3502 lines
133 KiB
Fortran
3502 lines
133 KiB
Fortran
*> \brief <b> DGESVD computes the singular value decomposition (SVD) for GE matrices</b>
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download DGESVD + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesvd.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgesvd.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgesvd.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT,
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* WORK, LWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER JOBU, JOBVT
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* INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ),
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* $ VT( LDVT, * ), WORK( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> DGESVD computes the singular value decomposition (SVD) of a real
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*> M-by-N matrix A, optionally computing the left and/or right singular
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*> vectors. The SVD is written
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*>
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*> A = U * SIGMA * transpose(V)
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*>
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*> where SIGMA is an M-by-N matrix which is zero except for its
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*> min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
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*> V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA
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*> are the singular values of A; they are real and non-negative, and
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*> are returned in descending order. The first min(m,n) columns of
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*> U and V are the left and right singular vectors of A.
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*>
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*> Note that the routine returns V**T, not V.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] JOBU
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*> \verbatim
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*> JOBU is CHARACTER*1
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*> Specifies options for computing all or part of the matrix U:
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*> = 'A': all M columns of U are returned in array U:
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*> = 'S': the first min(m,n) columns of U (the left singular
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*> vectors) are returned in the array U;
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*> = 'O': the first min(m,n) columns of U (the left singular
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*> vectors) are overwritten on the array A;
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*> = 'N': no columns of U (no left singular vectors) are
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*> computed.
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*> \endverbatim
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*>
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*> \param[in] JOBVT
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*> \verbatim
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*> JOBVT is CHARACTER*1
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*> Specifies options for computing all or part of the matrix
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*> V**T:
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*> = 'A': all N rows of V**T are returned in the array VT;
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*> = 'S': the first min(m,n) rows of V**T (the right singular
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*> vectors) are returned in the array VT;
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*> = 'O': the first min(m,n) rows of V**T (the right singular
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*> vectors) are overwritten on the array A;
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*> = 'N': no rows of V**T (no right singular vectors) are
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*> computed.
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*>
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*> JOBVT and JOBU cannot both be 'O'.
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*> \endverbatim
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*>
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*> \param[in] M
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*> \verbatim
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*> M is INTEGER
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*> The number of rows of the input matrix A. M >= 0.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The number of columns of the input matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is DOUBLE PRECISION array, dimension (LDA,N)
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*> On entry, the M-by-N matrix A.
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*> On exit,
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*> if JOBU = 'O', A is overwritten with the first min(m,n)
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*> columns of U (the left singular vectors,
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*> stored columnwise);
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*> if JOBVT = 'O', A is overwritten with the first min(m,n)
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*> rows of V**T (the right singular vectors,
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*> stored rowwise);
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*> if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
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*> are destroyed.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,M).
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*> \endverbatim
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*>
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*> \param[out] S
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*> \verbatim
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*> S is DOUBLE PRECISION array, dimension (min(M,N))
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*> The singular values of A, sorted so that S(i) >= S(i+1).
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*> \endverbatim
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*>
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*> \param[out] U
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*> \verbatim
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*> U is DOUBLE PRECISION array, dimension (LDU,UCOL)
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*> (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
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*> If JOBU = 'A', U contains the M-by-M orthogonal matrix U;
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*> if JOBU = 'S', U contains the first min(m,n) columns of U
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*> (the left singular vectors, stored columnwise);
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*> if JOBU = 'N' or 'O', U is not referenced.
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*> \endverbatim
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*>
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*> \param[in] LDU
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*> \verbatim
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*> LDU is INTEGER
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*> The leading dimension of the array U. LDU >= 1; if
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*> JOBU = 'S' or 'A', LDU >= M.
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*> \endverbatim
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*>
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*> \param[out] VT
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*> \verbatim
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*> VT is DOUBLE PRECISION array, dimension (LDVT,N)
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*> If JOBVT = 'A', VT contains the N-by-N orthogonal matrix
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*> V**T;
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*> if JOBVT = 'S', VT contains the first min(m,n) rows of
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*> V**T (the right singular vectors, stored rowwise);
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*> if JOBVT = 'N' or 'O', VT is not referenced.
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*> \endverbatim
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*>
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*> \param[in] LDVT
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*> \verbatim
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*> LDVT is INTEGER
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*> The leading dimension of the array VT. LDVT >= 1; if
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*> JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
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*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
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*> if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged
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*> superdiagonal elements of an upper bidiagonal matrix B
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*> whose diagonal is in S (not necessarily sorted). B
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*> satisfies A = U * B * VT, so it has the same singular values
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*> as A, and singular vectors related by U and VT.
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK.
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*> LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code):
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*> - PATH 1 (M much larger than N, JOBU='N')
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*> - PATH 1t (N much larger than M, JOBVT='N')
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*> LWORK >= MAX(1,3*MIN(M,N) + MAX(M,N),5*MIN(M,N)) for the other paths
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*> For good performance, LWORK should generally be larger.
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*>
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the WORK array, returns
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*> this value as the first entry of the WORK array, and no error
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*> message related to LWORK is issued by XERBLA.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit.
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*> < 0: if INFO = -i, the i-th argument had an illegal value.
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*> > 0: if DBDSQR did not converge, INFO specifies how many
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*> superdiagonals of an intermediate bidiagonal form B
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*> did not converge to zero. See the description of WORK
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*> above for details.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup doubleGEsing
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*
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* =====================================================================
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SUBROUTINE DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU,
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$ VT, LDVT, WORK, LWORK, INFO )
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*
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* -- LAPACK driver routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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CHARACTER JOBU, JOBVT
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INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ),
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$ VT( LDVT, * ), WORK( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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* ..
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* .. Local Scalars ..
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LOGICAL LQUERY, WNTUA, WNTUAS, WNTUN, WNTUO, WNTUS,
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$ WNTVA, WNTVAS, WNTVN, WNTVO, WNTVS
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INTEGER BDSPAC, BLK, CHUNK, I, IE, IERR, IR, ISCL,
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$ ITAU, ITAUP, ITAUQ, IU, IWORK, LDWRKR, LDWRKU,
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$ MAXWRK, MINMN, MINWRK, MNTHR, NCU, NCVT, NRU,
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$ NRVT, WRKBL
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INTEGER LWORK_DGEQRF, LWORK_DORGQR_N, LWORK_DORGQR_M,
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$ LWORK_DGEBRD, LWORK_DORGBR_P, LWORK_DORGBR_Q,
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$ LWORK_DGELQF, LWORK_DORGLQ_N, LWORK_DORGLQ_M
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DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM
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* ..
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* .. Local Arrays ..
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DOUBLE PRECISION DUM( 1 )
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* ..
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* .. External Subroutines ..
