#include "ESMF_LapackBlas.inc" !> \brief \b DGEBD2 ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DGEBD2 + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebd2.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebd2.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebd2.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) ! ! .. Scalar Arguments .. ! INTEGER INFO, LDA, M, N ! .. ! .. Array Arguments .. ! DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), ! $ TAUQ( * ), WORK( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DGEBD2 reduces a real general 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(M,N)) !> \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 DGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, 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, M, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ), & & TAUQ( * ), WORK( * ) ! .. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I ! .. ! .. External Subroutines .. EXTERNAL DLARF, DLARFG, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC MAX, MIN ! .. ! .. Executable Statements .. ! ! Test the input parameters ! INFO = 0 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 END IF IF( INFO.LT.0 ) THEN CALL XERBLA( 'DGEBD2', -INFO ) RETURN END IF ! IF( M.GE.N ) THEN ! ! Reduce to upper bidiagonal form ! DO 10 I = 1, N ! ! Generate elementary reflector H(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 ) A( I, I ) = ONE ! ! Apply H(i) to A(i:m,i+1:n) from the left ! IF( I.LT.N ) & & CALL DLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ), & & A( I, I+1 ), LDA, WORK ) A( I, I ) = D( I ) ! IF( I.LT.N ) THEN ! ! Generate elementary reflector G(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 ! ! Apply G(i) to A(i+1:m,i+1:n) from the right ! CALL DLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA, & & TAUP( I ), A( I+1, I+1 ), LDA, WORK ) A( I, I+1 ) = E( I ) ELSE TAUP( I ) = ZERO END IF 10 CONTINUE ELSE ! ! Reduce to lower bidiagonal form ! DO 20 I = 1, M ! ! Generate elementary reflector G(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 ) A( I, I ) = ONE ! ! Apply G(i) to A(i+1:m,i:n) from the right ! IF( I.LT.M ) & & CALL DLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, & & TAUP( I ), A( I+1, I ), LDA, WORK ) A( I, I ) = D( I ) ! IF( I.LT.M ) THEN ! ! Generate elementary reflector H(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 ! ! Apply H(i) to A(i+1:m,i+1:n) from the left ! CALL DLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ), & & A( I+1, I+1 ), LDA, WORK ) A( I+1, I ) = E( I ) ELSE TAUQ( I ) = ZERO END IF 20 CONTINUE END IF RETURN ! ! End of DGEBD2 ! END