SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO ) ! ! -- LAPACK routine (version 3.1) -- ! Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. ! November 2006 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ) ! .. ! ! Purpose ! ======= ! ! DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal ! form T by an orthogonal similarity transformation: Q' * A * Q = T. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored: ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! n-by-n upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading n-by-n lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! On exit, if UPLO = 'U', the diagonal and first superdiagonal ! of A are overwritten by the corresponding elements of the ! tridiagonal matrix T, and the elements above the first ! superdiagonal, with the array TAU, represent the orthogonal ! matrix Q as a product of elementary reflectors; if UPLO ! = 'L', the diagonal and first subdiagonal of A are over- ! written by the corresponding elements of the tridiagonal ! matrix T, and the elements below the first subdiagonal, with ! the array TAU, represent the orthogonal matrix Q as a product ! of elementary reflectors. See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! D (output) DOUBLE PRECISION array, dimension (N) ! The diagonal elements of the tridiagonal matrix T: ! D(i) = A(i,i). ! ! E (output) DOUBLE PRECISION array, dimension (N-1) ! The off-diagonal elements of the tridiagonal matrix T: ! E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. ! ! TAU (output) DOUBLE PRECISION array, dimension (N-1) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value. ! ! Further Details ! =============== ! ! If UPLO = 'U', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(n-1) . . . H(2) H(1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in ! A(1:i-1,i+1), and tau in TAU(i). ! ! If UPLO = 'L', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(1) H(2) . . . H(n-1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), ! and tau in TAU(i). ! ! The contents of A on exit are illustrated by the following examples ! with n = 5: ! ! if UPLO = 'U': if UPLO = 'L': ! ! ( d e v2 v3 v4 ) ( d ) ! ( d e v3 v4 ) ( e d ) ! ( d e v4 ) ( v1 e d ) ! ( d e ) ( v1 v2 e d ) ! ( d ) ( v1 v2 v3 e d ) ! ! where d and e denote diagonal and off-diagonal elements of T, and vi ! denotes an element of the vector defining H(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO, HALF PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, & HALF = 1.0D0 / 2.0D0 ) ! .. ! .. Local Scalars .. LOGICAL UPPER INTEGER I DOUBLE PRECISION ALPHA, TAUI ! .. ! .. External Subroutines .. ! EXTERNAL DAXPY, DLARFG, DSYMV, DSYR2, XERBLA ! .. ! .. External Functions .. ! LOGICAL LSAME ! DOUBLE PRECISION DDOT ! EXTERNAL LSAME, DDOT ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYTD2', -INFO ) RETURN END IF ! ! Quick return if possible ! IF( N.LE.0 ) & RETURN ! IF( UPPER ) THEN ! ! Reduce the upper triangle of A ! DO 10 I = N - 1, 1, -1 ! ! Generate elementary reflector H(i) = I - tau * v * v' ! to annihilate A(1:i-1,i+1) ! CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI ) E( I ) = A( I, I+1 ) ! IF( TAUI.NE.ZERO ) THEN ! ! Apply H(i) from both sides to A(1:i,1:i) ! A( I, I+1 ) = ONE ! ! Compute x := tau * A * v storing x in TAU(1:i) ! CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO, & TAU, 1 ) ! ! Compute w := x - 1/2 * tau * (x'*v) * v ! ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 ) CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 ) ! ! Apply the transformation as a rank-2 update: ! A := A - v * w' - w * v' ! CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A, & LDA ) ! A( I, I+1 ) = E( I ) END IF D( I+1 ) = A( I+1, I+1 ) TAU( I ) = TAUI 10 CONTINUE D( 1 ) = A( 1, 1 ) ELSE ! ! Reduce the lower triangle of A ! DO 20 I = 1, N - 1 ! ! Generate elementary reflector H(i) = I - tau * v * v' ! to annihilate A(i+2:n,i) ! CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, & TAUI ) E( I ) = A( I+1, I ) ! IF( TAUI.NE.ZERO ) THEN ! ! Apply H(i) from both sides to A(i+1:n,i+1:n) ! A( I+1, I ) = ONE ! ! Compute x := tau * A * v storing y in TAU(i:n-1) ! CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA, & A( I+1, I ), 1, ZERO, TAU( I ), 1 ) ! ! Compute w := x - 1/2 * tau * (x'*v) * v ! ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ), & 1 ) CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) ! ! Apply the transformation as a rank-2 update: ! A := A - v * w' - w * v' ! CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, & A( I+1, I+1 ), LDA ) ! A( I+1, I ) = E( I ) END IF D( I ) = A( I, I ) TAU( I ) = TAUI 20 CONTINUE D( N ) = A( N, N ) END IF ! RETURN ! ! End of DSYTD2 ! END SUBROUTINE DSYTD2