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EXTERNAL DBDSQR, DGEBRD, DGELQF, DGEMM, DGEQRF, DLACPY,
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$ DLASCL, DLASET, DORGBR, DORGLQ, DORGQR, DORMBR,
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$ XERBLA
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ILAENV
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DOUBLE PRECISION DLAMCH, DLANGE
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EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN, SQRT
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* ..
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* .. Executable Statements ..
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*
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* Test the input arguments
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*
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INFO = 0
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MINMN = MIN( M, N )
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WNTUA = LSAME( JOBU, 'A' )
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WNTUS = LSAME( JOBU, 'S' )
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WNTUAS = WNTUA .OR. WNTUS
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WNTUO = LSAME( JOBU, 'O' )
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WNTUN = LSAME( JOBU, 'N' )
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WNTVA = LSAME( JOBVT, 'A' )
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WNTVS = LSAME( JOBVT, 'S' )
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WNTVAS = WNTVA .OR. WNTVS
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WNTVO = LSAME( JOBVT, 'O' )
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WNTVN = LSAME( JOBVT, 'N' )
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LQUERY = ( LWORK.EQ.-1 )
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*
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IF( .NOT.( WNTUA .OR. WNTUS .OR. WNTUO .OR. WNTUN ) ) THEN
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INFO = -1
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ELSE IF( .NOT.( WNTVA .OR. WNTVS .OR. WNTVO .OR. WNTVN ) .OR.
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$ ( WNTVO .AND. WNTUO ) ) THEN
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INFO = -2
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ELSE IF( M.LT.0 ) THEN
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INFO = -3
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ELSE IF( N.LT.0 ) THEN
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INFO = -4
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ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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INFO = -6
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ELSE IF( LDU.LT.1 .OR. ( WNTUAS .AND. LDU.LT.M ) ) THEN
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INFO = -9
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ELSE IF( LDVT.LT.1 .OR. ( WNTVA .AND. LDVT.LT.N ) .OR.
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$ ( WNTVS .AND. LDVT.LT.MINMN ) ) THEN
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INFO = -11
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END IF
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*
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* Compute workspace
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* (Note: Comments in the code beginning "Workspace:" describe the
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* minimal amount of workspace needed at that point in the code,
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* as well as the preferred amount for good performance.
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* NB refers to the optimal block size for the immediately
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* following subroutine, as returned by ILAENV.)
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*
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IF( INFO.EQ.0 ) THEN
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MINWRK = 1
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MAXWRK = 1
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IF( M.GE.N .AND. MINMN.GT.0 ) THEN
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*
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* Compute space needed for DBDSQR
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*
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MNTHR = ILAENV( 6, 'DGESVD', JOBU // JOBVT, M, N, 0, 0 )
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BDSPAC = 5*N
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* Compute space needed for DGEQRF
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CALL DGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
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LWORK_DGEQRF = INT( DUM(1) )
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* Compute space needed for DORGQR
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CALL DORGQR( M, N, N, A, LDA, DUM(1), DUM(1), -1, IERR )
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LWORK_DORGQR_N = INT( DUM(1) )
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CALL DORGQR( M, M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
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LWORK_DORGQR_M = INT( DUM(1) )
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* Compute space needed for DGEBRD
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CALL DGEBRD( N, N, A, LDA, S, DUM(1), DUM(1),
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$ DUM(1), DUM(1), -1, IERR )
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LWORK_DGEBRD = INT( DUM(1) )
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* Compute space needed for DORGBR P
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CALL DORGBR( 'P', N, N, N, A, LDA, DUM(1),
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$ DUM(1), -1, IERR )
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LWORK_DORGBR_P = INT( DUM(1) )
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* Compute space needed for DORGBR Q
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CALL DORGBR( 'Q', N, N, N, A, LDA, DUM(1),
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$ DUM(1), -1, IERR )
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LWORK_DORGBR_Q = INT( DUM(1) )
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*
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IF( M.GE.MNTHR ) THEN
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IF( WNTUN ) THEN
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*
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* Path 1 (M much larger than N, JOBU='N')
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*
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MAXWRK = N + LWORK_DGEQRF
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MAXWRK = MAX( MAXWRK, 3*N + LWORK_DGEBRD )
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IF( WNTVO .OR. WNTVAS )
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$ MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_P )
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MAXWRK = MAX( MAXWRK, BDSPAC )
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MINWRK = MAX( 4*N, BDSPAC )
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ELSE IF( WNTUO .AND. WNTVN ) THEN
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*
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* Path 2 (M much larger than N, JOBU='O', JOBVT='N')
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*
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WRKBL = N + LWORK_DGEQRF
