#include "ESMF_LapackBlas.inc" !> \brief \b DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm). ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DSYTD2 + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytd2.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytd2.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytd2.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO ) ! ! .. Scalar Arguments .. ! CHARACTER UPLO ! INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. ! DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal !> form T by an orthogonal similarity transformation: Q**T * A * Q = T. !> \endverbatim ! ! Arguments: ! ========== ! !> \param[in] UPLO !> \verbatim !> UPLO is CHARACTER*1 !> Specifies whether the upper or lower triangular part of the !> symmetric matrix A is stored: !> = 'U': Upper triangular !> = 'L': Lower triangular !> \endverbatim !> !> \param[in] N !> \verbatim !> N is INTEGER !> The order of the matrix A. N >= 0. !> \endverbatim !> !> \param[in,out] A !> \verbatim !> A is 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. !> \endverbatim !> !> \param[in] LDA !> \verbatim !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !> \endverbatim !> !> \param[out] D !> \verbatim !> D is DOUBLE PRECISION array, dimension (N) !> The diagonal elements of the tridiagonal matrix T: !> D(i) = A(i,i). !> \endverbatim !> !> \param[out] E !> \verbatim !> E is 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'. !> \endverbatim !> !> \param[out] TAU !> \verbatim !> TAU is DOUBLE PRECISION array, dimension (N-1) !> The scalar factors of the elementary reflectors (see Further !> Details). !> \endverbatim !> !> \param[out] INFO !> \verbatim !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value. !> \endverbatim ! ! Authors: ! ======== ! !> \author Univ. of Tennessee !> \author Univ. of California Berkeley !> \author Univ. of Colorado Denver !> \author NAG Ltd. ! !> \date December 2016 ! !> \ingroup doubleSYcomputational ! !> \par Further Details: ! ===================== !> !> \verbatim !> !> 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**T !> !> 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**T !> !> 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). !> \endverbatim !> ! ===================================================================== SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO ) ! ! -- LAPACK computational routine (version 3.7.0) -- ! -- LAPACK is a software package provided by Univ. of Tennessee, -- ! -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- ! December 2016 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ) ! .. ! ! ===================================================================== ! ! .. 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**T ! 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**T * 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**T - w * v**T ! 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**T ! 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**T * 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**T - w * v**T ! 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