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of components of the vector. ! ! Input, real ( kind = 8 ) A(N), the vector to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' do i = 1, n write ( *, '(2x,i8,a,1x,g24.16)' ) i, ':', a(i) end do return end subroutine r8vec_print_part ( n, a, i_lo, i_hi, title ) !*****************************************************************************80 ! !! R8VEC_PRINT_PART prints "part" of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 September 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of the vector. ! ! Input, real ( kind = 8 ) A(N), the vector to be printed. ! ! Input, integer ( kind = 4 ) I_LO, I_HI, the first and last indices ! to print. The routine expects 1 <= I_LO <= I_HI <= N. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) i_hi integer ( kind = 4 ) i_lo character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' do i = max ( i_lo, 1 ), min ( i_hi, n ) write ( *, '(2x,i8,a,1x,g16.8)' ) i, ':', a(i) end do return end subroutine r8vec_print_some ( n, a, max_print, title ) !*****************************************************************************80 ! !! R8VEC_PRINT_SOME prints "some" of an R8VEC. ! ! Discussion: ! ! The user specifies MAX_PRINT, the maximum number of lines to print. ! ! If N, the size of the vector, is no more than MAX_PRINT, then ! the entire vector is printed, one entry per line. ! ! Otherwise, if possible, the first MAX_PRINT-2 entries are printed, ! followed by a line of periods suggesting an omission, ! and the last entry. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 19 December 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of the vector. ! ! Input, real ( kind = 8 ) A(N), the vector to be printed. ! ! Input, integer ( kind = 4 ) MAX_PRINT, the maximum number of lines ! to print. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) max_print character ( len = * ) title if ( max_print <= 0 ) then return end if if ( n <= 0 ) then return end if write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' if ( n <= max_print ) then do i = 1, n write ( *, '(2x,i8,a,1x,g14.6)' ) i, ':', a(i) end do else if ( 3 <= max_print ) then do i = 1, max_print - 2 write ( *, '(2x,i8,a,1x,g14.6)' ) i, ':', a(i) end do write ( *, '(a)' ) ' ........ ..............' i = n write ( *, '(2x,i8,a,1x,g14.6)' ) i, ':', a(i) else do i = 1, max_print - 1 write ( *, '(2x,i8,a,1x,g14.6)' ) i, ':', a(i) end do i = max_print write ( *, '(2x,i8,a,1x,g14.6,2x,a)' ) i, ':', a(i), '...more entries...' end if return end subroutine r8vec_print2 ( n, a ) !*****************************************************************************80 ! !! R8VEC_PRINT2 prints out an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 26 March 2006 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of A. ! ! Input, real ( kind = 8 ) A(N), the vector to be printed. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) amax real ( kind = 8 ) amin integer ( kind = 4 ) i character ( len = 11 ) iform logical ( kind = 4 ) integ integer ( kind = 4 ) lmax real ( kind = 8 ) r8_log_10 ! ! Check if all entries are integral. ! integ = .true. do i = 1, n if ( a(i) /= real ( int ( a(i) ), kind = 8 ) ) then integ = .false. exit end if end do ! ! Find the range of the array. ! amax = maxval ( abs ( a(1:n) ) ) amin = minval ( abs ( a(1:n) ) ) ! ! Use the information about the maximum size of an entry to ! compute an intelligent format for use with integer entries. ! ! Later, we might also do this for real vectors. ! lmax = int ( r8_log_10 ( amax ) ) if ( integ ) then write ( iform, '( ''(2x,i'', i2, '')'' )' ) lmax + 3 else iform = ' ' end if do i = 1, n if ( integ ) then write ( *, iform ) int ( a(i) ) else write ( *, '(2x,g14.6)' ) a(i) end if end do return end function r8vec_product ( n, a ) !*****************************************************************************80 ! !! R8VEC_PRODUCT returns the product of the entries of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! In FORTRAN90, this facility is offered by the built in ! PRODUCT function: ! ! R8VEC_PRODUCT ( N, A ) = PRODUCT ( A(1:N) ) ! ! In MATLAB, this facility is offered by the built in ! PROD function: ! ! R8VEC_PRODUCT ( N, A ) = PROD ( A(1:N) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 17 September 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, real ( kind = 8 ) R8VEC_PRODUCT, the product of the entries. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_product r8vec_product = product ( a(1:n) ) return end subroutine r8vec_range ( n, x, xmin, xmax, y, ymin, ymax ) !*****************************************************************************80 ! !! R8VEC_RANGE finds the range of Y's within a restricted X range. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The routine is given a set of pairs of points (X,Y), and a range ! XMIN to XMAX of valid X values. Over this range, it seeks ! YMIN and YMAX, the minimum and maximum values of Y for ! valid X's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 May 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) X(N), the X array. ! ! Input, real ( kind = 8 ) XMIN, XMAX, the range of X values to check. ! ! Input, real ( kind = 8 ) Y(N), the Y array. ! ! Output, real ( kind = 8 ) YMIN, YMAX, the range of Y values whose ! X value is within the X range. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i real ( kind = 8 ) x(n) real ( kind = 8 ) xmax real ( kind = 8 ) xmin real ( kind = 8 ) y(n) real ( kind = 8 ) ymax real ( kind = 8 ) ymin ymin = huge ( ymin ) ymax = - huge ( ymax ) do i = 1, n if ( xmin <= x(i) .and. x(i) <= xmax ) then ymin = min ( ymin, y(i) ) ymax = max ( ymax, y(i) ) end if end do return end subroutine r8vec_range_2 ( n, a, amin, amax ) !*****************************************************************************80 ! !! R8VEC_RANGE_2 updates a range to include a new array. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Given a range AMIN to AMAX, and an array A, the routine will ! decrease AMIN if necessary, or increase AMAX if necessary, so that ! every entry of A is between AMIN and AMAX. ! ! However, AMIN will not be increased, nor AMAX decreased. ! ! This routine may be used to compute the maximum and minimum of a ! collection of arrays one at a time. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 November 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Input/output, real ( kind = 8 ) AMIN, AMAX. On input, the ! current legal range of values for A. On output, AMIN and AMAX ! are either unchanged, or else "widened" so that all entries ! of A are within the range. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) amax real ( kind = 8 ) amin amax = max ( amax, maxval ( a(1:n) ) ) amin = min ( amin, minval ( a(1:n) ) ) return end subroutine r8vec_reverse ( n, a ) !*****************************************************************************80 ! !! R8VEC_REVERSE reverses the elements of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! In FORTRAN90, calling R8VEC_REVERSE is equivalent to ! ! A(1:N) = A(N:1:-1) ! ! Example: ! ! Input: ! ! N = 5, ! A = ( 11.0, 12.0, 13.0, 14.0, 15.0 ). ! ! Output: ! ! A = ( 15.0, 14.0, 13.0, 12.0, 11.0 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 September 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input/output, real ( kind = 8 ) A(N), the array to be reversed. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) a(1:n) = a(n:1:-1) return end subroutine r8vec_rotate ( n, a, m ) !*****************************************************************************80 ! !! R8VEC_ROTATE "rotates" the entries of an R8VEC in place. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! This routine rotates an array of real "objects", but the same ! logic can be used to permute an array of objects of any arithmetic ! type, or an array of objects of any complexity. The only temporary ! storage required is enough to store a single object. The number ! of data movements made is N + the number of cycles of order 2 or more, ! which is never more than N + N/2. ! ! Example: ! ! Input: ! ! N = 5, M = 2 ! A = ( 1.0, 2.0, 3.0, 4.0, 5.0 ) ! ! Output: ! ! A = ( 4.0, 5.0, 1.0, 2.0, 3.0 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 20 March 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of objects. ! ! Input, integer ( kind = 4 ) M, the number of positions to the right that ! each element should be moved. Elements that shift pass position ! N "wrap around" to the beginning of the array. ! ! Input/output, real ( kind = 8 ) A(N), the array to be rotated. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i4_modp integer ( kind = 4 ) iget integer ( kind = 4 ) iput integer ( kind = 4 ) istart integer ( kind = 4 ) m integer ( kind = 4 ) mcopy integer ( kind = 4 ) nset real ( kind = 8 ) temp ! ! Force M to be positive, between 0 and N-1. ! mcopy = i4_modp ( m, n ) if ( mcopy == 0 ) then return end if istart = 0 nset = 0 do istart = istart + 1 if ( n < istart ) then exit end if temp = a(istart) iget = istart ! ! Copy the new value into the vacated entry. ! do iput = iget iget = iget - mcopy if ( iget < 1 ) then iget = iget + n end if if ( iget == istart ) then exit end if a(iput) = a(iget) nset = nset + 1 end do a(iput) = temp nset = nset + 1 if ( n <= nset ) then exit end if end do return end function r8vec_rsquared ( n, y_data, y_model ) !*****************************************************************************80 ! !! R8VEC_RSQUARED computes the R^2 goodness of fit measurement. ! ! Discussion: ! ! We suppose a set of N data values Y_DATA are known, and that a ! formula or model has generated a corresponding set of Y_MODEL values. ! ! R^2 measures the extent to which the variation in Y_DATA is captured ! by the model data Y_MODEL. If the model is linear, then a value ! of R^2=1 is optimal, and R^2=0 is the worst possible. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 January 2019 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of values. ! ! Input, real ( kind = 8 ) Y_DATA(N), Y_MODEL(N), the data and model values. ! ! Output, real ( kind = 8 ) R8VEC_RSQUARED, the R^2 value. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) bot real ( kind = 8 ) r8vec_rsquared real ( kind = 8 ) top real ( kind = 8 ) y_average real ( kind = 8 ) y_data(n) real ( kind = 8 ) y_model(n) y_average = sum ( y_data(1:n) ) / real ( n, kind = 8 ) top = sum ( y_data(1:n) - y_model(1:n) ) ** 2 bot = sum ( y_data(1:n) - y_average ) ** 2 r8vec_rsquared = 1.0D+00 - top / bot return end function r8vec_rsquared_adjusted ( n, y_data, y_model, degree ) !*****************************************************************************80 ! !! R8VEC_RSQUARED_ADJUSTED computes the adjusted R^2 goodness of fit measurement. ! ! Discussion: ! ! We suppose a set of N data values Y_DATA are known, and that a ! formula or model has generated a corresponding set of Y_MODEL values. ! ! R^2 measures the extent to which the variation in Y_DATA is captured ! by the model data Y_MODEL. ! ! The adjusted value of R^2 accounts for the use of a polynomial model ! of degree higher than 1. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 16 January 2019 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of values. ! ! Input, real ( kind = 8 ) Y_DATA(N), Y_MODEL(N), the data and model values. ! ! Input, integer ( kind = 4 ) DEGREE, the polynomial degree of the model. ! ! Output, real ( kind = 8 ) R8VEC_RSQUARED_ADJUSTED, the adjusted R^2 value. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) bot integer ( kind = 4 ) degree real ( kind = 8 ) r8vec_rsquared_adjusted real ( kind = 8 ) top real ( kind = 8 ) y_average real ( kind = 8 ) y_data(n) real ( kind = 8 ) y_model(n) y_average = sum ( y_data(1:n) ) / real ( n, kind = 8 ) top = sum ( y_data(1:n) - y_model(1:n) ) ** 2 bot = sum ( y_data(1:n) - y_average ) ** 2 r8vec_rsquared_adjusted = 1.0D+00 - ( top / bot ) & * real ( n - 1, kind = 8 ) / real ( n - degree - 1, kind = 8 ) return end function r8vec_scalar_triple_product ( v1, v2, v3 ) !*****************************************************************************80 ! !! R8VEC_SCALAR_TRIPLE_PRODUCT computes the scalar triple product. ! ! Discussion: ! ! STRIPLE = V1 dot ( V2 x V3 ). ! ! STRIPLE is the volume of the parallelogram whose sides are ! formed by V1, V2 and V3. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 12 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) V1(3), V2(3), V3(3), the three vectors. ! ! Output, real ( kind = 8 ) R8VEC_SCALAR_TRIPLE_PRODUCT, the scalar ! triple product. ! implicit none real ( kind = 8 ) r8vec_scalar_triple_product real ( kind = 8 ) v1(3) real ( kind = 8 ) v2(3) real ( kind = 8 ) v3(3) r8vec_scalar_triple_product = & v1(1) * ( v2(2) * v3(3) - v2(3) * v3(2) ) & + v1(2) * ( v2(3) * v3(1) - v2(1) * v3(3) ) & + v1(3) * ( v2(1) * v3(2) - v2(2) * v3(1) ) return end subroutine r8vec_scale ( s, n, x ) !*****************************************************************************80 ! !! R8VEC_SCALE multiplies an R8VEC by a scale factor. ! ! Discussion: ! ! An R8VEC is a vector of R8 values. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 October 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) S, the scale factor. ! ! Input, integer ( kind = 4 ) N, the length of the vector. ! ! Input/output, real ( kind = 8 ) X(N), the vector to be scaled. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) s real ( kind = 8 ) x(n) x(1:n) = s * x(1:n) return end subroutine r8vec_scale_01 ( n, x, xs ) !*****************************************************************************80 ! !! R8VEC_SCALE_01 scales an R8VEC to [0,1]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) X(N), the vector to be scaled. ! ! Output, real ( kind = 8 ) XS(N), the scaled vector. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) x(n) real ( kind = 8 ) xmax real ( kind = 8 ) xmin real ( kind = 8 ) xs(n) xs(1:n) = x(1:n) xmin = minval ( xs ) xmax = maxval ( xs ) if ( 0.0D+00 < xmax - xmin ) then xs = ( xs - xmin ) / ( xmax - xmin ) else xs = 0.5D+00 end if return end subroutine r8vec_scale_ab ( n, x, a, b, xs ) !*****************************************************************************80 ! !! R8VEC_SCALE_AB scales an R8VEC to [A,B]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) X(N), the vector to be scaled. ! ! Input, real ( kind = 8 ) A, B, the limits. ! ! Output, real ( kind = 8 ) XS(N), the scaled vector. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b real ( kind = 8 ) x(n) real ( kind = 8 ) xmax real ( kind = 8 ) xmin real ( kind = 8 ) xs(n) xs(1:n) = x(1:n) xmin = minval ( xs ) xmax = maxval ( xs ) if ( 0.0D+00 < xmax - xmin ) then xs = a + ( b - a ) * ( xs - xmin ) / ( xmax - xmin ) else xs = ( a + b ) / 2.0D+00 end if return end subroutine r8vec_search_binary_a ( n, a, aval, indx ) !*****************************************************************************80 ! !! R8VEC_SEARCH_BINARY_A searches an ascending sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Binary search is used. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 July 2000 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Donald Kreher, Douglas Simpson, ! Algorithm 1.9, ! Combinatorial Algorithms, ! CRC Press, 1998, page 26. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements in the array. ! ! Input, real ( kind = 8 ) A(N), the array to be searched. The array must ! be sorted in ascending order. ! ! Input, real ( kind = 8 ) AVAL, the value to be searched for. ! ! Output, integer ( kind = 4 ) INDX, the result of the search. ! -1, AVAL does not occur in the array. ! I, A(I) = AVAL. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) aval integer ( kind = 4 ) high integer ( kind = 4 ) indx integer ( kind = 4 ) low integer ( kind = 4 ) mid indx = -1 low = 1 high = n do while ( low <= high ) mid = ( low + high ) / 2 if ( a(mid) == aval ) then indx = mid exit else if ( a(mid) < aval ) then low = mid + 1 else if ( aval < a(mid) ) then high = mid - 1 end if end do return end subroutine r8vec_shift ( shift, n, x ) !*****************************************************************************80 ! !! R8VEC_SHIFT performs a shift on an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8 values. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 March 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) SHIFT, the amount by which each entry is to ! be shifted. ! ! Input, integer ( kind = 4 ) N, the length of the vector. ! ! Input/output, real ( kind = 8 ) X(N), the vector to be shifted. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) ihi integer ( kind = 4 ) ilo integer ( kind = 4 ) shift real ( kind = 8 ) x(n) real ( kind = 8 ) y(n) y(1:n) = x(1:n) x(1:n) = 0.0D+00 ilo = max ( 1, 1 + shift ) ihi = min ( n, n + shift ) x(ilo:ihi) = y(ilo-shift:ihi-shift) return end subroutine r8vec_shift_circular ( shift, n, x ) !*****************************************************************************80 ! !! R8VEC_SHIFT_CIRCULAR performs a circular shift on an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8 values. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 11 March 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) SHIFT, the amount by which each entry is to ! be shifted. ! ! Input, integer ( kind = 4 ) N, the length of the vector. ! ! Input/output, real ( kind = 8 ) X(N), the vector to be shifted. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) i4_wrap integer ( kind = 4 ) j integer ( kind = 4 ) shift real ( kind = 8 ) x(n) real ( kind = 8 ) y(n) y(1:n) = x(1:n) do i = 1, n j = i4_wrap ( i - shift, 1, n ) x(i) = y(j) end do return end subroutine r8vec_sign3_running ( n, v, s ) !*****************************************************************************80 ! !! R8VEC_SIGN3_RUNNING computes the running threeway sign of an R8VEC. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 22 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items. ! ! Input, real ( kind = 8 ) V(N), the data. ! ! Output, real ( kind = 8 ) S(N+1), the running threeway sign. S(I) is: ! -1.0, if the sum of the first I-1 values in V is negative ! 0.0, if zero ! +1.0, if positive. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i real ( kind = 8 ) s(n+1) real ( kind = 8 ) v(n) ! ! Sum. ! s(1) = 0.0D+00 do i = 2, n + 1 s(i) = s(i-1) + v(i-1) end do do i = 1, n + 1 if ( s(i) < 0.0D+00 ) then s(i) = -1.0D+00 else if ( s(i) == 0.0D+00 ) then s(i) = 0.0D+00 else if ( 0.0D+00 < s(i) ) then s(i) = +1.0D+00 end if end do return end subroutine r8vec_smooth ( n, x, s, z ) !*****************************************************************************80 ! !! r8vec_smooth smooths an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Except for the beginning and ending entries, the vector values ! are replaced by averages of 2*S+1 neighbors. ! ! Example: ! ! S = 2 ! ! Z(1) = X(1) ! Z(2) = ( X(1) + X(2) + X(3) ) / 3 ! Z(3) = ( X(1) + X(2) + X(3) + X(4) + X(5) ) / 5 ! Z(4) = ( X(2) + X(3) + X(4) + X(5) + X(6) ) / 5 ! ... ! Z(N-2) = ( X(N-4) + X(N-3) + X(N-2) + X(N-1) + X(N) ) / 5 ! Z(N-1) = ( X(N-2) + X(N-1) + X(N) ) / 3 ! Z(N) = X(N) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 April 2019 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the dimension of X. ! ! Input, real ( kind = 8 ) X(N), the vector to be smoothed. ! ! Output, real ( kind = 8 ) Z(N), the smoothed vector. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) j integer ( kind = 4 ) s real ( kind = 8 ) x(n) real ( kind = 8 ) y(n) real ( kind = 8 ) z(n) ! ! Work with temporary vector Y, so that user can identify X and Z. ! do j = 1, s y(j) = sum ( x(1:2*j-1) ) / real ( 2 * j - 1, kind = 8 ) end do do j = s + 1, n - s y(j) = sum ( x(j-s:j+s) ) / real ( 2 * s + 1, kind = 8 ) end do do j = s, 1, -1 y(n+1-j) = sum ( x(n+1-(2*j-1):n) ) / real ( 2 * j - 1, kind = 8 ) end do z(1:n) = y(1:n) return end subroutine r8vec_softmax ( n, x, s ) !*****************************************************************************80 ! !! R8VEC_SOFTMAX evaluates the SOFTMAX function. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 August 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the vector. ! ! Input, real ( kind = 8 ) X(N), the vector. ! ! Output, real ( kind = 8 ) VALUE(N), the function values. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) bottom integer ( kind = 4 ) i real ( kind = 8 ) s(n) real ( kind = 8 ) x(n) call r8vec_max_index ( n, x, i ) bottom = sum ( exp ( x(1:n) - x(i) ) ) s(1:n) = exp ( x(1:n) - x(i) ) / bottom return end subroutine r8vec_sort_bubble_a ( n, a ) !*****************************************************************************80 ! !! R8VEC_SORT_BUBBLE_A ascending sorts an R8VEC using bubble sort. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Bubble sort is simple to program, but inefficient. It should not ! be used for large arrays. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 February 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input/output, real ( kind = 8 ) A(N). ! On input, an unsorted array. ! On output, the array has been sorted. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) t do i = 1, n - 1 do j = i + 1, n if ( a(j) < a(i) ) then t = a(i) a(i) = a(j) a(j) = t end if end do end do return end subroutine r8vec_sort_bubble_d ( n, a ) !*****************************************************************************80 ! !! R8VEC_SORT_BUBBLE_D descending sorts an R8VEC using bubble sort. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Bubble sort is simple to program, but inefficient. It should not ! be used for large arrays. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 31 January 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input/output, real ( kind = 8 ) A(N). ! On input, an unsorted array. ! On output, the array has been sorted. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) t do i = 1, n - 1 do j = i + 1, n if ( a(i) < a(j) ) then t = a(i) a(i) = a(j) a(j) = t end if end do end do return end subroutine r8vec_sort_heap_a ( n, a ) !*****************************************************************************80 ! !! R8VEC_SORT_HEAP_A ascending sorts an R8VEC using heap sort. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 July 2003 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input/output, real ( kind = 8 ) A(N). ! On input, the array to be sorted; ! On output, the array has been sorted. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) n1 real ( kind = 8 ) temp if ( n <= 1 ) then return end if ! ! 1: Put A into descending heap form. ! call r8vec_heap_d ( n, a ) ! ! 2: Sort A. ! ! The largest object in the heap is in A(1). ! Move it to position A(N). ! temp = a(1) a(1) = a(n) a(n) = temp ! ! Consider the diminished heap of size N1. ! do n1 = n - 1, 2, -1 ! ! Restore the heap structure of A(1) through A(N1). ! call r8vec_heap_d ( n1, a ) ! ! Take the largest object from A(1) and move it to A(N1). ! temp = a(1) a(1) = a(n1) a(n1) = temp end do return end subroutine r8vec_sort_heap_d ( n, a ) !*****************************************************************************80 ! !! R8VEC_SORT_HEAP_D descending sorts an R8VEC using heap sort. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 July 2003 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input/output, real ( kind = 8 ) A(N). ! On input, the array to be sorted; ! On output, the array has been sorted. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) n1 real ( kind = 8 ) t if ( n <= 1 ) then return end if ! ! 1: Put A into ascending heap form. ! call r8vec_heap_a ( n, a ) ! ! 2: Sort A. ! ! The smallest object in the heap is in A(1). ! Move it to position A(N). ! t = a(1) a(1) = a(n) a(n) = t ! ! Consider the diminished heap of size N1. ! do n1 = n - 1, 2, -1 ! ! Restore the heap structure of A(1) through A(N1). ! call r8vec_heap_a ( n1, a ) ! ! Take the smallest object from A(1) and move it to A(N1). ! t = a(1) a(1) = a(n1) a(n1) = t end do return end subroutine r8vec_sort_heap_index_a ( n, a, indx ) !*****************************************************************************80 ! !! R8VEC_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The sorting is not actually carried out. Rather an index array is ! created which defines the sorting. This array may be used to sort ! or index the array, or to sort or index related arrays keyed on the ! original array. ! ! Once the index array is computed, the sorting can be carried out ! "implicitly: ! ! A(INDX(I:N)) is sorted, ! ! or explicitly, by the call ! ! call r8vec_permute ( n, indx, a ) ! ! after which A(1:N) is sorted. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 March 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), an array to be index-sorted. ! ! Output, integer ( kind = 4 ) INDX(N), the sort index. The ! I-th element of the sorted array is A(INDX(I)). ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) aval integer ( kind = 4 ) i integer ( kind = 4 ) indx(n) integer ( kind = 4 ) indxt integer ( kind = 4 ) ir integer ( kind = 4 ) j integer ( kind = 4 ) l if ( n < 1 ) then return end if do i = 1, n indx(i) = i end do if ( n == 1 ) then return end if l = n / 2 + 1 ir = n do if ( 1 < l ) then l = l - 1 indxt = indx(l) aval = a(indxt) else indxt = indx(ir) aval = a(indxt) indx(ir) = indx(1) ir = ir - 1 if ( ir == 1 ) then indx(1) = indxt exit end if end if i = l j = l + l do while ( j <= ir ) if ( j < ir ) then if ( a(indx(j)) < a(indx(j+1)) ) then j = j + 1 end if end if if ( aval < a(indx(j)) ) then indx(i) = indx(j) i = j j = j + j else j = ir + 1 end if end do indx(i) = indxt end do return end subroutine r8vec_sort_heap_index_d ( n, a, indx ) !*****************************************************************************80 ! !! R8VEC_SORT_HEAP_INDEX_D does an indexed heap descending sort of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The sorting is not actually carried out. Rather an index array is ! created which defines the sorting. This array may be used to sort ! or index the array, or to sort or index related arrays keyed on the ! original array. ! ! Once the index array is computed, the sorting can be carried out ! "implicitly: ! ! A(INDX(1:N)) is sorted, ! ! or explicitly, by the call ! ! call r8vec_permute ( n, indx, a ) ! ! after which A(1:N) is sorted. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 July 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), an array to be index-sorted. ! ! Output, integer ( kind = 4 ) INDX(N), the sort index. The ! I-th element of the sorted array is A(INDX(I)). ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) aval integer ( kind = 4 ) i integer ( kind = 4 ) indx(n) integer ( kind = 4 ) indxt integer ( kind = 4 ) ir integer ( kind = 4 ) j integer ( kind = 4 ) l if ( n < 1 ) then return end if do i = 1, n indx(i) = i end do if ( n == 1 ) then return end if l = n / 2 + 1 ir = n do if ( 1 < l ) then l = l - 1 indxt = indx(l) aval = a(indxt) else indxt = indx(ir) aval = a(indxt) indx(ir) = indx(1) ir = ir - 1 if ( ir == 1 ) then indx(1) = indxt exit end if end if i = l j = l + l do while ( j <= ir ) if ( j < ir ) then if ( a(indx(j+1)) < a(indx(j)) ) then j = j + 1 end if end if if ( a(indx(j)) < aval ) then indx(i) = indx(j) i = j j = j + j else j = ir + 1 end if end do indx(i) = indxt end do return end subroutine r8vec_sort_heap_mask_a ( n, a, mask_num, mask, indx ) !*****************************************************************************80 ! !! R8VEC_SORT_HEAP_MASK_A: indexed heap ascending sort of a masked R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! An array A is given. An array MASK of indices into A is given. ! The routine produces a vector INDX, which is a permutation of the ! entries of MASK, so that: ! ! A(MASK(INDX(I)) <= A(MASK(INDX(J)) ! ! whenever ! ! I <= J ! ! In other words, only the elements of A that are indexed by MASK ! are to be considered, and the only thing that happens is that ! a rearrangment of the indices in MASK is returned that orders the ! masked elements. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 25 September 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), an array to be index-sorted. ! ! Input, integer ( kind = 4 ) MASK_NUM, the number of mask elements. ! ! Input, integer ( kind = 4 ) MASK(MASK_NUM), the mask array. This is ! simply a list of indices of A. The entries of MASK should ! be unique, and each one should be between 1 and N. ! ! Output, integer ( kind = 4 ) INDX(MASK_NUM), the sort index. There are ! MASK_NUM elements of A selected by MASK. If we want to list those ! elements in order, then the I-th element is A(MASK(INDX(I))). ! implicit none integer ( kind = 4 ) mask_num integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) aval integer ( kind = 4 ) i integer ( kind = 4 ) indx(mask_num) integer ( kind = 4 ) indxt integer ( kind = 4 ) ir integer ( kind = 4 ) j integer ( kind = 4 ) l integer ( kind = 4 ) mask(mask_num) if ( n < 1 ) then return end if if ( mask_num < 1 ) then return end if if ( mask_num == 1 ) then indx(1) = 1 return end if call i4vec_indicator1 ( mask_num, indx ) l = mask_num / 2 + 1 ir = mask_num do if ( 1 < l ) then l = l - 1 indxt = indx(l) aval = a(mask(indxt)) else indxt = indx(ir) aval = a(mask(indxt)) indx(ir) = indx(1) ir = ir - 1 if ( ir == 1 ) then indx(1) = indxt exit end if end if i = l j = l + l do while ( j <= ir ) if ( j < ir ) then if ( a(mask(indx(j))) < a(mask(indx(j+1))) ) then j = j + 1 end if end if if ( aval < a(mask(indx(j))) ) then indx(i) = indx(j) i = j j = j + j else j = ir + 1 end if end do indx(i) = indxt end do return end subroutine r8vec_sort_insert_a ( n, a ) !*****************************************************************************80 ! !! R8VEC_SORT_INSERT_A ascending sorts an R8VEC using an insertion sort. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 July 2000 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Donald Kreher, Douglas Simpson, ! Algorithm 1.1, ! Combinatorial Algorithms, ! CRC Press, 1998, page 11. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items in the vector. ! N must be positive. ! ! Input/output, real ( kind = 8 ) A(N). ! On input, the array to be sorted; ! On output, the array has been sorted. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) x do i = 2, n x = a(i) j = i - 1 do while ( 1 <= j ) if ( a(j) <= x ) then exit end if a(j+1) = a(j) j = j - 1 end do a(j+1) = x end do return end subroutine r8vec_sort_insert_index_a ( n, a, indx ) !*****************************************************************************80 ! !! R8VEC_SORT_INSERT_INDEX_A ascending index sorts an R8VEC using insertion. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 February 2001 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Donald Kreher, Douglas Simpson, ! Algorithm 1.1, ! Combinatorial Algorithms, ! CRC Press, 1998, page 11. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items in the vector. ! N must be positive. ! ! Input, real ( kind = 8 ) A(N), the array to be sorted. ! ! Output, integer ( kind = 4 ) INDX(N), the sorted indices. The array ! is sorted when listed from A(INDX(1)) through A(INDX(N)). ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) indx(n) integer ( kind = 4 ) j real ( kind = 8 ) x if ( n < 1 ) then return end if do i = 1, n indx(i) = i end do do i = 2, n x = a(i) j = i - 1 do while ( 1 <= j ) if ( a(indx(j)) <= x ) then exit end if indx(j+1) = indx(j) j = j - 1 end do indx(j+1) = i end do return end subroutine r8vec_sort_insert_index_d ( n, a, indx ) !*****************************************************************************80 ! !! R8VEC_SORT_INSERT_INDEX_D descending index sorts an R8VEC using insertion. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 07 February 2001 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Donald Kreher, Douglas Simpson, ! Algorithm 1.1, ! Combinatorial Algorithms, ! CRC Press, 1998, page 11. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items in the vector. ! N must be positive. ! ! Input, real ( kind = 8 ) A(N), the array to be sorted. ! ! Output, integer ( kind = 4 ) INDX(N), the sorted indices. The array ! is sorted when listed from A(INDX(1)) through A(INDX(N)). ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) indx(n) integer ( kind = 4 ) j real ( kind = 8 ) x if ( n < 1 ) then return end if do i = 1, n indx(i) = i end do do i = 2, n x = a(i) j = i - 1 do while ( 1 <= j ) if ( x <= a(indx(j)) ) then exit end if indx(j+1) = indx(j) j = j - 1 end do indx(j+1) = i end do return end subroutine r8vec_sort_quick_a ( n, a ) !*****************************************************************************80 ! !! R8VEC_SORT_QUICK_A ascending sorts an R8VEC using quick sort. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Example: ! ! Input: ! ! N = 7 ! A = ( 6, 7, 3, 2, 9, 1, 8 ) ! ! Output: ! ! A = ( 1, 2, 3, 6, 7, 8, 9 ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input/output, real ( kind = 8 ) A(N). ! On input, the array to be sorted. ! On output, the array has been sorted. ! implicit none integer ( kind = 4 ), parameter :: level_max = 30 integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) base integer ( kind = 4 ) l_segment integer ( kind = 4 ) level integer ( kind = 4 ) n_segment integer ( kind = 4 ) rsave(level_max) integer ( kind = 4 ) r_segment if ( n < 1 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_SORT_QUICK_A - Fatal error!' write ( *, '(a)' ) ' N < 1.' stop 1 end if if ( n == 1 ) then return end if level = 1 rsave(level) = n + 1 base = 1 n_segment = n do ! ! Partition the segment. ! call r8vec_part_quick_a ( n_segment, a(base), l_segment, r_segment ) ! ! If the left segment has more than one element, we need to partition it. ! if ( 1 < l_segment ) then if ( level_max < level ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_SORT_QUICK_A - Fatal error!' write ( *, '(a,i8)' ) ' Exceeding recursion maximum of ', level_max stop 1 end if level = level + 1 n_segment = l_segment rsave(level) = r_segment + base - 1 ! ! The left segment and the middle segment are sorted. ! Must the right segment be partitioned? ! else if ( r_segment < n_segment ) then n_segment = n_segment + 1 - r_segment base = base + r_segment - 1 ! ! Otherwise, we back up a level if there is an earlier one. ! else do if ( level <= 1 ) then return end if base = rsave(level) n_segment = rsave(level-1) - rsave(level) level = level - 1 if ( 0 < n_segment ) then exit end if end do end if end do return end subroutine r8vec_sort_shell_a ( n, a ) !*****************************************************************************80 ! !! R8VEC_SORT_SHELL_A ascending sorts an R8VEC using Shell's sort. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 February 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input/output, real ( kind = 8 ) A(N). ! On input, an array to be sorted. ! On output, the sorted array. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) asave integer ( kind = 4 ) i integer ( kind = 4 ) ifree integer ( kind = 4 ) inc integer ( kind = 4 ) ipow integer ( kind = 4 ) j integer ( kind = 4 ) k integer ( kind = 4 ) maxpow if ( n <= 1 ) then return end if ! ! Determine the smallest MAXPOW so that ! N <= ( 3**MAXPOW - 1 ) / 2 ! maxpow = 1 do while ( 3 ** maxpow < 2 * n + 1 ) maxpow = maxpow + 1 end do if ( 1 < maxpow ) then maxpow = maxpow - 1 end if ! ! Now sort groups of size ( 3^IPOW - 1 ) / 2. ! do ipow = maxpow, 1, -1 inc = ( 3**ipow - 1 ) / 2 ! ! Sort the values with indices equal to K mod INC. ! do k = 1, inc ! ! Insertion sort of the items with index ! INC+K, 2*INC+K, 3*INC+K, ... ! do i = inc+k, n, inc asave = a(i) ifree = i j = i - inc do if ( j < 1 ) then exit end if if ( a(j) <= asave ) then exit end if ifree = j a(j+inc) = a(j) j = j - inc end do a(ifree) = asave end do end do end do return end subroutine r8vec_sort2_a ( n, x, y ) !*****************************************************************************80 ! !! R8VEC_SORT2_A ascending sorts an R8VEC and adjusts an associated R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The routine sorts the elements of X, and whenever ! an element of X is moved, the corresponding element of ! Y is moved in the same way. This action means that after ! the sorting, every element of X is still paired to the ! same Y value. ! ! If you have more than one array associated with X, or ! an integer array, or some other complication, you may want to ! look at doing an "indexed sort" instead. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 June 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, length of input array. ! ! Input/output, real ( kind = 8 ) X(N). On input, an unsorted array. ! On output, X has been sorted. ! ! Input/output, real ( kind = 8 ) Y(N), an array which is to be ! shifted corresponding to the shifts made in X. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) indx integer ( kind = 4 ) isgn integer ( kind = 4 ) j real ( kind = 8 ) t real ( kind = 8 ) x(n) real ( kind = 8 ) y(n) if ( n <= 1 ) then return end if i = 0 indx = 0 isgn = 0 j = 0 do call sort_heap_external ( n, indx, i, j, isgn ) if ( 0 < indx ) then t = x(i) x(i) = x(j) x(j) = t t = y(i) y(i) = y(j) y(j) = t else if ( indx < 0 ) then if ( x(i) <= x(j) ) then isgn = -1 else isgn = + 1 end if else if ( indx == 0 ) then exit end if end do return end subroutine r8vec_sorted_merge_a ( na, a, nb, b, nc, c ) !*****************************************************************************80 ! !! R8VEC_SORTED_MERGE_A merges two ascending sorted R8VEC's. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The elements of A and B should be sorted in ascending order. ! ! The elements in the output array C will also be in ascending order, ! and unique. ! ! The output vector C may share storage with A or B. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 April 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) NA, the dimension of A. ! ! Input, real ( kind = 8 ) A(NA), the first sorted array. ! ! Input, integer ( kind = 4 ) NB, the dimension of B. ! ! Input, real ( kind = 8 ) B(NB), the second sorted array. ! ! Output, integer ( kind = 4 ) NC, the number of elements in the output ! array. Note that C should usually be dimensioned at least NA+NB in the ! calling routine. ! ! Output, real ( kind = 8 ) C(NC), the merged unique sorted array. ! implicit none integer ( kind = 4 ) na integer ( kind = 4 ) nb real ( kind = 8 ) a(na) real ( kind = 8 ) b(nb) real ( kind = 8 ) c(na+nb) real ( kind = 8 ) d(na+nb) integer ( kind = 4 ) j integer ( kind = 4 ) ja integer ( kind = 4 ) jb integer ( kind = 4 ) na2 integer ( kind = 4 ) nb2 integer ( kind = 4 ) nc integer ( kind = 4 ) order na2 = na nb2 = nb ja = 0 jb = 0 nc = 0 call r8vec_order_type ( na2, a, order ) if ( order < 0 .or. 2 < order ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_SORTED_MERGE_A - Fatal error!' write ( *, '(a)' ) ' The input array A is not ascending sorted!' stop 1 end if call r8vec_order_type ( nb2, b, order ) if ( order < 0 .or. 2 < order ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_SORTED_MERGE_A - Fatal error!' write ( *, '(a)' ) ' The input array B is not ascending sorted!' stop 1 end if do ! ! If we've used up all the entries of A, stick the rest of B on the end. ! if ( na2 <= ja ) then do j = 1, nb2 - jb jb = jb + 1 if ( nc == 0 ) then nc = nc + 1 d(nc) = b(jb) else if ( d(nc) < b(jb) ) then nc = nc + 1 d(nc) = b(jb) end if end do c(1:nc) = d(1:nc) exit ! ! If we've used up all the entries of B, stick the rest of A on the end. ! else if ( nb2 <= jb ) then do j = 1, na2 - ja ja = ja + 1 if ( nc == 0 ) then nc = nc + 1 d(nc) = a(ja) else if ( d(nc) < a(ja) ) then nc = nc + 1 d(nc) = a(ja) end if end do c(1:nc) = d(1:nc) exit ! ! Otherwise, if the next entry of A is smaller, that's our candidate. ! else if ( a(ja+1) <= b(jb+1) ) then ja = ja + 1 if ( nc == 0 ) then nc = nc + 1 d(nc) = a(ja) else if ( d(nc) < a(ja) ) then nc = nc + 1 d(nc) = a(ja) end if ! ! ...or if the next entry of B is the smaller, consider that. ! else jb = jb + 1 if ( nc == 0 ) then nc = nc + 1 d(nc) = b(jb) else if ( d(nc) < b(jb) ) then nc = nc + 1 d(nc) = b(jb) end if end if end do return end function r8vec_sorted_nearest ( n, a, value ) !*****************************************************************************80 ! !! R8VEC_SORTED_NEAREST returns the nearest element in a sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 April 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input, real ( kind = 8 ) A(N), a sorted vector. ! ! Input, real ( kind = 8 ) VALUE, the value whose nearest vector ! entry is sought. ! ! Output, integer ( kind = 4 ) R8VEC_SORTED_NEAREST, the index of the nearest ! entry in the vector. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) r8vec_sorted_nearest integer ( kind = 4 ) hi integer ( kind = 4 ) lo integer ( kind = 4 ) mid real ( kind = 8 ) value if ( n < 1 ) then r8vec_sorted_nearest = -1 return end if if ( n == 1 ) then r8vec_sorted_nearest = 1 return end if if ( a(1) < a(n) ) then if ( value < a(1) ) then r8vec_sorted_nearest = 1 return else if ( a(n) < value ) then r8vec_sorted_nearest = n return end if ! ! Seek an interval containing the value. ! lo = 1 hi = n do while ( lo < hi - 1 ) mid = ( lo + hi ) / 2 if ( value == a(mid) ) then r8vec_sorted_nearest = mid return else if ( value < a(mid) ) then hi = mid else lo = mid end if end do ! ! Take the nearest. ! if ( abs ( value - a(lo) ) < abs ( value - a(hi) ) ) then r8vec_sorted_nearest = lo else r8vec_sorted_nearest = hi end if return ! ! A descending sorted vector A. ! else if ( value < a(n) ) then r8vec_sorted_nearest = n return else if ( a(1) < value ) then r8vec_sorted_nearest = 1 return end if ! ! Seek an interval containing the value. ! lo = n hi = 1 do while ( lo < hi - 1 ) mid = ( lo + hi ) / 2 if ( value == a(mid) ) then r8vec_sorted_nearest = mid return else if ( value < a(mid) ) then hi = mid else lo = mid end if end do ! ! Take the nearest. ! if ( abs ( value - a(lo) ) < abs ( value - a(hi) ) ) then r8vec_sorted_nearest = lo else r8vec_sorted_nearest = hi end if return end if return end subroutine r8vec_sorted_range ( n, r, r_lo, r_hi, i_lo, i_hi ) !*****************************************************************************80 ! !! R8VEC_SORTED_RANGE searches a sorted vector for elements in a range. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 September 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items in the vector. ! ! Input, real ( kind = 8 ) R(N), the sorted vector. ! ! Input, real ( kind = 8 ) R_LO, R_HI, the limits of the range. ! ! Output, integer ( kind = 4 ) I_LO, I_HI, the range of indices ! so that I_LO <= I <= I_HI => R_LO <= R(I) <= R_HI. If no ! values in R lie in the range, then I_HI < I_LO will be returned. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i_hi integer ( kind = 4 ) i_lo integer ( kind = 4 ) i1 integer ( kind = 4 ) i2 integer ( kind = 4 ) j1 integer ( kind = 4 ) j2 real ( kind = 8 ) r(n) real ( kind = 8 ) r_hi real ( kind = 8 ) r_lo ! ! Cases we can handle immediately. ! if ( r(n) < r_lo ) then i_lo = 0 i_hi = - 1 return end if if ( r_hi < r(1) ) then i_lo = 0 i_hi = - 1 return end if ! ! Are there are least two intervals? ! if ( n == 1 ) then if ( r_lo <= r(1) .and. r(1) <= r_hi ) then i_lo = 1 i_hi = 1 else i_lo = 0 i_hi = -1 end if return end if ! ! Bracket R_LO. ! if ( r_lo <= r(1) ) then i_lo = 1 else ! ! R_LO is in one of the intervals spanned by R(J1) to R(J2). ! Examine the intermediate interval [R(I1), R(I1+1)]. ! Does R_LO lie here, or below or above? ! j1 = 1 j2 = n i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 do if ( r_lo < r(i1) ) then j2 = i1 i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 else if ( r(i2) < r_lo ) then j1 = i2 i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 else i_lo = i1 exit end if end do end if ! ! Bracket R_HI ! if ( r(n) <= r_hi ) then i_hi = n else j1 = i_lo j2 = n i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 do if ( r_hi < r(i1) ) then j2 = i1 i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 else if ( r(i2) < r_hi ) then j1 = i2 i1 = ( j1 + j2 - 1 ) / 2 i2 = i1 + 1 else i_hi = i2 exit end if end do end if ! ! We expect to have computed the largest I_LO and smallest I_HI such that ! R(I_LO) <= R_LO <= R_HI <= R(I_HI) ! but what we want is actually ! R_LO <= R(I_LO) <= R(I_HI) <= R_HI ! which we can usually get simply by incrementing I_LO and decrementing I_HI. ! if ( r(i_lo) < r_lo ) then i_lo = i_lo + 1 if ( n < i_lo ) then i_hi = i_lo - 1 end if end if if ( r_hi < r(i_hi) ) then i_hi = i_hi - 1 if ( i_hi < 1 ) then i_lo = i_hi + 1 end if end if return end subroutine r8vec_sorted_split ( n, a, split, i_lt, i_gt ) !*****************************************************************************80 ! !! R8VEC_SORTED_SPLIT "splits" a sorted R8VEC, given a splitting value. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Given a splitting value SPLIT, the routine seeks indices ! I_LT and I_GT so that ! ! A(I_LT) < SPLIT < A(I_GT), ! ! and if there are intermediate index values between I_LT and ! I_GT, then those entries of A are exactly equal to SPLIT. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 November 2000 ! ! Author: ! ! John Burkardt ! ! Parameters ! ! Input, integer ( kind = 4 ) N, the number of entries in A. ! ! Input, real ( kind = 8 ) A(N), a sorted array. ! ! Input, real ( kind = 8 ) SPLIT, a value to which the entries in A are ! to be compared. ! ! Output, integer ( kind = 4 ) I_LT: ! 0 if no entries are less than SPLIT; ! N if all entries are less than SPLIT; ! otherwise, the index of the last entry in A less than SPLIT. ! ! Output, integer ( kind = 4 ) I_GT: ! 1 if all entries are greater than SPLIT; ! N+1 if no entries are greater than SPLIT; ! otherwise the index of the first entry in A greater than SPLIT. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) hi integer ( kind = 4 ) i integer ( kind = 4 ) i_gt integer ( kind = 4 ) i_lt integer ( kind = 4 ) lo integer ( kind = 4 ) mid real ( kind = 8 ) split if ( n < 1 ) then i_lt = -1 i_gt = -1 return end if if ( split < a(1) ) then i_lt = 0 i_gt = 1 return end if if ( a(n) < split ) then i_lt = n i_gt = n + 1 return end if lo = 1 hi = n do if ( lo + 1 == hi ) then i_lt = lo exit end if mid = ( lo + hi ) / 2 if ( split <= a(mid) ) then hi = mid else lo = mid end if end do do i = i_lt + 1, n if ( split < a(i) ) then i_gt = i return end if end do i_gt = n + 1 return end subroutine r8vec_sorted_undex ( x_num, x_val, x_unique_num, tol, undx, xdnu ) !*****************************************************************************80 ! !! R8VEC_SORTED_UNDEX returns unique sorted indexes for a sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The goal of this routine is to determine a vector UNDX, ! which points, to the unique elements of X, in sorted order, ! and a vector XDNU, which identifies, for each entry of X, the index of ! the unique sorted element of X. ! ! This is all done with index vectors, so that the elements of ! X are never moved. ! ! Assuming X is already sorted, we examine the entries of X in order, ! noting the unique entries, creating the entries of XDNU and ! UNDX as we go. ! ! Once this process has been completed, the vector X could be ! replaced by a compressed vector XU, containing the unique entries ! of X in sorted order, using the formula ! ! XU(I) = X(UNDX(I)). ! ! We could then, if we wished, reconstruct the entire vector X, or ! any element of it, by index, as follows: ! ! X(I) = XU(XDNU(I)). ! ! We could then replace X by the combination of XU and XDNU. ! ! Later, when we need the I-th entry of X, we can locate it as ! the XDNU(I)-th entry of XU. ! ! Here is an example of a vector X, the sort and inverse sort ! index vectors, and the unique sort and inverse unique sort vectors ! and the compressed unique sorted vector. ! ! Here is an example of a vector X, the unique sort and ! inverse unique sort vectors and the compressed unique sorted vector. ! ! I X XU Undx Xdnu ! ----+------+------+-----+-----+ ! 1 | 11.0 | 11.0 1 1 ! 2 | 11.0 | 22.0 5 1 ! 3 | 11.0 | 33.0 8 1 ! 4 | 11.0 | 55.0 9 1 ! 5 | 22.0 | 2 ! 6 | 22.0 | 2 ! 7 | 22.0 | 2 ! 8 | 33.0 | 3 ! 9 | 55.0 | ! ! INDX(2) = 3 means that sorted item(2) is X(3). ! XDNI(2) = 5 means that X(2) is sorted item(5). ! ! UNDX(3) = 4 means that unique sorted item(3) is at X(4). ! XDNU(8) = 2 means that X(8) is at unique sorted item(2). ! ! XU(XDNU(I))) = X(I). ! XU(I) = X(UNDX(I)). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 October 2008 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) X_NUM, the number of data values. ! ! Input, real ( kind = 8 ) X_VAL(X_NUM), the data values. ! ! Input, integer ( kind = 4 ) X_UNIQUE_NUM, the number of unique values ! in X_VAL. This value is only required for languages in which the size of ! UNDX must be known in advance. ! ! Input, real ( kind = 8 ) TOL, a tolerance for equality. ! ! Output, integer ( kind = 4 ) UNDX(X_UNIQUE_NUM), the UNDX vector. ! ! Output, integer ( kind = 4 ) XDNU(X_NUM), the XDNU vector. ! implicit none integer ( kind = 4 ) x_num integer ( kind = 4 ) x_unique_num integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) tol integer ( kind = 4 ) undx(x_unique_num) real ( kind = 8 ) x_val(x_num) integer ( kind = 4 ) xdnu(x_num) ! ! Walk through the sorted array X. ! i = 1 j = 1 undx(j) = i xdnu(i) = j do i = 2, x_num if ( tol < abs ( x_val(i) - x_val(undx(j)) ) ) then j = j + 1 undx(j) = i end if xdnu(i) = j end do return end subroutine r8vec_sorted_unique ( n, a, tol, unique_num ) !