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WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
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WRKBL = MAX( WRKBL, BDSPAC )
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MAXWRK = MAX( N*N + WRKBL, N*N + M*N + N )
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MINWRK = MAX( 3*N + M, BDSPAC )
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ELSE IF( WNTUO .AND. WNTVAS ) THEN
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*
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* Path 3 (M much larger than N, JOBU='O', JOBVT='S' or
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* 'A')
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*
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WRKBL = N + LWORK_DGEQRF
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WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
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WRKBL = MAX( WRKBL, BDSPAC )
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MAXWRK = MAX( N*N + WRKBL, N*N + M*N + N )
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MINWRK = MAX( 3*N + M, BDSPAC )
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ELSE IF( WNTUS .AND. WNTVN ) THEN
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*
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* Path 4 (M much larger than N, JOBU='S', JOBVT='N')
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*
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WRKBL = N + LWORK_DGEQRF
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WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
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WRKBL = MAX( WRKBL, BDSPAC )
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MAXWRK = N*N + WRKBL
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MINWRK = MAX( 3*N + M, BDSPAC )
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ELSE IF( WNTUS .AND. WNTVO ) THEN
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*
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* Path 5 (M much larger than N, JOBU='S', JOBVT='O')
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*
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WRKBL = N + LWORK_DGEQRF
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WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
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WRKBL = MAX( WRKBL, BDSPAC )
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MAXWRK = 2*N*N + WRKBL
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MINWRK = MAX( 3*N + M, BDSPAC )
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ELSE IF( WNTUS .AND. WNTVAS ) THEN
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*
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* Path 6 (M much larger than N, JOBU='S', JOBVT='S' or
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* 'A')
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*
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WRKBL = N + LWORK_DGEQRF
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WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
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WRKBL = MAX( WRKBL, BDSPAC )
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MAXWRK = N*N + WRKBL
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MINWRK = MAX( 3*N + M, BDSPAC )
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ELSE IF( WNTUA .AND. WNTVN ) THEN
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*
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* Path 7 (M much larger than N, JOBU='A', JOBVT='N')
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*
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WRKBL = N + LWORK_DGEQRF
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WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
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WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
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WRKBL = MAX( WRKBL, BDSPAC )
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MAXWRK = N*N + WRKBL
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MINWRK = MAX( 3*N + M, BDSPAC )
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ELSE IF( WNTUA .AND. WNTVO ) THEN
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*
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* Path 8 (M much larger than N, JOBU='A', JOBVT='O')
|
|
*
|
|
WRKBL = N + LWORK_DGEQRF
|
|
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
|
|
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
|
|
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
|
|
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
|
|
WRKBL = MAX( WRKBL, BDSPAC )
|
|
MAXWRK = 2*N*N + WRKBL
|
|
MINWRK = MAX( 3*N + M, BDSPAC )
|
|
ELSE IF( WNTUA .AND. WNTVAS ) THEN
|
|
*
|
|
* Path 9 (M much larger than N, JOBU='A', JOBVT='S' or
|
|
* 'A')
|
|
*
|
|
WRKBL = N + LWORK_DGEQRF
|
|
WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
|
|
WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
|
|
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
|
|
WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
|
|
WRKBL = MAX( WRKBL, BDSPAC )
|
|
MAXWRK = N*N + WRKBL
|
|
MINWRK = MAX( 3*N + M, BDSPAC )
|
|
END IF
|
|
ELSE
|
|
*
|
|
* Path 10 (M at least N, but not much larger)
|
|
*
|
|
CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
|
|
$ DUM(1), DUM(1), -1, IERR )
|
|
LWORK_DGEBRD = INT( DUM(1) )
|
|
MAXWRK = 3*N + LWORK_DGEBRD
|
|
IF( WNTUS .OR. WNTUO ) THEN
|
|
CALL DORGBR( 'Q', M, N, N, A, LDA, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_DORGBR_Q = INT( DUM(1) )
|
|
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_Q )
|
|
END IF
|
|
IF( WNTUA ) THEN
|
|
CALL DORGBR( 'Q', M, M, N, A, LDA, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_DORGBR_Q = INT( DUM(1) )
|
|
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_Q )
|
|
END IF
|
|
IF( .NOT.WNTVN ) THEN
|
|
MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_P )
|
|
END IF
|
|
MAXWRK = MAX( MAXWRK, BDSPAC )
|
|
MINWRK = MAX( 3*N + M, BDSPAC )
|
|
END IF
|
|
ELSE IF( MINMN.GT.0 ) THEN
|
|
*
|
|
* Compute space needed for DBDSQR
|
|
*
|
|
MNTHR = ILAENV( 6, 'DGESVD', JOBU // JOBVT, M, N, 0, 0 )
|
|
BDSPAC = 5*M
|
|
* Compute space needed for DGELQF
|
|
CALL DGELQF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
|
|
LWORK_DGELQF = INT( DUM(1) )
|
|
* Compute space needed for DORGLQ
|
|
CALL DORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1, IERR )
|
|
LWORK_DORGLQ_N = INT( DUM(1) )
|
|
CALL DORGLQ( M, N, M, A, LDA, DUM(1), DUM(1), -1, IERR )
|
|
LWORK_DORGLQ_M = INT( DUM(1) )
|
|
* Compute space needed for DGEBRD
|
|
CALL DGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
|
|
$ DUM(1), DUM(1), -1, IERR )
|
|
LWORK_DGEBRD = INT( DUM(1) )
|
|
* Compute space needed for DORGBR P
|
|
CALL DORGBR( 'P', M, M, M, A, N, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_DORGBR_P = INT( DUM(1) )
|
|
* Compute space needed for DORGBR Q
|
|
CALL DORGBR( 'Q', M, M, M, A, N, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_DORGBR_Q = INT( DUM(1) )
|
|
IF( N.GE.MNTHR ) THEN
|
|
IF( WNTVN ) THEN
|
|
*
|
|
* Path 1t(N much larger than M, JOBVT='N')
|
|
*
|
|
MAXWRK = M + LWORK_DGELQF
|
|
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DGEBRD )
|
|
IF( WNTUO .OR. WNTUAS )
|
|
$ MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_Q )
|
|
MAXWRK = MAX( MAXWRK, BDSPAC )
|
|
MINWRK = MAX( 4*M, BDSPAC )
|
|
ELSE IF( WNTVO .AND. WNTUN ) THEN
|
|
*
|
|
* Path 2t(N much larger than M, JOBU='N', JOBVT='O')
|
|
*
|
|
WRKBL = M + LWORK_DGELQF
|
|
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
|
|
WRKBL = MAX( WRKBL, BDSPAC )
