#include "ESMF_LapackBlas.inc" !> \brief \b DLABRD ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DLABRD + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlabrd.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlabrd.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlabrd.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, ! LDY ) ! ! .. Scalar Arguments .. ! INTEGER LDA, LDX, LDY, M, N, NB ! .. ! .. Array Arguments .. ! DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), ! $ TAUQ( * ), X( LDX, * ), Y( LDY, * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DLABRD reduces the first NB rows and columns of a real general !> m by n matrix A to upper or lower bidiagonal form by an orthogonal !> transformation Q**T * A * P, and returns the matrices X and Y which !> are needed to apply the transformation to the unreduced part of A. !> !> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower !> bidiagonal form. !> !> This is an auxiliary routine called by DGEBRD !> \endverbatim ! ! Arguments: ! ========== ! !> \param[in] M !> \verbatim !> M is INTEGER !> The number of rows in the matrix A. !> \endverbatim !> !> \param[in] N !> \verbatim !> N is INTEGER !> The number of columns in the matrix A. !> \endverbatim !> !> \param[in] NB !> \verbatim !> NB is INTEGER !> The number of leading rows and columns of A to be reduced. !> \endverbatim !> !> \param[in,out] A !> \verbatim !> A is DOUBLE PRECISION array, dimension (LDA,N) !> On entry, the m by n general matrix to be reduced. !> On exit, the first NB rows and columns of the matrix are !> overwritten; the rest of the array is unchanged. !> If m >= n, elements on and below the diagonal in the first NB !> columns, with the array TAUQ, represent the orthogonal !> matrix Q as a product of elementary reflectors; and !> elements above the diagonal in the first NB rows, with the !> array TAUP, represent the orthogonal matrix P as a product !> of elementary reflectors. !> If m < n, elements below the diagonal in the first NB !> columns, with the array TAUQ, represent the orthogonal !> matrix Q as a product of elementary reflectors, and !> elements on and above the diagonal in the first NB rows, !> with the array TAUP, represent the orthogonal matrix P 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,M). !> \endverbatim !> !> \param[out] D !> \verbatim !> D is DOUBLE PRECISION array, dimension (NB) !> The diagonal elements of the first NB rows and columns of !> the reduced matrix. D(i) = A(i,i). !> \endverbatim !> !> \param[out] E !> \verbatim !> E is DOUBLE PRECISION array, dimension (NB) !> The off-diagonal elements of the first NB rows and columns of !> the reduced matrix. !> \endverbatim !> !> \param[out] TAUQ !> \verbatim !> TAUQ is DOUBLE PRECISION array dimension (NB) !> The scalar factors of the elementary reflectors which !> represent the orthogonal matrix Q. See Further Details. !> \endverbatim !> !> \param[out] TAUP !> \verbatim !> TAUP is DOUBLE PRECISION array, dimension (NB) !> The scalar factors of the elementary reflectors which !> represent the orthogonal matrix P. See Further Details. !> \endverbatim !> !> \param[out] X !> \verbatim !> X is DOUBLE PRECISION array, dimension (LDX,NB) !> The m-by-nb matrix X required to update the unreduced part !> of A. !> \endverbatim !> !> \param[in] LDX !> \verbatim !> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,M). !> \endverbatim !> !> \param[out] Y !> \verbatim !> Y is DOUBLE PRECISION array, dimension (LDY,NB) !> The n-by-nb matrix Y required to update the unreduced part !> of A. !> \endverbatim !> !> \param[in] LDY !> \verbatim !> LDY is INTEGER !> The leading dimension of the array Y. LDY >= max(1,N). !> \endverbatim ! ! Authors: ! ======== ! !> \author Univ. of Tennessee !> \author Univ. of California Berkeley !> \author Univ. of Colorado Denver !> \author NAG Ltd. ! !> \date November 2011 ! !> \ingroup doubleOTHERauxiliary ! !> \par Further Details: ! ===================== !> !> \verbatim !> !> The matrices Q and P are represented as products of elementary !> reflectors: !> !> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) !> !> Each H(i) and G(i) has the form: !> !> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T !> !> where tauq and taup are real scalars, and v and u are real vectors. !> !> If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in !> A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in !> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). !> !> If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in !> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in !> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). !> !> The elements of the vectors v and u together form the m-by-nb matrix !> V and the nb-by-n matrix U**T which are needed, with X and Y, to apply !> the transformation to the unreduced part of the matrix, using a block !> update of the form: A := A - V*Y**T - X*U**T. !> !> The contents of A on exit are illustrated by the following examples !> with nb = 2: !> !> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): !> !> ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) !> ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) !> ( v1 v2 a a a ) ( v1 1 a a a a ) !> ( v1 v2 a a a ) ( v1 v2 a a a a ) !> ( v1 v2 a a a ) ( v1 v2 a a a a ) !> ( v1 v2 a a a ) !> !> where a denotes an element of the original matrix which is unchanged, !