*****************************************************************************80 ! !! R8VEC_SORTED_UNIQUE keeps the unique elements in a sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 April 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input/output, real ( kind = 8 ) A(N). ! On input, the sorted array of N elements; ! On output, the sorted unique array of UNIQUE_NUM elements. ! ! Input, real ( kind = 8 ) TOL, a nonnegative tolerance for equality. ! Set it to 0.0 for the strictest test. ! ! Output, integer ( kind = 4 ) UNIQUE_NUM, the number of unique elements ! of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) unique_num real ( kind = 8 ) tol if ( n <= 0 ) then unique_num = 0 return end if unique_num = 1 do i = 2, n if ( tol < abs ( a(i) - a(unique_num) ) ) then unique_num = unique_num + 1 a(unique_num) = a(i) end if end do return end subroutine r8vec_sorted_unique_count ( n, a, tol, unique_num ) !*****************************************************************************80 ! !! R8VEC_SORTED_UNIQUE_COUNT counts the unique elements in a sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Because the array is sorted, this algorithm is O(N). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 April 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input, real ( kind = 8 ) A(N), the sorted array to examine. ! ! Input, real ( kind = 8 ) TOL, a nonnegative tolerance for equality. ! Set it to 0.0 for the strictest test. ! ! Output, integer ( kind = 4 ) UNIQUE_NUM, the number of unique elements ! of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) unique_num real ( kind = 8 ) tol if ( n < 1 ) then unique_num = 0 return end if unique_num = 1 do i = 2, n if ( tol < abs ( a(i-1) - a(i) ) ) then unique_num = unique_num + 1 end if end do return end subroutine r8vec_sorted_unique_hist ( n, a, tol, maxuniq, unique_num, & auniq, acount ) !*****************************************************************************80 ! !! R8VEC_SORTED_UNIQUE_HIST histograms the unique elements of a sorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 April 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input, real ( kind = 8 ) A(N), the array to examine. The elements of A ! should have been sorted. ! ! Input, real ( kind = 8 ) TOL, a nonnegative tolerance for equality. ! Set it to 0.0 for the strictest test. ! ! Input, integer ( kind = 4 ) MAXUNIQ, the maximum number of unique elements ! that can be handled. If there are more than MAXUNIQ unique ! elements in A, the excess will be ignored. ! ! Output, integer ( kind = 4 ) UNIQUE_NUM, the number of unique elements ! of A. ! ! Output, real ( kind = 8 ) AUNIQ(UNIQUE_NUM), the unique elements of A. ! ! Output, integer ( kind = 4 ) ACOUNT(UNIQUE_NUM), the number of times ! each element of AUNIQ occurs in A. ! implicit none integer ( kind = 4 ) maxuniq integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) acount(maxuniq) real ( kind = 8 ) auniq(maxuniq) integer ( kind = 4 ) i integer ( kind = 4 ) unique_num real ( kind = 8 ) tol ! ! Start taking statistics. ! unique_num = 0 do i = 1, n if ( i == 1 ) then unique_num = 1 auniq(unique_num) = a(1) acount(unique_num) = 1 else if ( abs ( a(i) - auniq(unique_num) ) <= tol ) then acount(unique_num) = acount(unique_num) + 1 else if ( unique_num < maxuniq ) then unique_num = unique_num + 1 auniq(unique_num) = a(i) acount(unique_num) = 1 end if end do return end subroutine r8vec_split ( n, a, split, isplit ) !*****************************************************************************80 ! !! R8VEC_SPLIT "splits" an unsorted R8VEC based on a splitting value. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! If the vector is already sorted, it is simpler to do a binary search ! on the data than to call this routine. ! ! The vector is not assumed to be sorted before input, and is not ! sorted during processing. If sorting is not needed, then it is ! more efficient to use this routine. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 21 October 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input/output, real ( kind = 8 ) A(N), the array to split. On output, ! all the entries of A that are less than or equal to SPLIT ! are in A(1:ISPLIT). ! ! Input, real ( kind = 8 ) SPLIT, the value used to split the vector. ! It is not necessary that any value of A actually equal SPLIT. ! ! Output, integer ( kind = 4 ) ISPLIT, indicates the position of the last ! entry of the split vector that is less than or equal to SPLIT. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) i1 integer ( kind = 4 ) i2 integer ( kind = 4 ) i3 integer ( kind = 4 ) isplit integer ( kind = 4 ) j1 integer ( kind = 4 ) j2 integer ( kind = 4 ) j3 real ( kind = 8 ) split ! ! Partition the vector into A1, A2, A3, where ! A1 = A(I1:J1) holds values <= SPLIT, ! A2 = A(I2:J2) holds untested values, ! A3 = A(I3:J3) holds values > SPLIT. ! i1 = 1 j1 = 0 i2 = 1 j2 = n i3 = n + 1 j3 = n ! ! Pick the next item from A2, and move it into A1 or A3. ! Adjust indices appropriately. ! do i = 1, n if ( a(i2) <= split ) then i2 = i2 + 1 j1 = j1 + 1 else call r8_swap ( a(i2), a(i3-1) ) i3 = i3 - 1 j2 = j2 - 1 end if end do isplit = j1 return end subroutine r8vec_standardize ( n, x, xs ) !*****************************************************************************80 ! !! R8VEC_STANDARDIZE standarizes an R8VEC. ! ! Discussion: ! ! The output vector will have 0 mean and unit standard deviation. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 October 2018 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) X(N), the vector to be standardized. ! ! Output, real ( kind = 8 ) XS(N), the standardized vector. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) mu real ( kind = 8 ) sigma real ( kind = 8 ) x(n) real ( kind = 8 ) xs(n) call r8vec_mean ( n, x, mu ) call r8vec_std_sample ( n, x, sigma ) if ( sigma /= 0.0D+00 ) then xs(1:n) = ( x(1:n) - mu ) / sigma else xs(1:n) = 0.0D+00 end if return end subroutine r8vec_std ( n, a, std ) !*****************************************************************************80 ! !! R8VEC_STD returns the standard deviation of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The standard deviation of a vector X of length N is defined as ! ! mean ( X(1:n) ) = sum ( X(1:n) ) / n ! ! std ( X(1:n) ) = sqrt ( sum ( ( X(1:n) - mean )^2 ) / n ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 February 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! N should be at least 2. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, real ( kind = 8 ) STD, the standard deviation of the vector. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) mean real ( kind = 8 ) std if ( n < 2 ) then std = 0.0D+00 else mean = sum ( a(1:n) ) / real ( n, kind = 8 ) std = sum ( ( a(1:n) - mean ) ** 2 ) std = sqrt ( std / real ( n, kind = 8 ) ) end if return end subroutine r8vec_std_sample ( n, a, std ) !*****************************************************************************80 ! !! R8VEC_STD_SAMPLE returns the sample standard deviation of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The standard deviation of a vector X of length N is defined as ! ! mean ( X(1:n) ) = sum ( X(1:n) ) / n ! ! std ( X(1:n) ) = sqrt ( sum ( ( X(1:n) - mean )^2 ) / ( n - 1 ) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 06 February 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! N should be at least 2. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, real ( kind = 8 ) STD, the sample standard deviation of the vector. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) mean real ( kind = 8 ) std if ( n < 2 ) then std = 0.0D+00 else mean = sum ( a(1:n) ) / real ( n, kind = 8 ) std = sum ( ( a(1:n) - mean ) ** 2 ) std = sqrt ( std / real ( n - 1, kind = 8 ) ) end if return end subroutine r8vec_std_update ( nm1, mean_nm1, std_nm1, xn, n, mean_n, & std_n ) !*****************************************************************************80 ! !! R8VEC_STD_UPDATE updates standard deviation of an R8VEC with one new value. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 November 2017 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) NM1, the number of entries in the old vector. ! ! Input, real ( kind = 8 ) MEAN_NM1, the mean of the old vector. ! ! Input, real ( kind = 8 ) STD_NM1, the standard deviation of the old vector. ! ! Input, real ( kind = 8 ) XN, the new N-th entry of the vector. ! ! Output, integer ( kind = 4 ) N, the number of entries in the new vector. ! ! Output, real ( kind = 8 ) MEAN_N, the mean of the new vector. ! ! Output, real ( kind = 8 ) STD_N, the standard deviation of the new vector. ! implicit none real ( kind = 8 ) mean_n real ( kind = 8 ) mean_nm1 integer ( kind = 4 ) n integer ( kind = 4 ) nm1 real ( kind = 8 ) std_n real ( kind = 8 ) std_nm1 real ( kind = 8 ) xn if ( nm1 <= 0 ) then n = 1 mean_n = xn std_n = 0.0D+00 else n = nm1 + 1 mean_n = mean_nm1 + ( xn - mean_nm1 ) / real ( n, kind = 8 ) std_n = sqrt ( ( std_nm1 ** 2 * real ( nm1, kind = 8 ) & + ( xn - mean_nm1 ) * ( xn - mean_n ) ) / real ( n, kind = 8 ) ) end if return end subroutine r8vec_step ( x0, n, x, fx ) !*****************************************************************************80 ! !! R8VEC_STEP evaluates a unit step function. ! ! Discussion: ! ! F(X) = 0 if X < X0 ! 1 if X0 <= X ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 May 2013 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) X0, the location of the jump. ! ! Input, integer ( kind = 4 ) N, the number of argument values. ! ! Output, real ( kind = 8 ) X(N), the arguments. ! ! Output, real ( kind = 8 ) FX(N), the function values. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) fx(n) real ( kind = 8 ) x(n) real ( kind = 8 ) x0 where ( x < x0 ) fx = 0.0D+00 else where fx = 1.0D+00 end where return end subroutine r8vec_stutter ( n, a, m, am ) !*****************************************************************************80 ! !! R8VEC_STUTTER makes a "stuttering" copy of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Applying a stuttering factor M of 3, the vector A = ( 1, 5, 8 ) becomes ! AM = ( 1, 1, 1, 5, 5, 5, 8, 8, 8 ). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 23 March 2011 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the size of the input vector. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Input, integer ( kind = 4 ) M, the "stuttering factor". ! ! Output, real ( kind = 8 ) AM(M*N), the stuttering vector. ! implicit none integer ( kind = 4 ) m integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) am(m*n) integer ( kind = 4 ) i integer ( kind = 4 ) jhi integer ( kind = 4 ) jlo do i = 1, n jlo = m * ( i - 1 ) + 1 jhi = m * i am(jlo:jhi) = a(i) end do return end function r8vec_sum ( n, a ) !*****************************************************************************80 ! !! R8VEC_SUM returns the sum of the entries of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! In FORTRAN90, this facility is offered by the built in ! SUM function: ! ! R8VEC_SUM ( N, A ) = SUM ( A(1:N) ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 09 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Output, real ( kind = 8 ) R8VEC_SUM, the sum of the entries. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) r8vec_sum r8vec_sum = sum ( a(1:n) ) return end subroutine r8vec_sum_running ( n, v, s ) !*****************************************************************************80 ! !! R8VEC_SUM_RUNNING computes the running sum of an R8VEC. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 22 February 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items. ! ! Input, real ( kind = 8 ) V(N), the data. ! ! Output, real ( kind = 8 ) S(N+1), the running sum. S(I) is the sum ! of the first I-1 values in V. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i real ( kind = 8 ) s(n+1) real ( kind = 8 ) v(n) ! ! Sum. ! s(1) = 0.0D+00 do i = 2, n + 1 s(i) = s(i-1) + v(i-1) end do return end subroutine r8vec_swap ( n, a1, a2 ) !*****************************************************************************80 ! !! R8VEC_SWAP swaps the entries of two R8VECs. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the arrays. ! ! Input/output, real ( kind = 8 ) A1(N), A2(N), the vectors to swap. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) real ( kind = 8 ) a3(n) a3(1:n) = a1(1:n) a1(1:n) = a2(1:n) a2(1:n) = a3(1:n) return end subroutine r8vec_transpose_print ( n, a, title ) !*****************************************************************************80 ! !! R8VEC_TRANSPOSE_PRINT prints an R8VEC "transposed". ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Example: ! ! A = (/ 1.0, 2.1, 3.2, 4.3, 5.4, 6.5, 7.6, 8.7, 9.8, 10.9, 11.0 /) ! TITLE = 'My vector: ' ! ! My vector: 1.0 2.1 3.2 4.3 5.4 ! 6.5 7.6 8.7 9.8 10.9 ! 11.0 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 August 2016 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of components of the vector. ! ! Input, real ( kind = 8 ) A(N), the vector to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) ihi integer ( kind = 4 ) ilo integer ( kind = 4 ) j character ( len = * ) title integer ( kind = 4 ) title_length title_length = len_trim ( title ) do ilo = 1, n, 5 if ( ilo == 1 ) then write ( *, '(a)', advance = 'NO' ) trim ( title ) else do i = 1, title_length write ( *, '(1x)', advance = 'NO' ) end do end if write ( *, '(2x)', advance = 'NO' ) ihi = min ( ilo + 5 - 1, n ) do j = ilo, ihi write ( *, '(g14.6)', advance = 'NO' ) a(j) end do write ( *, '(a)' ) end do return end subroutine r8vec_undex ( x_num, x_val, x_unique_num, tol, undx, xdnu ) !*****************************************************************************80 ! !! R8VEC_UNDEX returns unique sorted indexes for an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The goal of this routine is to determine a vector UNDX, ! which points, to the unique elements of X, in sorted order, ! and a vector XDNU, which identifies, for each entry of X, the index of ! the unique sorted element of X. ! ! This is all done with index vectors, so that the elements of ! X are never moved. ! ! The first step of the algorithm requires the indexed sorting ! of X, which creates arrays INDX and XDNI. (If all the entries ! of X are unique, then these arrays are the same as UNDX and XDNU.) ! ! We then use INDX to examine the entries of X in sorted order, ! noting the unique entries, creating the entries of XDNU and ! UNDX as we go. ! ! Once this process has been completed, the vector X could be ! replaced by a compressed vector XU, containing the unique entries ! of X in sorted order, using the formula ! ! XU(1:X_UNIQUE_NUM) = X(UNDX(1:X_UNIQUE_NUM)). ! ! We could then, if we wished, reconstruct the entire vector X, or ! any element of it, by index, as follows: ! ! X(I) = XU(XDNU(I)). ! ! We could then replace X by the combination of XU and XDNU. ! ! Later, when we need the I-th entry of X, we can locate it as ! the XDNU(I)-th entry of XU. ! ! Here is an example of a vector X, the sort and inverse sort ! index vectors, and the unique sort and inverse unique sort vectors ! and the compressed unique sorted vector. ! ! I X Indx Xdni XU Undx Xdnu ! ----+-----+-----+-----+--------+-----+-----+ ! 1 | 11. 1 1 | 11, 1 1 ! 2 | 22. 3 5 | 22, 2 2 ! 3 | 11. 6 2 | 33, 4 1 ! 4 | 33. 9 8 | 55, 5 3 ! 5 | 55. 2 9 | 4 ! 6 | 11. 7 3 | 1 ! 7 | 22. 8 6 | 2 ! 8 | 22. 4 7 | 2 ! 9 | 11. 5 4 | 1 ! ! INDX(2) = 3 means that sorted item(2) is X(3). ! XDNI(2) = 5 means that X(2) is sorted item(5). ! ! UNDX(3) = 4 means that unique sorted item(3) is at X(4). ! XDNU(8) = 2 means that X(8) is at unique sorted item(2). ! ! XU(XDNU(I))) = X(I). ! XU(I) = X(UNDX(I)). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 October 2008 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) X_NUM, the number of data values. ! ! Input, real ( kind = 8 ) X_VAL(X_NUM), the data values. ! ! Input, integer ( kind = 4 ) X_UNIQUE_NUM, the number of unique values ! in X_VAL. This value is only required for languages in which the size of ! UNDX must be known in advance. ! ! Input, real ( kind = 8 ) TOL, a tolerance for equality. ! ! Output, integer ( kind = 4 ) UNDX(X_UNIQUE_NUM), the UNDX vector. ! ! Output, integer ( kind = 4 ) XDNU(X_NUM), the XDNU vector. ! implicit none integer ( kind = 4 ) x_num integer ( kind = 4 ) x_unique_num integer ( kind = 4 ) i integer ( kind = 4 ) indx(x_num) integer ( kind = 4 ) j real ( kind = 8 ) tol integer ( kind = 4 ) undx(x_unique_num) real ( kind = 8 ) x_val(x_num) integer ( kind = 4 ) xdnu(x_num) ! ! Implicitly sort the array. ! call r8vec_sort_heap_index_a ( x_num, x_val, indx ) ! ! Walk through the implicitly sorted array X. ! i = 1 j = 1 undx(j) = indx(i) xdnu(indx(i)) = j do i = 2, x_num if ( tol < abs ( x_val(indx(i)) - x_val(undx(j)) ) ) then j = j + 1 undx(j) = indx(i) end if xdnu(indx(i)) = j end do return end subroutine r8vec_uniform_01 ( n, seed, r ) !*****************************************************************************80 ! !! R8VEC_UNIFORM_01 returns a unit pseudorandom R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 August 2014 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Paul Bratley, Bennett Fox, Linus Schrage, ! A Guide to Simulation, ! Springer Verlag, pages 201-202, 1983. ! ! Bennett Fox, ! Algorithm 647: ! Implementation and Relative Efficiency of Quasirandom ! Sequence Generators, ! ACM Transactions on Mathematical Software, ! Volume 12, Number 4, pages 362-376, 1986. ! ! Peter Lewis, Allen Goodman, James Miller ! A Pseudo-Random Number Generator for the System/360, ! IBM Systems Journal, ! Volume 8, pages 136-143, 1969. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input/output, integer ( kind = 4 ) SEED, the "seed" value, which ! should NOT be 0. On output, SEED has been updated. ! ! Output, real ( kind = 8 ) R(N), the vector of pseudorandom values. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ), parameter :: i4_huge = 2147483647 integer ( kind = 4 ) k integer ( kind = 4 ) seed real ( kind = 8 ) r(n) if ( seed == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_UNIFORM_01 - Fatal error!' write ( *, '(a)' ) ' Input value of SEED = 0.' stop 1 end if do i = 1, n k = seed / 127773 seed = 16807 * ( seed - k * 127773 ) - k * 2836 if ( seed < 0 ) then seed = seed + i4_huge end if r(i) = real ( seed, kind = 8 ) * 4.656612875D-10 end do return end subroutine r8vec_uniform_ab ( n, a, b, seed, r ) !*****************************************************************************80 ! !! R8VEC_UNIFORM_AB returns a scaled pseudorandom R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Each dimension ranges from A to B. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 31 May 2007 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Paul Bratley, Bennett Fox, Linus Schrage, ! A Guide to Simulation, ! Second Edition, ! Springer, 1987, ! ISBN: 0387964673, ! LC: QA76.9.C65.B73. ! ! Bennett Fox, ! Algorithm 647: ! Implementation and Relative Efficiency of Quasirandom ! Sequence Generators, ! ACM Transactions on Mathematical Software, ! Volume 12, Number 4, December 1986, pages 362-376. ! ! Pierre L'Ecuyer, ! Random Number Generation, ! in Handbook of Simulation, ! edited by Jerry Banks, ! Wiley, 1998, ! ISBN: 0471134031, ! LC: T57.62.H37. ! ! Peter Lewis, Allen Goodman, James Miller, ! A Pseudo-Random Number Generator for the System/360, ! IBM Systems Journal, ! Volume 8, Number 2, 1969, pages 136-143. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A, B, the lower and upper limits. ! ! Input/output, integer ( kind = 4 ) SEED, the "seed" value, which ! should NOT be 0. On output, SEED has been updated. ! ! Output, real ( kind = 8 ) R(N), the vector of pseudorandom values. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a real ( kind = 8 ) b integer ( kind = 4 ) i integer ( kind = 4 ), parameter :: i4_huge = 2147483647 integer ( kind = 4 ) k integer ( kind = 4 ) seed real ( kind = 8 ) r(n) if ( seed == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_UNIFORM_AB - Fatal error!' write ( *, '(a)' ) ' Input value of SEED = 0.' stop 1 end if do i = 1, n k = seed / 127773 seed = 16807 * ( seed - k * 127773 ) - k * 2836 if ( seed < 0 ) then seed = seed + i4_huge end if r(i) = a + ( b - a ) * real ( seed, kind = 8 ) * 4.656612875D-10 end do return end subroutine r8vec_uniform_abvec ( n, a, b, seed, r ) !*****************************************************************************80 ! !! R8VEC_UNIFORM_ABVEC returns a scaled pseudorandom R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Dimension I ranges from A(I) to B(I). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 01 October 2012 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Paul Bratley, Bennett Fox, Linus Schrage, ! A Guide to Simulation, ! Second Edition, ! Springer, 1987, ! ISBN: 0387964673, ! LC: QA76.9.C65.B73. ! ! Bennett Fox, ! Algorithm 647: ! Implementation and Relative Efficiency of Quasirandom ! Sequence Generators, ! ACM Transactions on Mathematical Software, ! Volume 12, Number 4, December 1986, pages 362-376. ! ! Pierre L'Ecuyer, ! Random Number Generation, ! in Handbook of Simulation, ! edited by Jerry Banks, ! Wiley, 1998, ! ISBN: 0471134031, ! LC: T57.62.H37. ! ! Peter Lewis, Allen Goodman, James Miller, ! A Pseudo-Random Number Generator for the System/360, ! IBM Systems Journal, ! Volume 8, Number 2, 1969, pages 136-143. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) A(N), B(N), the lower and upper limits ! for each dimension. ! ! Input/output, integer ( kind = 4 ) SEED, the "seed" value, which ! should NOT be 0. On output, SEED has been updated. ! ! Output, real ( kind = 8 ) R(N), the vector of pseudorandom values. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) b(n) integer ( kind = 4 ) i integer ( kind = 4 ), parameter :: i4_huge = 2147483647 integer ( kind = 4 ) k integer ( kind = 4 ) seed real ( kind = 8 ) r(n) if ( seed == 0 ) then write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'R8VEC_UNIFORM_ABVEC - Fatal error!' write ( *, '(a)' ) ' Input value of SEED = 0.' stop 1 end if do i = 1, n k = seed / 127773 seed = 16807 * ( seed - k * 127773 ) - k * 2836 if ( seed < 0 ) then seed = seed + i4_huge end if r(i) = a(i) + ( b(i) - a(i) ) * real ( seed, kind = 8 ) * 4.656612875D-10 end do return end subroutine r8vec_uniform_unit ( m, seed, w ) !*****************************************************************************80 ! !! R8VEC_UNIFORM_UNIT generates a uniformly random unit vector. ! ! Discussion: ! ! An R8MAT is an MxN array of R8's, stored by (I,J) -> [I+J*M]. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 04 October 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) M, the spatial dimension. ! ! Input/output, integer ( kind = 4 ) SEED, a seed for the random number ! generator. ! ! Output, real ( kind = 8 ) W(M), a random direction vector, ! with unit norm. ! implicit none integer ( kind = 4 ) m real ( kind = 8 ) norm integer ( kind = 4 ) seed real ( kind = 8 ) w(m) ! ! Get M values from a standard normal distribution. ! call r8vec_normal_01 ( m, seed, w ) ! ! Compute the length of the vector. ! norm = sqrt ( sum ( w(1:m) ** 2 ) ) ! ! Normalize the vector. ! w(1:m) = w(1:m) / norm return end subroutine r8vec_unique_count ( n, a, tol, unique_num ) !*****************************************************************************80 ! !! R8VEC_UNIQUE_COUNT counts the unique elements in an unsorted R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Because the array is unsorted, this algorithm is O(N^2). ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 08 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input, real ( kind = 8 ) A(N), the unsorted array to examine. ! ! Input, real ( kind = 8 ) TOL, a nonnegative tolerance for equality. ! Set it to 0.0 for the strictest test. ! ! Output, integer ( kind = 4 ) UNIQUE_NUM, the number of unique elements ! of A. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) j integer ( kind = 4 ) unique_num real ( kind = 8 ) tol unique_num = 0 do i = 1, n unique_num = unique_num + 1 do j = 1, i - 1 if ( abs ( a(i) - a(j) ) <= tol ) then unique_num = unique_num - 1 exit end if end do end do return end subroutine r8vec_unique_index ( n, a, tol, unique_index ) !