|
|
MAXWRK = MAX( M*M + WRKBL, M*M + M*N + M )
|
|
MINWRK = MAX( 3*M + N, BDSPAC )
|
|
ELSE IF( WNTVO .AND. WNTUAS ) THEN
|
|
*
|
|
* Path 3t(N much larger than M, JOBU='S' or 'A',
|
|
* JOBVT='O')
|
|
*
|
|
WRKBL = M + LWORK_DGELQF
|
|
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
|
|
WRKBL = MAX( WRKBL, BDSPAC )
|
|
MAXWRK = MAX( M*M + WRKBL, M*M + M*N + M )
|
|
MINWRK = MAX( 3*M + N, BDSPAC )
|
|
ELSE IF( WNTVS .AND. WNTUN ) THEN
|
|
*
|
|
* Path 4t(N much larger than M, JOBU='N', JOBVT='S')
|
|
*
|
|
WRKBL = M + LWORK_DGELQF
|
|
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
|
|
WRKBL = MAX( WRKBL, BDSPAC )
|
|
MAXWRK = M*M + WRKBL
|
|
MINWRK = MAX( 3*M + N, BDSPAC )
|
|
ELSE IF( WNTVS .AND. WNTUO ) THEN
|
|
*
|
|
* Path 5t(N much larger than M, JOBU='O', JOBVT='S')
|
|
*
|
|
WRKBL = M + LWORK_DGELQF
|
|
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
|
|
WRKBL = MAX( WRKBL, BDSPAC )
|
|
MAXWRK = 2*M*M + WRKBL
|
|
MINWRK = MAX( 3*M + N, BDSPAC )
|
|
ELSE IF( WNTVS .AND. WNTUAS ) THEN
|
|
*
|
|
* Path 6t(N much larger than M, JOBU='S' or 'A',
|
|
* JOBVT='S')
|
|
*
|
|
WRKBL = M + LWORK_DGELQF
|
|
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
|
|
WRKBL = MAX( WRKBL, BDSPAC )
|
|
MAXWRK = M*M + WRKBL
|
|
MINWRK = MAX( 3*M + N, BDSPAC )
|
|
ELSE IF( WNTVA .AND. WNTUN ) THEN
|
|
*
|
|
* Path 7t(N much larger than M, JOBU='N', JOBVT='A')
|
|
*
|
|
WRKBL = M + LWORK_DGELQF
|
|
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
|
|
WRKBL = MAX( WRKBL, BDSPAC )
|
|
MAXWRK = M*M + WRKBL
|
|
MINWRK = MAX( 3*M + N, BDSPAC )
|
|
ELSE IF( WNTVA .AND. WNTUO ) THEN
|
|
*
|
|
* Path 8t(N much larger than M, JOBU='O', JOBVT='A')
|
|
*
|
|
WRKBL = M + LWORK_DGELQF
|
|
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
|
|
WRKBL = MAX( WRKBL, BDSPAC )
|
|
MAXWRK = 2*M*M + WRKBL
|
|
MINWRK = MAX( 3*M + N, BDSPAC )
|
|
ELSE IF( WNTVA .AND. WNTUAS ) THEN
|
|
*
|
|
* Path 9t(N much larger than M, JOBU='S' or 'A',
|
|
* JOBVT='A')
|
|
*
|
|
WRKBL = M + LWORK_DGELQF
|
|
WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
|
|
WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
|
|
WRKBL = MAX( WRKBL, BDSPAC )
|
|
MAXWRK = M*M + WRKBL
|
|
MINWRK = MAX( 3*M + N, BDSPAC )
|
|
END IF
|
|
ELSE
|
|
*
|
|
* Path 10t(N greater than M, but not much larger)
|
|
*
|
|
CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
|
|
$ DUM(1), DUM(1), -1, IERR )
|
|
LWORK_DGEBRD = INT( DUM(1) )
|
|
MAXWRK = 3*M + LWORK_DGEBRD
|
|
IF( WNTVS .OR. WNTVO ) THEN
|
|
* Compute space needed for DORGBR P
|
|
CALL DORGBR( 'P', M, N, M, A, N, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_DORGBR_P = INT( DUM(1) )
|
|
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_P )
|
|
END IF
|
|
IF( WNTVA ) THEN
|
|
CALL DORGBR( 'P', N, N, M, A, N, DUM(1),
|
|
$ DUM(1), -1, IERR )
|
|
LWORK_DORGBR_P = INT( DUM(1) )
|
|
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_P )
|
|
END IF
|
|
IF( .NOT.WNTUN ) THEN
|
|
MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_Q )
|
|
END IF
|
|
MAXWRK = MAX( MAXWRK, BDSPAC )
|
|
MINWRK = MAX( 3*M + N, BDSPAC )
|
|
END IF
|
|
END IF
|
|
MAXWRK = MAX( MAXWRK, MINWRK )
|
|
WORK( 1 ) = MAXWRK
|
|
*
|
|
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
|
|
INFO = -13
|
|
END IF
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'DGESVD', -INFO )
|
|
RETURN
|
|
ELSE IF( LQUERY ) THEN
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if possible
|
|
*
|
|
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Get machine constants
|
|
*
|
|
EPS = DLAMCH( 'P' )
|
|
SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS
|
|
BIGNUM = ONE / SMLNUM
|
|
*
|
|
* Scale A if max element outside range [SMLNUM,BIGNUM]
|
|
*
|
|
ANRM = DLANGE( 'M', M, N, A, LDA, DUM )
|
|
ISCL = 0
|
|
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
|
|
ISCL = 1
|
|
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR )
|
|
ELSE IF( ANRM.GT.BIGNUM ) THEN
|
|
ISCL = 1
|
|
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, IERR )
|
|
END IF
|
|
*
|
|
IF( M.GE.N ) THEN
|
|
*
|
|
* A has at least as many rows as columns. If A has sufficiently
|
|
* more rows than columns, first reduce using the QR
|
|
* decomposition (if sufficient workspace available)
|
|
*
|
|
IF( M.GE.MNTHR ) THEN
|
|
*
|
|
IF( WNTUN ) THEN
|
|
*
|
|
* Path 1 (M much larger than N, JOBU='N')
|
|
* No left singular vectors to be computed
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Zero out below R
|
|
*
|
|
IF( N .GT. 1 ) THEN
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
|
|
$ LDA )
|
|
END IF
|
|
IE = 1
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in A
|
|
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
|
|
$ IERR )
|
|
NCVT = 0
|
|
IF( WNTVO .OR. WNTVAS ) THEN
|
|
*
|
|
* If right singular vectors desired, generate P'.
|
|
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
NCVT = N
|
|
END IF
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of A in A if desired
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, NCVT, 0, 0, S, WORK( IE ), A, LDA,
|
|
$ DUM, 1, DUM, 1, WORK( IWORK ), INFO )
|
|
*
|
|
* If right singular vectors desired in VT, copy them there
|
|
*
|
|
IF( WNTVAS )
|
|
$ CALL DLACPY( 'F', N, N, A, LDA, VT, LDVT )
|
|
*
|
|
ELSE IF( WNTUO .AND. WNTVN ) THEN
|
|
*
|
|
* Path 2 (M much larger than N, JOBU='O', JOBVT='N')
|
|
* N left singular vectors to be overwritten on A and
|
|
* no right singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N, WORK(IR) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + N*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N, WORK(IR) is N by N
|
|
*
|
|
LDWRKU = LDA
|
|
LDWRKR = N
|
|
ELSE
|
|
*
|
|
* WORK(IU) is LDWRKU by N, WORK(IR) is N by N
|
|
*
|
|
LDWRKU = ( LWORK-N*N-N ) / N
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IR) and zero out below it
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ),
|
|
$ LDWRKR )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
|
|
*
|
|
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IR)
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left vectors bidiagonalizing R
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
|
|
*
|
|
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IR)
|
|
* (Workspace: need N*N + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM, 1,
|
|
$ WORK( IR ), LDWRKR, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
IU = IE + N
|
|
*
|
|
* Multiply Q in A by left singular vectors of R in
|
|
* WORK(IR), storing result in WORK(IU) and copying to A
|
|
* (Workspace: need N*N + 2*N, prefer N*N + M*N + N)
|
|
*
|
|
DO 10 I = 1, M, LDWRKU
|
|
CHUNK = MIN( M-I+1, LDWRKU )
|
|
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
|
|
$ LDA, WORK( IR ), LDWRKR, ZERO,
|
|
$ WORK( IU ), LDWRKU )
|
|
CALL DLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU,
|
|
$ A( I, 1 ), LDA )
|
|
10 CONTINUE
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
IE = 1
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize A
|
|
* (Workspace: need 3*N + M, prefer 3*N + (M + N)*NB)
|
|
*
|
|
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left vectors bidiagonalizing A
|
|
* (Workspace: need 4*N, prefer 3*N + N*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in A
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ), DUM, 1,
|
|
$ A, LDA, DUM, 1, WORK( IWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUO .AND. WNTVAS ) THEN
|
|
*
|
|
* Path 3 (M much larger than N, JOBU='O', JOBVT='S' or 'A')
|
|
* N left singular vectors to be overwritten on A and
|
|
* N right singular vectors to be computed in VT
|
|
*
|
|
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + N*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = LDA
|
|
LDWRKR = N
|
|
ELSE
|
|
*
|
|
* WORK(IU) is LDWRKU by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = ( LWORK-N*N-N ) / N