> vi denotes an element of the vector defining H(i), and ui an element !> of the vector defining G(i). !> \endverbatim !> ! ===================================================================== SUBROUTINE DLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, & & LDY ) ! ! -- LAPACK auxiliary routine (version 3.4.0) -- ! -- LAPACK is a software package provided by Univ. of Tennessee, -- ! -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- ! November 2011 ! ! .. Scalar Arguments .. INTEGER LDA, LDX, LDY, M, N, NB ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), & & TAUQ( * ), X( LDX, * ), Y( LDY, * ) ! .. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) ! .. ! .. Local Scalars .. INTEGER I ! .. ! .. External Subroutines .. EXTERNAL DGEMV, DLARFG, DSCAL ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( M.LE.0 .OR. N.LE.0 ) & & RETURN ! IF( M.GE.N ) THEN ! ! Reduce to upper bidiagonal form ! DO 10 I = 1, NB ! ! Update A(i:m,i) ! CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ), & & LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 ) CALL DGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ), & & LDX, A( 1, I ), 1, ONE, A( I, I ), 1 ) ! ! Generate reflection Q(i) to annihilate A(i+1:m,i) ! CALL DLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1, & & TAUQ( I ) ) D( I ) = A( I, I ) IF( I.LT.N ) THEN A( I, I ) = ONE ! ! Compute Y(i+1:n,i) ! CALL DGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ), & & LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 ) CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ), LDA, & & A( I, I ), 1, ZERO, Y( 1, I ), 1 ) CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ), & & LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) CALL DGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ), LDX, & & A( I, I ), 1, ZERO, Y( 1, I ), 1 ) CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ), & & LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 ) ! ! Update A(i,i+1:n) ! CALL DGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ), & & LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA ) CALL DGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ), & & LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA ) ! ! Generate reflection P(i) to annihilate A(i,i+2:n) ! CALL DLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ), & & LDA, TAUP( I ) ) E( I ) = A( I, I+1 ) A( I, I+1 ) = ONE ! ! Compute X(i+1:m,i) ! CALL DGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ), & & LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 ) CALL DGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ), LDY, & & A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 ) CALL DGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ), & & LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) CALL DGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ), & & LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 ) CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ), & & LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 ) END IF 10 CONTINUE ELSE ! ! Reduce to lower bidiagonal form ! DO 20 I = 1, NB ! ! Update A(i,i:n) ! CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ), & & LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA ) CALL DGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ), LDA, & & X( I, 1 ), LDX, ONE, A( I, I ), LDA ) ! ! Generate reflection P(i) to annihilate A(i,i+1:n) ! CALL DLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA, & & TAUP( I ) ) D( I ) = A( I, I ) IF( I.LT.M ) THEN A( I, I ) = ONE ! ! Compute X(i+1:m,i) ! CALL DGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ), & & LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 ) CALL DGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ), LDY, & & A( I, I ), LDA, ZERO, X( 1, I ), 1 ) CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ), & & LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) CALL DGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ), & & LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 ) CALL DGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ), & & LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 ) CALL DSCAL( M-I, TAUP( I ), X( I+1, I ), 1 ) ! ! Update A(i+1:m,i) ! CALL DGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ), & & LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 ) CALL DGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ), & & LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 ) ! ! Generate reflection Q(i) to annihilate A(i+2:m,i) ! CALL DLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1, & & TAUQ( I ) ) E( I ) = A( I+1, I ) A( I+1, I ) = ONE ! ! Compute Y(i+1:n,i) ! CALL DGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ), & & LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 ) CALL DGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ), LDA, & & A( I+1, I ), 1, ZERO, Y( 1, I ), 1 ) CALL DGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ), & & LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) CALL DGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ), LDX, & & A( I+1, I ), 1, ZERO, Y( 1, I ), 1 ) CALL DGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ), LDA, & & Y( 1, I ), 1, ONE, Y( I+1, I ), 1 ) CALL DSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 ) END IF 20 CONTINUE END IF RETURN ! ! End of DLABRD ! END