*****************************************************************************80 ! !! R8VEC_UNIQUE_INDEX indexes the unique occurrence of values in an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! For element A(I) of the vector, UNIQUE_INDEX(I) is the uniqueness index ! of A(I). That is, if A_UNIQUE contains the unique elements of A, ! gathered in order, then ! ! A_UNIQUE ( UNIQUE_INDEX(I) ) = A(I) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 24 August 2008 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of elements of A. ! ! Input, real ( kind = 8 ) A(N), the array. ! ! Input, real ( kind = 8 ) TOL, a tolerance for equality. ! ! Output, integer ( kind = 4 ) UNIQUE_INDEX(N), the unique index. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) integer ( kind = 4 ) i integer ( kind = 4 ) j real ( kind = 8 ) tol integer ( kind = 4 ) unique_index(n) integer ( kind = 4 ) unique_num unique_index(1:n) = -1 unique_num = 0 do i = 1, n if ( unique_index(i) == -1 ) then unique_num = unique_num + 1 unique_index(i) = unique_num do j = i + 1, n if ( abs ( a(i) - a(j) ) <= tol ) then unique_index(j) = unique_num end if end do end if end do return end subroutine r8vec_variance ( n, a, variance ) !*****************************************************************************80 ! !! R8VEC_VARIANCE returns the variance of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The variance of a vector X of length N is defined as ! ! mean ( X(1:n) ) = sum ( X(1:n) ) / n ! ! var ( X(1:n) ) = sum ( ( X(1:n) - mean )^2 ) / n ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 30 November 2017 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! N should be at least 2. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, real ( kind = 8 ) VARIANCE, the variance of the vector. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) mean real ( kind = 8 ) variance if ( n < 2 ) then variance = 0.0D+00 else mean = sum ( a(1:n) ) / real ( n, kind = 8 ) variance = sum ( ( a(1:n) - mean ) ** 2 ) variance = variance / real ( n, kind = 8 ) end if return end subroutine r8vec_variance_circular ( n, x, variance_circular ) !*****************************************************************************80 ! !! R8VEC_VARIANCE_CIRCULAR returns the circular variance of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Input, real ( kind = 8 ) X(N), the vector whose variance is desired. ! ! Output, real ( kind = 8 ) VARIANCE_CIRCULAR, the circular variance ! of the vector entries. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) mean real ( kind = 8 ) variance_circular real ( kind = 8 ) x(n) call r8vec_mean ( n, x, mean ) variance_circular = & ( sum ( cos ( x(1:n) - mean ) ) ) ** 2 & + ( sum ( sin ( x(1:n) - mean ) ) ) ** 2 variance_circular = sqrt ( variance_circular ) / real ( n, kind = 8 ) variance_circular = 1.0D+00 - variance_circular return end subroutine r8vec_variance_sample ( n, a, variance ) !*****************************************************************************80 ! !! R8VEC_VARIANCE_SAMPLE returns the sample_variance of an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! The variance of a vector X of length N is defined as ! ! mean ( X(1:n) ) = sum ( X(1:n) ) / n ! ! var ( X(1:n) ) = sum ( ( X(1:n) - mean )^2 ) / ( n - 1 ) ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 14 February 1999 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! N should be at least 2. ! ! Input, real ( kind = 8 ) A(N), the vector. ! ! Output, real ( kind = 8 ) VARIANCE, the sample variance of the vector. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) mean real ( kind = 8 ) variance if ( n < 2 ) then variance = 0.0D+00 else mean = sum ( a(1:n) ) / real ( n, kind = 8 ) variance = sum ( ( a(1:n) - mean ) ** 2 ) variance = variance / real ( n - 1, kind = 8 ) end if return end subroutine r8vec_variance_update ( nm1, mean_nm1, variance_nm1, xn, n, mean_n, & variance_n ) !*****************************************************************************80 ! !! R8VEC_VARIANCE_UPDATE updates the variance of an R8VEC with one new value. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 29 November 2017 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) NM1, the number of entries in the old vector. ! ! Input, real ( kind = 8 ) MEAN_NM1, the mean of the old vector. ! ! Input, real ( kind = 8 ) VARIANCE_NM1, the variance of the old vector. ! ! Input, real ( kind = 8 ) XN, the new N-th entry of the vector. ! ! Output, integer ( kind = 4 ) N, the number of entries in the new vector. ! ! Output, real ( kind = 8 ) MEAN_N, the mean of the new vector. ! ! Output, real ( kind = 8 ) VARIANCE_N, the variance of the new vector. ! implicit none real ( kind = 8 ) mean_n real ( kind = 8 ) mean_nm1 integer ( kind = 4 ) n integer ( kind = 4 ) nm1 real ( kind = 8 ) variance_n real ( kind = 8 ) variance_nm1 real ( kind = 8 ) xn if ( nm1 <= 0 ) then n = 1 mean_n = xn variance_n = 0.0D+00 else n = nm1 + 1 mean_n = mean_nm1 + ( xn - mean_nm1 ) / real ( n, kind = 8 ) variance_n = ( variance_nm1 * real ( nm1, kind = 8 ) & + ( xn - mean_nm1 ) * ( xn - mean_n ) ) / real ( n, kind = 8 ) end if return end subroutine r8vec_vector_triple_product ( v1, v2, v3, v ) !*****************************************************************************80 ! !! R8VEC_VECTOR_TRIPLE_PRODUCT computes the vector triple product. ! ! Discussion: ! ! VTRIPLE = V1 x ( V2 x V3 ) ! ! VTRIPLE is a vector perpendicular to V1, lying in the plane ! spanned by V2 and V3. The norm of VTRIPLE is the product ! of the norms of V1, V2 and V3. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 27 October 2010 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, real ( kind = 8 ) V1(3), V2(3), V3(3), three vectors. ! ! Output, real ( kind = 8 ) V(3), the vector triple product. ! implicit none real ( kind = 8 ) v(3) real ( kind = 8 ) v1(3) real ( kind = 8 ) v2(3) real ( kind = 8 ) v3(3) real ( kind = 8 ) v4(3) call r8vec_cross_product_3d ( v2, v3, v4 ) call r8vec_cross_product_3d ( v1, v4, v ) return end subroutine r8vec_write ( n, r, output_file ) !*****************************************************************************80 ! !! R8VEC_WRITE writes an R8VEC to a file. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 31 July 2006 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the order of the matrix. ! ! Input, real ( kind = 8 ) R(N), the vector to be written. ! ! Input, character ( len = * ) OUTPUT_FILE, the name of the file to which ! the information is to be written. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i character ( len = * ) output_file integer ( kind = 4 ) output_unit real ( kind = 8 ) r(n) call get_unit ( output_unit ) open ( unit = output_unit, file = output_file, status = 'replace' ) do i = 1, n write ( output_unit, '(2x,g16.8)' ) r(i) end do close ( unit = output_unit ) return end subroutine r8vec_zeros ( n, a ) !*****************************************************************************80 ! !! R8VEC_ZEROS zeroes out an R8VEC. ! ! Discussion: ! ! An R8VEC is a vector of R8's. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 March 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the vector. ! ! Output, real ( kind = 8 ) A(N), the vector to be zeroed. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) a(1:n) = 0.0D+00 return end subroutine r8vec2_compare ( n, a1, a2, i, j, isgn ) !*****************************************************************************80 ! !! R8VEC2_COMPARE compares two entries in an R8VEC2. ! ! Discussion: ! ! An R8VEC2 is a dataset consisting of N pairs of R8's, stored ! as two separate vectors A1 and A2. ! ! The lexicographic ordering is used. ! ! Example: ! ! A1(I) A2(I) A1(J) A2(J) ISGN ! ----------- ----------- ---- ! 1.0 5.0 < 1.0 6.0 -1 ! 1.0 5.0 < 2.0 8.0 -1 ! 1.0 5.0 < 9.0 1.0 -1 ! 1.0 5.0 = 1.0 5.0 0 ! 1.0 5.0 > 0.0 2.0 +1 ! 1.0 5.0 > 0.0 5.0 +1 ! 1.0 5.0 > 1.0 3.0 +1 ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of data items. ! ! Input, real ( kind = 8 ) A1(N), A2(N), the two components of each item. ! ! Input, integer ( kind = 4 ) I, J, the items to be compared. ! ! Output, integer ( kind = 4 ) ISGN, the results of the comparison: ! -1, item I < item J, ! 0, item I = item J, ! +1, item I > item J. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) integer ( kind = 4 ) i integer ( kind = 4 ) isgn integer ( kind = 4 ) j isgn = 0 if ( a1(i) < a1(j) ) then isgn = -1 else if ( a1(i) == a1(j) ) then if ( a2(i) < a2(j) ) then isgn = -1 else if ( a2(i) < a2(j) ) then isgn = 0 else if ( a2(j) < a2(i) ) then isgn = +1 end if else if ( a1(j) < a1(i) ) then isgn = +1 end if return end subroutine r8vec2_print ( n, a1, a2, title ) !*****************************************************************************80 ! !! R8VEC2_PRINT prints an R8VEC2. ! ! Discussion: ! ! An R8VEC2 is a dataset consisting of N pairs of R8's, stored ! as two separate vectors A1 and A2. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of components of the vector. ! ! Input, real ( kind = 8 ) A1(N), A2(N), the vectors to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) integer ( kind = 4 ) i character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' do i = 1, n write ( *, '(2x,i4,2x,g14.6,2x,g14.6)' ) i, a1(i), a2(i) end do return end subroutine r8vec2_print_some ( n, x1, x2, max_print, title ) !*****************************************************************************80 ! !! R8VEC2_PRINT_SOME prints "some" of an R8VEC2. ! ! Discussion: ! ! An R8VEC2 is a dataset consisting of N pairs of R8's, stored ! as two separate vectors A1 and A2. ! ! The user specifies MAX_PRINT, the maximum number of lines to print. ! ! If N, the size of the vectors, is no more than MAX_PRINT, then ! the entire vectors are printed, one entry of each per line. ! ! Otherwise, if possible, the first MAX_PRINT-2 entries are printed, ! followed by a line of periods suggesting an omission, ! and the last entry. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 10 September 2009 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries of the vectors. ! ! Input, real ( kind = 8 ) X1(N), X2(N), the vector to be printed. ! ! Input, integer ( kind = 4 ) MAX_PRINT, the maximum number of lines ! to print. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) max_print character ( len = * ) title real ( kind = 8 ) x1(n) real ( kind = 8 ) x2(n) if ( max_print <= 0 ) then return end if if ( n <= 0 ) then return end if write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' if ( n <= max_print ) then do i = 1, n write ( *, '(2x,i8,2x,g14.6,2x,g14.6)' ) i, x1(i), x2(i) end do else if ( 3 <= max_print ) then do i = 1, max_print - 2 write ( *, '(2x,i8,2x,g14.6,2x,g14.6)' ) i, x1(i), x2(i) end do write ( *, '(a)' ) ' ...... .............. ..............' i = n write ( *, '(2x,i8,2x,g14.6,2x,g14.6)' ) i, x1(i), x2(i) else do i = 1, max_print - 1 write ( *, '(2x,i8,2x,g14.6,2x,g14.6)' ) i, x1(i), x2(i) end do i = max_print write ( *, '(2x,i8,2x,g14.6,2x,g14.6,2x,a)' ) i, x1(i), x2(i), & '...more entries...' end if return end subroutine r8vec2_sort_a ( n, a1, a2 ) !*****************************************************************************80 ! !! R8VEC2_SORT_A ascending sorts an R8VEC2. ! ! Discussion: ! ! An R8VEC2 is a dataset consisting of N pairs of R8's, stored ! as two separate vectors A1 and A2. ! ! Each item to be sorted is a pair (I,J), with the I ! and J values stored in separate vectors A1 and A2. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items of data. ! ! Input/output, real ( kind = 8 ) A1(N), A2(N), the data to be sorted. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) integer ( kind = 4 ) i integer ( kind = 4 ) indx integer ( kind = 4 ) isgn integer ( kind = 4 ) j if ( n <= 1 ) then return end if ! ! Initialize. ! i = 0 indx = 0 isgn = 0 j = 0 ! ! Call the external heap sorter. ! do call sort_heap_external ( n, indx, i, j, isgn ) ! ! Interchange the I and J objects. ! if ( 0 < indx ) then call r8_swap ( a1(i), a1(j) ) call r8_swap ( a2(i), a2(j) ) ! ! Compare the I and J objects. ! else if ( indx < 0 ) then call r8vec2_compare ( n, a1, a2, i, j, isgn ) else if ( indx == 0 ) then exit end if end do return end subroutine r8vec2_sort_d ( n, a1, a2 ) !*****************************************************************************80 ! !! R8VEC2_SORT_D descending sorts an R8VEC2. ! ! Discussion: ! ! An R8VEC2 is a dataset consisting of N pairs of R8's, stored ! as two separate vectors A1 and A2. ! ! Each item to be sorted is a pair (I,J), with the I ! and J values stored in separate vectors A1 and A2. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items of data. ! ! Input/output, real ( kind = 8 ) A1(N), A2(N), the data to be sorted. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) integer ( kind = 4 ) i integer ( kind = 4 ) indx integer ( kind = 4 ) isgn integer ( kind = 4 ) j if ( n <= 1 ) then return end if ! ! Initialize. ! i = 0 indx = 0 isgn = 0 j = 0 ! ! Call the external heap sorter. ! do call sort_heap_external ( n, indx, i, j, isgn ) ! ! Interchange the I and J objects. ! if ( 0 < indx ) then call r8_swap ( a1(i), a1(j) ) call r8_swap ( a2(i), a2(j) ) ! ! Compare the I and J objects. ! Reverse the value of ISGN to effect a descending sort. ! else if ( indx < 0 ) then call r8vec2_compare ( n, a1, a2, i, j, isgn ) isgn = -isgn else if ( indx == 0 ) then exit end if end do return end subroutine r8vec2_sort_heap_index_a ( n, x, y, indx ) !*****************************************************************************80 ! !! R8VEC2_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R8VEC2. ! ! Discussion: ! ! An R8VEC2 is a dataset consisting of N pairs of R8's, stored ! as two separate vectors A1 and A2. ! ! The sorting is not actually carried out. Rather an index array is ! created which defines the sorting. This array may be used to sort ! or index the array, or to sort or index related arrays keyed on the ! original array. ! ! ( X(I), Y(I) ) < ( X(J), Y(J) ) if: ! ! * X(I) < X(J), or ! ! * X(I) = X(J), and Y(I) < Y(J). ! ! Once the index array is computed, the sorting can be carried out ! "implicitly: ! ! ( X(INDX(1:N)), Y(INDX(1:N) ), is sorted, ! ! or explicitly, by the call ! ! call r8vec_permute ( n, indx, x ) ! call r8vec_permute ( n, indx, y ) ! ! after which ( X(1:N), Y(1:N) ), is sorted. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) X(N),Y(N), pairs of X, Y coordinates of points. ! ! Output, integer ( kind = 4 ) INDX(N), the sort index. The ! I-th element of the sorted array has coordinates ( X(INDX(I)), Y(INDX(I) ). ! implicit none integer ( kind = 4 ) n integer ( kind = 4 ) i integer ( kind = 4 ) indx(n) integer ( kind = 4 ) indxt integer ( kind = 4 ) ir integer ( kind = 4 ) j integer ( kind = 4 ) l real ( kind = 8 ) x(n) real ( kind = 8 ) xval real ( kind = 8 ) y(n) real ( kind = 8 ) yval if ( n < 1 ) then return end if call i4vec_indicator1 ( n, indx ) if ( n == 1 ) then return end if l = n / 2 + 1 ir = n do if ( 1 < l ) then l = l - 1 indxt = indx(l) xval = x(indxt) yval = y(indxt) else indxt = indx(ir) xval = x(indxt) yval = y(indxt) indx(ir) = indx(1) ir = ir - 1 if ( ir == 1 ) then indx(1) = indxt exit end if end if i = l j = l + l do while ( j <= ir ) if ( j < ir ) then if ( x(indx(j)) < x(indx(j+1)) .or. & ( x(indx(j)) == x(indx(j+1)) .and. y(indx(j)) < y(indx(j+1)) ) ) then j = j + 1 end if end if if ( xval < x(indx(j)) .or. & ( xval == x(indx(j)) .and. yval < y(indx(j)) ) ) then indx(i) = indx(j) i = j j = j + j else j = ir + 1 end if end do indx(i) = indxt end do return end subroutine r8vec2_sorted_unique ( n, a1, a2, unique_num ) !*****************************************************************************80 ! !! R8VEC2_SORTED_UNIQUE keeps unique elements in a sorted R8VEC2. ! ! Discussion: ! ! An R8VEC2 is a dataset consisting of N pairs of R8's, stored ! as two separate vectors A1 and A2. ! ! Item I is stored as the pair A1(I), A2(I). ! ! The items must have been sorted, or at least it must be the ! case that equal items are stored in adjacent vector locations. ! ! If the items were not sorted, then this routine will only ! replace a string of equal values by a single representative. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items. ! ! Input/output, real ( kind = 8 ) A1(N), A2(N). ! On input, the array of N items. ! On output, an array of unique items. ! ! Output, integer ( kind = 4 ) UNIQUE_NUM, the number of unique items. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) integer ( kind = 4 ) itest integer ( kind = 4 ) unique_num if ( n <= 0 ) then unique_num = 0 return end if unique_num = 1 do itest = 2, n if ( a1(itest) /= a1(unique_num) .or. a2(itest) /= a2(unique_num) ) then unique_num = unique_num + 1 a1(unique_num) = a1(itest) a2(unique_num) = a2(itest) end if end do return end subroutine r8vec2_sorted_unique_index ( n, a1, a2, unique_num, indx ) !*****************************************************************************80 ! !! R8VEC2_SORTED_UNIQUE_INDEX indexes unique elements in a sorted R8VEC2. ! ! Discussion: ! ! An R8VEC2 is a dataset consisting of N pairs of R8's, stored ! as two separate vectors A1 and A2. ! ! Item I is stored as the pair A1(I), A2(I). ! ! The items must have been sorted, or at least it should be the ! case that equal items are stored in adjacent vector locations. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 02 November 2005 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items. ! ! Input, real ( kind = 8 ) A1(N), A2(N), the array of N items. ! ! Output, integer ( kind = 4 ) UNIQUE_NUM, the number of unique items. ! ! Output, integer ( kind = 4 ) INDX(N), contains in entries 1 through ! UNIQUE_NUM an index array of the unique items. To build new arrays ! with no repeated elements: ! B1(1:UNIQUE_NUM) = A1(INDX(1:UNIQUE_NUM)) ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) integer ( kind = 4 ) indx(n) integer ( kind = 4 ) itest integer ( kind = 4 ) unique_num if ( n <= 0 ) then unique_num = 0 return end if unique_num = 1 indx(1) = 1 do itest = 2, n if ( a1(itest-1) /= a1(itest) .or. a2(itest-1) /= a2(itest) ) then unique_num = unique_num + 1 indx(unique_num) = itest end if end do indx(unique_num+1:n) = 0 return end subroutine r8vec2_sum_max_index ( n, a, b, sum_max_index ) !*****************************************************************************80 ! !! R8VEC2_SUM_MAX_INDEX returns the index of the maximum sum of two R8VEC's. ! ! Discussion: ! ! An R8VEC2 is a dataset consisting of N pairs of R8's, stored ! as two separate vectors A1 and A2. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 13 December 2004 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of entries in the array. ! ! Input, real ( kind = 8 ) A(N), B(N), two arrays whose sum ! is to be examined. ! ! Output, integer ( kind = 4 ) SUM_MAX_INDEX, the index of the largest ! entry in A+B. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a(n) real ( kind = 8 ) b(n) integer ( kind = 4 ) i real ( kind = 8 ) sum_max integer ( kind = 4 ) sum_max_index if ( n <= 0 ) then sum_max_index = -1 else sum_max_index = 1 sum_max = a(1) + b(1) do i = 2, n if ( sum_max < a(i) + b(i) ) then sum_max = a(i) + b(i) sum_max_index = i end if end do end if return end subroutine r8vec3_print ( n, a1, a2, a3, title ) !*****************************************************************************80 ! !! R8VEC3_PRINT prints an R8VEC3. ! ! Discussion: ! ! An R8VEC3 is a dataset consisting of N triples of R8's, stored ! as three separate vectors A1, A2, A3. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 22 September 2012 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of components of the vector. ! ! Input, real ( kind = 8 ) A1(N), A2(N), A3(N), the vectors to be printed. ! ! Input, character ( len = * ) TITLE, a title. ! implicit none integer ( kind = 4 ) n real ( kind = 8 ) a1(n) real ( kind = 8 ) a2(n) real ( kind = 8 ) a3(n) integer ( kind = 4 ) i character ( len = * ) title write ( *, '(a)' ) ' ' write ( *, '(a)' ) trim ( title ) write ( *, '(a)' ) ' ' do i = 1, n write ( *, '(i8,3g14.6)' ) i, a1(i), a2(i), a3(i) end do return end subroutine sort_heap_external ( n, indx, i, j, isgn ) !*****************************************************************************80 ! !! SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order. ! ! Discussion: ! ! The actual list of data is not passed to the routine. Hence this ! routine may be used to sort integers, reals, numbers, names, ! dates, shoe sizes, and so on. After each call, the routine asks ! the user to compare or interchange two items, until a special ! return value signals that the sorting is completed. ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 05 February 2004 ! ! Author: ! ! Original FORTRAN77 version by Albert Nijenhuis, Herbert Wilf. ! FORTRAN90 version by John Burkardt. ! ! Reference: ! ! Albert Nijenhuis, Herbert Wilf, ! Combinatorial Algorithms for Computers and Calculators, ! Academic Press, 1978, ! ISBN: 0-12-519260-6, ! LC: QA164.N54. ! ! Parameters: ! ! Input, integer ( kind = 4 ) N, the number of items to be sorted. ! ! Input/output, integer ( kind = 4 ) INDX, the main communication signal. ! The user must set INDX to 0 before the first call. ! Thereafter, the user should not change the value of INDX until ! the sorting is done. ! On return, if INDX is ! * greater than 0, ! > interchange items I and J; ! > call again. ! * less than 0, ! > compare items I and J; ! > set ISGN = -1 if I < J, ISGN = +1 if J < I; ! > call again. ! * equal to 0, the sorting is done. ! ! Output, integer ( kind = 4 ) I, J, the indices of two items. ! On return with INDX positive, elements I and J should be interchanged. ! On return with INDX negative, elements I and J should be compared, and ! the result reported in ISGN on the next call. ! ! Input, integer ( kind = 4 ) ISGN, results of comparison of elements ! I and J. (Used only when the previous call returned INDX less than 0). ! ISGN <= 0 means I is less than or equal to J; ! 0 <= ISGN means I is greater than or equal to J. ! implicit none integer ( kind = 4 ) i integer ( kind = 4 ), save :: i_save = 0 integer ( kind = 4 ) indx integer ( kind = 4 ) isgn integer ( kind = 4 ) j integer ( kind = 4 ), save :: j_save = 0 integer ( kind = 4 ), save :: k = 0 integer ( kind = 4 ), save :: k1 = 0 integer ( kind = 4 ) n integer ( kind = 4 ), save :: n1 = 0 ! ! INDX = 0: This is the first call. ! if ( indx == 0 ) then i_save = 0 j_save = 0 k = n / 2 k1 = k n1 = n ! ! INDX < 0: The user is returning the results of a comparison. ! else if ( indx < 0 ) then if ( indx == -2 ) then if ( isgn < 0 ) then i_save = i_save + 1 end if j_save = k1 k1 = i_save indx = -1 i = i_save j = j_save return end if if ( 0 < isgn ) then indx = 2 i = i_save j = j_save return end if if ( k <= 1 ) then if ( n1 == 1 ) then i_save = 0 j_save = 0 indx = 0 else i_save = n1 n1 = n1 - 1 j_save = 1 indx = 1 end if i = i_save j = j_save return end if k = k - 1 k1 = k ! ! 0 < INDX, the user was asked to make an interchange. ! else if ( indx == 1 ) then k1 = k end if do i_save = 2 * k1 if ( i_save == n1 ) then j_save = k1 k1 = i_save indx = -1 i = i_save j = j_save return else if ( i_save <= n1 ) then j_save = i_save + 1 indx = -2 i = i_save j = j_save return end if if ( k <= 1 ) then exit end if k = k - 1 k1 = k end do if ( n1 == 1 ) then i_save = 0 j_save = 0 indx = 0 i = i_save j = j_save else i_save = n1 n1 = n1 - 1 j_save = 1 indx = 1 i = i_save j = j_save end if return end subroutine timestamp ( ) !*****************************************************************************80 ! !! timestamp prints the current YMDHMS date as a time stamp. ! ! Example: ! ! 31 May 2001 9:45:54.872 AM ! ! Licensing: ! ! This code is distributed under the GNU LGPL license. ! ! Modified: ! ! 18 May 2013 ! ! Author: ! ! John Burkardt ! implicit none character ( len = 8 ) ampm integer ( kind = 4 ) d integer ( kind = 4 ) h integer ( kind = 4 ) m integer ( kind = 4 ) mm character ( len = 9 ), parameter, dimension(12) :: month = (/ & 'January ', 'February ', 'March ', 'April ', & 'May ', 'June ', 'July ', 'August ', & 'September', 'October ', 'November ', 'December ' /) integer ( kind = 4 ) n integer ( kind = 4 ) s integer ( kind = 4 ) values(8) integer ( kind = 4 ) y call date_and_time ( values = values ) y = values(1) m = values(2) d = values(3) h = values(5) n = values(6) s = values(7) mm = values(8) if ( h < 12 ) then ampm = 'AM' else if ( h == 12 ) then if ( n == 0 .and. s == 0 ) then ampm = 'Noon' else ampm = 'PM' end if else h = h - 12 if ( h < 12 ) then ampm = 'PM' else if ( h == 12 ) then if ( n == 0 .and. s == 0 ) then ampm = 'Midnight' else ampm = 'AM' end if end if end if write ( *, '(i2,1x,a,1x,i4,2x,i2,a1,i2.2,a1,i2.2,a1,i3.3,1x,a)' ) & d, trim ( month(m) ), y, h, ':', n, ':', s, '.', mm, trim ( ampm ) return end