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to VT, zeroing out below it
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
|
|
IF( N.GT.1 )
|
|
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ VT( 2, 1 ), LDVT )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
|
|
*
|
|
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in VT, copying result to WORK(IR)
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', N, N, VT, LDVT, WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate left vectors bidiagonalizing R in WORK(IR)
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
|
|
*
|
|
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing R in VT
|
|
* (Workspace: need N*N + 4*N-1, prefer N*N + 3*N + (N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IR) and computing right
|
|
* singular vectors of R in VT
|
|
* (Workspace: need N*N + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT, LDVT,
|
|
$ WORK( IR ), LDWRKR, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
IU = IE + N
|
|
*
|
|
* Multiply Q in A by left singular vectors of R in
|
|
* WORK(IR), storing result in WORK(IU) and copying to A
|
|
* (Workspace: need N*N + 2*N, prefer N*N + M*N + N)
|
|
*
|
|
DO 20 I = 1, M, LDWRKU
|
|
CHUNK = MIN( M-I+1, LDWRKU )
|
|
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
|
|
$ LDA, WORK( IR ), LDWRKR, ZERO,
|
|
$ WORK( IU ), LDWRKU )
|
|
CALL DLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU,
|
|
$ A( I, 1 ), LDA )
|
|
20 CONTINUE
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to VT, zeroing out below it
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
|
|
IF( N.GT.1 )
|
|
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ VT( 2, 1 ), LDVT )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in VT
|
|
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in A by left vectors bidiagonalizing R
|
|
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
|
|
*
|
|
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
|
|
$ WORK( ITAUQ ), A, LDA, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing R in VT
|
|
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in A and computing right
|
|
* singular vectors of A in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT, LDVT,
|
|
$ A, LDA, DUM, 1, WORK( IWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUS ) THEN
|
|
*
|
|
IF( WNTVN ) THEN
|
|
*
|
|
* Path 4 (M much larger than N, JOBU='S', JOBVT='N')
|
|
* N left singular vectors to be computed in U and
|
|
* no right singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.WRKBL+LDA*N ) THEN
|
|
*
|
|
* WORK(IR) is LDA by N
|
|
*
|
|
LDWRKR = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IR) is N by N
|
|
*
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IR), zeroing out below it
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ),
|
|
$ LDWRKR )
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ WORK( IR+1 ), LDWRKR )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
|
|
*
|
|
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IR)
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left vectors bidiagonalizing R in WORK(IR)
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
|
|
*
|
|
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IR)
|
|
* (Workspace: need N*N + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM,
|
|
$ 1, WORK( IR ), LDWRKR, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply Q in A by left singular vectors of R in
|
|
* WORK(IR), storing result in U
|
|
* (Workspace: need N*N)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
|
|
$ WORK( IR ), LDWRKR, ZERO, U, LDU )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Zero out below R in A
|
|
*
|
|
IF( N .GT. 1 ) THEN
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ A( 2, 1 ), LDA )
|
|
END IF
|
|
*
|
|
* Bidiagonalize R in A
|
|
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left vectors bidiagonalizing R
|
|
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
|
|
*
|
|
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ), DUM,
|
|
$ 1, U, LDU, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVO ) THEN
|
|
*
|
|
* Path 5 (M much larger than N, JOBU='S', JOBVT='O')
|
|
* N left singular vectors to be computed in U and
|
|
* N right singular vectors to be overwritten on A
|
|
*
|
|
IF( LWORK.GE.2*N*N+MAX( 4*N, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+2*LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.WRKBL+( LDA + N )*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = N
|
|
ELSE
|
|
*
|
|
* WORK(IU) is N by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = N
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IU), zeroing out below it
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ WORK( IU+1 ), LDWRKU )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
|
|
*
|
|
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IU), copying result to
|
|
* WORK(IR)
|
|
* (Workspace: need 2*N*N + 4*N,
|
|
* prefer 2*N*N+3*N+2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IU)
|
|
* (Workspace: need 2*N*N + 4*N, prefer 2*N*N + 3*N + N*NB)
|
|
*
|
|
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IR)
|
|
* (Workspace: need 2*N*N + 4*N-1,
|
|
* prefer 2*N*N+3*N+(N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IU) and computing
|
|
* right singular vectors of R in WORK(IR)
|
|
* (Workspace: need 2*N*N + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ),
|
|
$ WORK( IR ), LDWRKR, WORK( IU ),
|
|
$ LDWRKU, DUM, 1, WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply Q in A by left singular vectors of R in
|
|
* WORK(IU), storing result in U
|
|
* (Workspace: need N*N)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
|
|
$ WORK( IU ), LDWRKU, ZERO, U, LDU )
|
|
*
|
|
* Copy right singular vectors of R to A
|
|
* (Workspace: need N*N)
|
|
*
|
|
CALL DLACPY( 'F', N, N, WORK( IR ), LDWRKR, A,
|
|
$ LDA )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Zero out below R in A
|
|
*
|
|
IF( N .GT. 1 ) THEN
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ A( 2, 1 ), LDA )
|
|
END IF
|
|
*
|
|
* Bidiagonalize R in A
|
|
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left vectors bidiagonalizing R
|
|
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
|
|
*
|
|
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing R in A
|
|
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in A
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), A,
|
|
$ LDA, U, LDU, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVAS ) THEN
|
|
*
|
|
* Path 6 (M much larger than N, JOBU='S', JOBVT='S'
|
|
* or 'A')
|
|
* N left singular vectors to be computed in U and
|
|
* N right singular vectors to be computed in VT
|
|
*
|
|
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IU) is N by N
|
|
*
|
|
LDWRKU = N
|
|
END IF
|
|
ITAU = IU + LDWRKU*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IU), zeroing out below it
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ WORK( IU+1 ), LDWRKU )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
|
|
*
|
|
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IU), copying result to VT
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT,
|
|
$ LDVT )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IU)
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
|
|
*
|
|
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in VT
|
|
* (Workspace: need N*N + 4*N-1,
|
|
* prefer N*N+3*N+(N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IU) and computing
|
|
* right singular vectors of R in VT
|
|
* (Workspace: need N*N + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
|
|
$ LDVT, WORK( IU ), LDWRKU, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply Q in A by left singular vectors of R in
