#include "ESMF_LapackBlas.inc" !> \brief \b DGEBRD ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DGEBRD + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebrd.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebrd.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebrd.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, ! INFO ) ! ! .. Scalar Arguments .. ! INTEGER INFO, LDA, LWORK, M, N ! .. ! .. Array Arguments .. ! DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), ! $ TAUQ( * ), WORK( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DGEBRD reduces a general real M-by-N matrix A to upper or lower !> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. !> !> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. !> \endverbatim ! ! Arguments: ! ========== ! !> \param[in] M !> \verbatim !> M is INTEGER !> The number of rows in the matrix A. M >= 0. !> \endverbatim !> !> \param[in] N !> \verbatim !> N is INTEGER !> The number of columns in the matrix A. N >= 0. !> \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, !> if m >= n, the diagonal and the first superdiagonal are !> overwritten with the upper bidiagonal matrix B; the !> elements below the diagonal, with the array TAUQ, represent !> the orthogonal matrix Q as a product of elementary !> reflectors, and the elements above the first superdiagonal, !> with the array TAUP, represent the orthogonal matrix P as !> a product of elementary reflectors; !> if m < n, the diagonal and the first subdiagonal are !> overwritten with the lower bidiagonal matrix B; the !> elements below the first subdiagonal, with the array TAUQ, !> represent the orthogonal matrix Q as a product of !> elementary reflectors, and the elements above the diagonal, !> 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 (min(M,N)) !> The diagonal elements of the bidiagonal matrix B: !> D(i) = A(i,i). !> \endverbatim !> !> \param[out] E !> \verbatim !> E is DOUBLE PRECISION array, dimension (min(M,N)-1) !> The off-diagonal elements of the bidiagonal matrix B: !> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; !> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. !> \endverbatim !> !> \param[out] TAUQ !> \verbatim !> TAUQ is DOUBLE PRECISION array dimension (min(M,N)) !> 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 (min(M,N)) !> The scalar factors of the elementary reflectors which !> represent the orthogonal matrix P. See Further Details. !> \endverbatim !> !> \param[out] WORK !> \verbatim !> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !> \endverbatim !> !> \param[in] LWORK !> \verbatim !> LWORK is INTEGER !> The length of the array WORK. LWORK >= max(1,M,N). !> For optimum performance LWORK >= (M+N)*NB, where NB !> is the optimal blocksize. !> !> If LWORK = -1, then a workspace query is assumed; the routine !> only calculates the optimal size of the WORK array, returns !> this value as the first entry of the WORK array, and no error !> message related to LWORK is issued by XERBLA. !> \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 November 2011 ! !> \ingroup doubleGEcomputational ! !> \par Further Details: ! ===================== !> !> \verbatim !> !> The matrices Q and P are represented as products of elementary !> reflectors: !> !> If m >= n, !> !> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) !> !> 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; !> v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); !> u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); !> tauq is stored in TAUQ(i) and taup in TAUP(i). !> !> If m < n, !> !> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) !> !> 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; !> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); !> u(1:i-1) = 0, u(i) = 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). !> !> The contents of A on exit are illustrated by the following examples: !> !> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): !> !> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) !> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) !> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) !> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) !> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) !> ( v1 v2 v3 v4 v5 ) !> !> where d and e denote diagonal and off-diagonal elements of B, vi !> denotes an element of the vector defining H(i), and ui an element of !> the vector defining G(i). !> \endverbatim !> ! ===================================================================== SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, & & INFO ) ! ! -- LAPACK computational 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 INFO, LDA, LWORK, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), & & TAUQ( * ), WORK( * ) ! .. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. LOGICAL LQUERY INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB, & & NBMIN, NX DOUBLE PRECISION WS ! .. ! .. External Subroutines .. EXTERNAL DGEBD2, DGEMM, DLABRD, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN ! .. ! .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) ) LWKOPT = ( M+N )*NB WORK( 1 ) = DBLE( LWKOPT ) LQUERY = ( LWORK.EQ.-1 ) IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -4 ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN INFO = -10 END IF IF( INFO.LT.0 ) THEN CALL XERBLA( 'DGEBRD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF ! ! Quick return if possible ! MINMN = MIN( M, N ) IF( MINMN.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF ! WS = MAX( M, N ) LDWRKX = M LDWRKY = N ! IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN ! ! Set the crossover point NX. ! NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) ) ! ! Determine when to switch from blocked to unblocked code. ! IF( NX.LT.MINMN ) THEN WS = ( M+N )*NB IF( LWORK.LT.WS ) THEN ! ! Not enough work space for the optimal NB, consider using ! a smaller block size. ! NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 ) IF( LWORK.GE.( M+N )*NBMIN ) THEN NB = LWORK / ( M+N ) ELSE NB = 1 NX = MINMN END IF END IF END IF ELSE NX = MINMN END IF ! DO 30 I = 1, MINMN - NX, NB ! ! Reduce rows and columns i:i+nb-1 to bidiagonal form and return ! the matrices X and Y which are needed to update the unreduced ! part of the matrix ! CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ), & & TAUQ( I ), TAUP( I ), WORK, LDWRKX, & & WORK( LDWRKX*NB+1 ), LDWRKY ) ! ! Update the trailing submatrix A(i+nb:m,i+nb:n), using an update ! of the form A := A - V*Y**T - X*U**T ! CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1, & & NB, -ONE, A( I+NB, I ), LDA, & & WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE, & & A( I+NB, I+NB ), LDA ) CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1, & & NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA, & & ONE, A( I+NB, I+NB ), LDA ) ! ! Copy diagonal and off-diagonal elements of B back into A ! IF( M.GE.N ) THEN DO 10 J = I, I + NB - 1 A( J, J ) = D( J ) A( J, J+1 ) = E( J ) 10 CONTINUE ELSE DO 20 J = I, I + NB - 1 A( J, J ) = D( J ) A( J+1, J ) = E( J ) 20 CONTINUE END IF 30 CONTINUE ! ! Use unblocked code to reduce the remainder of the matrix ! CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ), & & TAUQ( I ), TAUP( I ), WORK, IINFO ) WORK( 1 ) = WS RETURN ! ! End of DGEBRD ! END