|
|
* WORK(IU), storing result in U
|
|
* (Workspace: need N*N)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
|
|
$ WORK( IU ), LDWRKU, ZERO, U, LDU )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to VT, zeroing out below it
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
|
|
IF( N.GT.1 )
|
|
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ VT( 2, 1 ), LDVT )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in VT
|
|
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left bidiagonalizing vectors
|
|
* in VT
|
|
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
|
|
*
|
|
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in VT
|
|
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT,
|
|
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUA ) THEN
|
|
*
|
|
IF( WNTVN ) THEN
|
|
*
|
|
* Path 7 (M much larger than N, JOBU='A', JOBVT='N')
|
|
* M left singular vectors to be computed in U and
|
|
* no right singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.WRKBL+LDA*N ) THEN
|
|
*
|
|
* WORK(IR) is LDA by N
|
|
*
|
|
LDWRKR = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IR) is N by N
|
|
*
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Copy R to WORK(IR), zeroing out below it
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ),
|
|
$ LDWRKR )
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ WORK( IR+1 ), LDWRKR )
|
|
*
|
|
* Generate Q in U
|
|
* (Workspace: need N*N + N + M, prefer N*N + N + M*NB)
|
|
*
|
|
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IR)
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IR)
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
|
|
*
|
|
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IR)
|
|
* (Workspace: need N*N + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM,
|
|
$ 1, WORK( IR ), LDWRKR, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply Q in U by left singular vectors of R in
|
|
* WORK(IR), storing result in A
|
|
* (Workspace: need N*N)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
|
|
$ WORK( IR ), LDWRKR, ZERO, A, LDA )
|
|
*
|
|
* Copy left singular vectors of A from A to U
|
|
*
|
|
CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (Workspace: need N + M, prefer N + M*NB)
|
|
*
|
|
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Zero out below R in A
|
|
*
|
|
IF( N .GT. 1 ) THEN
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ A( 2, 1 ), LDA )
|
|
END IF
|
|
*
|
|
* Bidiagonalize R in A
|
|
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left bidiagonalizing vectors
|
|
* in A
|
|
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
|
|
*
|
|
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ), DUM,
|
|
$ 1, U, LDU, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVO ) THEN
|
|
*
|
|
* Path 8 (M much larger than N, JOBU='A', JOBVT='O')
|
|
* M left singular vectors to be computed in U and
|
|
* N right singular vectors to be overwritten on A
|
|
*
|
|
IF( LWORK.GE.2*N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+2*LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.WRKBL+( LDA + N )*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = N
|
|
ELSE
|
|
*
|
|
* WORK(IU) is N by N and WORK(IR) is N by N
|
|
*
|
|
LDWRKU = N
|
|
IR = IU + LDWRKU*N
|
|
LDWRKR = N
|
|
END IF
|
|
ITAU = IR + LDWRKR*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (Workspace: need 2*N*N + N + M, prefer 2*N*N + N + M*NB)
|
|
*
|
|
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IU), zeroing out below it
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ WORK( IU+1 ), LDWRKU )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IU), copying result to
|
|
* WORK(IR)
|
|
* (Workspace: need 2*N*N + 4*N,
|
|
* prefer 2*N*N+3*N+2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IU)
|
|
* (Workspace: need 2*N*N + 4*N, prefer 2*N*N + 3*N + N*NB)
|
|
*
|
|
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IR)
|
|
* (Workspace: need 2*N*N + 4*N-1,
|
|
* prefer 2*N*N+3*N+(N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IU) and computing
|
|
* right singular vectors of R in WORK(IR)
|
|
* (Workspace: need 2*N*N + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ),
|
|
$ WORK( IR ), LDWRKR, WORK( IU ),
|
|
$ LDWRKU, DUM, 1, WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply Q in U by left singular vectors of R in
|
|
* WORK(IU), storing result in A
|
|
* (Workspace: need N*N)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
|
|
$ WORK( IU ), LDWRKU, ZERO, A, LDA )
|
|
*
|
|
* Copy left singular vectors of A from A to U
|
|
*
|
|
CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Copy right singular vectors of R from WORK(IR) to A
|
|
*
|
|
CALL DLACPY( 'F', N, N, WORK( IR ), LDWRKR, A,
|
|
$ LDA )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (Workspace: need N + M, prefer N + M*NB)
|
|
*
|
|
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Zero out below R in A
|
|
*
|
|
IF( N .GT. 1 ) THEN
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ A( 2, 1 ), LDA )
|
|
END IF
|
|
*
|
|
* Bidiagonalize R in A
|
|
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left bidiagonalizing vectors
|
|
* in A
|
|
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
|
|
*
|
|
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in A
|
|
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in A
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), A,
|
|
$ LDA, U, LDU, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVAS ) THEN
|
|
*
|
|
* Path 9 (M much larger than N, JOBU='A', JOBVT='S'
|
|
* or 'A')
|
|
* M left singular vectors to be computed in U and
|
|
* N right singular vectors to be computed in VT
|
|
*
|
|
IF( LWORK.GE.N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+LDA*N ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IU) is N by N
|
|
*
|
|
LDWRKU = N
|
|
END IF
|
|
ITAU = IU + LDWRKU*N
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (Workspace: need N*N + N + M, prefer N*N + N + M*NB)
|
|
*
|
|
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R to WORK(IU), zeroing out below it
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ WORK( IU+1 ), LDWRKU )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in WORK(IU), copying result to VT
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT,
|
|
$ LDVT )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IU)
|
|
* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
|
|
*
|
|
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in VT
|
|
* (Workspace: need N*N + 4*N-1,
|
|
* prefer N*N+3*N+(N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of R in WORK(IU) and computing
|
|
* right singular vectors of R in VT
|
|
* (Workspace: need N*N + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
|
|
$ LDVT, WORK( IU ), LDWRKU, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply Q in U by left singular vectors of R in
|
|
* WORK(IU), storing result in A
|
|
* (Workspace: need N*N)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
|
|
$ WORK( IU ), LDWRKU, ZERO, A, LDA )
|
|
*
|
|
* Copy left singular vectors of A from A to U
|
|
*
|
|
CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R, copying result to U
|
|
* (Workspace: need 2*N, prefer N + N*NB)
|
|
*
|
|
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
*
|
|
* Generate Q in U
|
|
* (Workspace: need N + M, prefer N + M*NB)
|
|
*
|
|
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy R from A to VT, zeroing out below it
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
|
|
IF( N.GT.1 )
|
|
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
|
|
$ VT( 2, 1 ), LDVT )
|
|
IE = ITAU
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in VT
|
|
* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
|
|
*
|
|
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply Q in U by left bidiagonalizing vectors
|
|
* in VT
|
|
* (Workspace: need 3*N + M, prefer 3*N + M*NB)
|
|
*
|
|
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
|
|
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in VT
|
|
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT,
|
|
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
ELSE
|
|
*
|
|
* M .LT. MNTHR
|
|
*
|
|
* Path 10 (M at least N, but not much larger)
|
|
* Reduce to bidiagonal form without QR decomposition
|
|
*
|
|
IE = 1
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize A
|
|
* (Workspace: need 3*N + M, prefer 3*N + (M + N)*NB)
|
|
*
|
|
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
|
|
$ IERR )
|
|
IF( WNTUAS ) THEN
|
|
*
|
|
* If left singular vectors desired in U, copy result to U
|
|
* and generate left bidiagonalizing vectors in U
|
|
* (Workspace: need 3*N + NCU, prefer 3*N + NCU*NB)
|
|
*
|
|
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
|
|
IF( WNTUS )
|
|
$ NCU = N
|
|
IF( WNTUA )
|
|
$ NCU = M
|
|
CALL DORGBR( 'Q', M, NCU, N, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTVAS ) THEN
|
|
*
|
|
* If right singular vectors desired in VT, copy result to
|
|
* VT and generate right bidiagonalizing vectors in VT
|
|
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
|
|
*
|
|
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
|
|
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTUO ) THEN
|
|
*
|
|
* If left singular vectors desired in A, generate left
|
|
* bidiagonalizing vectors in A
|
|
* (Workspace: need 4*N, prefer 3*N + N*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTVO ) THEN
|
|
*
|
|
* If right singular vectors desired in A, generate right
|
|
* bidiagonalizing vectors in A
|
|
* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IWORK = IE + N
|
|
IF( WNTUAS .OR. WNTUO )
|
|
$ NRU = M
|
|
IF( WNTUN )
|
|
$ NRU = 0
|
|
IF( WNTVAS .OR. WNTVO )
|
|
$ NCVT = N
|
|
IF( WNTVN )
|
|
$ NCVT = 0
|
|
IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in U and computing right singular
|
|
* vectors in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), VT,
|
|
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ), INFO )
|
|
ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in U and computing right singular
|
|
* vectors in A
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), A, LDA,
|
|
$ U, LDU, DUM, 1, WORK( IWORK ), INFO )
|
|
ELSE
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in A and computing right singular
|
|
* vectors in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), VT,
|
|
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ), INFO )
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
ELSE
|
|
*
|
|
* A has more columns than rows. If A has sufficiently more
|
|
* columns than rows, first reduce using the LQ decomposition (if
|
|
* sufficient workspace available)
|
|
*
|
|
IF( N.GE.MNTHR ) THEN
|
|
*
|
|
IF( WNTVN ) THEN
|
|
*
|
|
* Path 1t(N much larger than M, JOBVT='N')
|
|
* No right singular vectors to be computed
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Zero out above L
|
|
*
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), LDA )
|
|
IE = 1
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in A
|
|
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
|
|
$ IERR )
|
|
IF( WNTUO .OR. WNTUAS ) THEN
|
|
*
|
|
* If left singular vectors desired, generate Q
|
|
* (Workspace: need 4*M, prefer 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IWORK = IE + M
|
|
NRU = 0
|
|
IF( WNTUO .OR. WNTUAS )
|
|
$ NRU = M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left singular
|
|
* vectors of A in A if desired
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, 0, NRU, 0, S, WORK( IE ), DUM, 1, A,
|
|
$ LDA, DUM, 1, WORK( IWORK ), INFO )
|
|
*
|
|
* If left singular vectors desired in U, copy them there
|
|
*
|
|
IF( WNTUAS )
|
|
$ CALL DLACPY( 'F', M, M, A, LDA, U, LDU )
|
|
*
|
|
ELSE IF( WNTVO .AND. WNTUN ) THEN
|
|
*
|
|
* Path 2t(N much larger than M, JOBU='N', JOBVT='O')
|
|
* M right singular vectors to be overwritten on A and
|
|
* no left singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is LDA by M
|
|
*
|
|
LDWRKU = LDA
|
|
CHUNK = N
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + M*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = LDA
|
|
CHUNK = N
|
|
LDWRKR = M
|
|
ELSE
|
|
*
|
|
* WORK(IU) is M by CHUNK and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = M
|
|
CHUNK = ( LWORK-M*M-M ) / M
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IR) and zero out above it
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ), LDWRKR )
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
|
|
$ WORK( IR+LDWRKR ), LDWRKR )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
|
|
*
|
|
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IR)
|
|
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing L
|
|
* (Workspace: need M*M + 4*M-1, prefer M*M + 3*M + (M-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of L in WORK(IR)
|
|
* (Workspace: need M*M + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
|
|
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
IU = IE + M
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IR) by Q
|
|
* in A, storing result in WORK(IU) and copying to A
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M*N + M)
|
|
*
|
|
DO 30 I = 1, N, CHUNK
|
|
BLK = MIN( N-I+1, CHUNK )
|
|
CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IR ),
|
|
$ LDWRKR, A( 1, I ), LDA, ZERO,
|
|
$ WORK( IU ), LDWRKU )
|
|
CALL DLACPY( 'F', M, BLK, WORK( IU ), LDWRKU,
|
|
$ A( 1, I ), LDA )
|
|
30 CONTINUE
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
IE = 1
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize A
|
|
* (Workspace: need 3*M + N, prefer 3*M + (M + N)*NB)
|
|
*
|
|
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing A
|
|
* (Workspace: need 4*M, prefer 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of A in A
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'L', M, N, 0, 0, S, WORK( IE ), A, LDA,
|
|
$ DUM, 1, DUM, 1, WORK( IWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVO .AND. WNTUAS ) THEN
|
|
*
|
|
* Path 3t(N much larger than M, JOBU='S' or 'A', JOBVT='O')
|
|
* M right singular vectors to be overwritten on A and
|
|
* M left singular vectors to be computed in U
|
|
*
|
|
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is LDA by M
|
|
*
|
|
LDWRKU = LDA
|
|
CHUNK = N
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + M*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = LDA
|
|
CHUNK = N
|
|
LDWRKR = M
|
|
ELSE
|
|
*
|
|
* WORK(IU) is M by CHUNK and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = M
|
|
CHUNK = ( LWORK-M*M-M ) / M
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to U, zeroing about above it
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
|
|
$ LDU )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
|
|
*
|
|
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in U, copying result to WORK(IR)
|
|
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', M, M, U, LDU, WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate right vectors bidiagonalizing L in WORK(IR)
|
|
* (Workspace: need M*M + 4*M-1, prefer M*M + 3*M + (M-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left vectors bidiagonalizing L in U
|
|
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of L in U, and computing right
|
|
* singular vectors of L in WORK(IR)
|
|
* (Workspace: need M*M + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
|
|
$ WORK( IR ), LDWRKR, U, LDU, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
IU = IE + M
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IR) by Q
|
|
* in A, storing result in WORK(IU) and copying to A
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M*N + M))
|
|
*
|
|
DO 40 I = 1, N, CHUNK
|
|
BLK = MIN( N-I+1, CHUNK )
|
|
CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IR ),
|
|
$ LDWRKR, A( 1, I ), LDA, ZERO,
|
|
$ WORK( IU ), LDWRKU )
|
|
CALL DLACPY( 'F', M, BLK, WORK( IU ), LDWRKU,
|
|
$ A( 1, I ), LDA )
|
|
40 CONTINUE
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to U, zeroing out above it
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
|
|
$ LDU )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in U
|
|
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right vectors bidiagonalizing L by Q in A
|
|
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
|
|
*
|
|
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
|
|
$ WORK( ITAUP ), A, LDA, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left vectors bidiagonalizing L in U
|
|
* (Workspace: need 4*M, prefer 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in A
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), A, LDA,
|
|
$ U, LDU, DUM, 1, WORK( IWORK ), INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVS ) THEN
|
|
*
|
|
IF( WNTUN ) THEN
|
|
*
|
|
* Path 4t(N much larger than M, JOBU='N', JOBVT='S')
|
|
* M right singular vectors to be computed in VT and
|
|
* no left singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.WRKBL+LDA*M ) THEN
|
|
*
|
|
* WORK(IR) is LDA by M
|
|
*
|
|
LDWRKR = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IR) is M by M
|
|
*
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IR), zeroing out above it
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ),
|
|
$ LDWRKR )
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
|
|
$ WORK( IR+LDWRKR ), LDWRKR )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
|
|
*
|
|
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IR)
|
|
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right vectors bidiagonalizing L in
|
|
* WORK(IR)
|
|
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + (M-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of L in WORK(IR)
|
|
* (Workspace: need M*M + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
|
|
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IR) by
|
|
* Q in A, storing result in VT
|
|
* (Workspace: need M*M)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IR ),
|
|
$ LDWRKR, A, LDA, ZERO, VT, LDVT )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy result to VT
|
|
*
|
|
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Zero out above L in A
|
|
*
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
|
|
$ LDA )
|
|
*
|
|
* Bidiagonalize L in A
|
|
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right vectors bidiagonalizing L by Q in VT
|
|
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
|
|
*
|
|
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of A in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, N, 0, 0, S, WORK( IE ), VT,
|
|
$ LDVT, DUM, 1, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUO ) THEN
|
|
*
|
|
* Path 5t(N much larger than M, JOBU='O', JOBVT='S')
|
|
* M right singular vectors to be computed in VT and
|
|
* M left singular vectors to be overwritten on A
|
|
*
|
|
IF( LWORK.GE.2*M*M+MAX( 4*M, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+2*LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by M and WORK(IR) is LDA by M
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.WRKBL+( LDA + M )*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by M and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = M
|
|
ELSE
|
|
*
|
|
* WORK(IU) is M by M and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = M
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IU), zeroing out below it
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
|
|
$ WORK( IU+LDWRKU ), LDWRKU )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
|
|
*
|
|
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IU), copying result to
|
|
* WORK(IR)
|
|
* (Workspace: need 2*M*M + 4*M,
|
|
* prefer 2*M*M+3*M+2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IU)
|
|
* (Workspace: need 2*M*M + 4*M-1,
|
|
* prefer 2*M*M+3*M+(M-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IR)
|
|
* (Workspace: need 2*M*M + 4*M, prefer 2*M*M + 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of L in WORK(IR) and computing
|
|
* right singular vectors of L in WORK(IU)
|
|
* (Workspace: need 2*M*M + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
|
|
$ WORK( IU ), LDWRKU, WORK( IR ),
|
|
$ LDWRKR, DUM, 1, WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IU) by
|
|
* Q in A, storing result in VT
|
|
* (Workspace: need M*M)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
|
|
$ LDWRKU, A, LDA, ZERO, VT, LDVT )
|
|
*
|
|
* Copy left singular vectors of L to A
|
|
* (Workspace: need M*M)
|
|
*
|
|
CALL DLACPY( 'F', M, M, WORK( IR ), LDWRKR, A,
|
|
$ LDA )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Zero out above L in A
|
|
*
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
|
|
$ LDA )
|
|
*
|
|
* Bidiagonalize L in A
|
|
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right vectors bidiagonalizing L by Q in VT
|
|
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
|
|
*
|
|
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors of L in A
|
|
* (Workspace: need 4*M, prefer 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, compute left
|
|
* singular vectors of A in A and compute right
|
|
* singular vectors of A in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
|
|
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUAS ) THEN
|
|
*
|
|
* Path 6t(N much larger than M, JOBU='S' or 'A',
|
|
* JOBVT='S')
|
|
* M right singular vectors to be computed in VT and
|
|
* M left singular vectors to be computed in U
|
|
*
|
|
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by N
|
|
*
|
|
LDWRKU = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IU) is LDA by M
|
|
*
|
|
LDWRKU = M
|
|
END IF
|
|
ITAU = IU + LDWRKU*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IU), zeroing out above it
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
|
|
$ WORK( IU+LDWRKU ), LDWRKU )
|
|
*
|
|
* Generate Q in A
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
|
|
*
|
|
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IU), copying result to U
|
|
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, U,
|
|
$ LDU )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IU)
|
|
* (Workspace: need M*M + 4*M-1,
|
|
* prefer M*M+3*M+(M-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in U
|
|
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of L in U and computing right
|
|
* singular vectors of L in WORK(IU)
|
|
* (Workspace: need M*M + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
|
|
$ WORK( IU ), LDWRKU, U, LDU, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IU) by
|
|
* Q in A, storing result in VT
|
|
* (Workspace: need M*M)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
|
|
$ LDWRKU, A, LDA, ZERO, VT, LDVT )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to U, zeroing out above it
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
|
|
$ LDU )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in U
|
|
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right bidiagonalizing vectors in U by Q
|
|
* in VT
|
|
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
|
|
*
|
|
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in U
|
|
* (Workspace: need 4*M, prefer 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
|
|
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTVA ) THEN
|
|
*
|
|
IF( WNTUN ) THEN
|
|
*
|
|
* Path 7t(N much larger than M, JOBU='N', JOBVT='A')
|
|
* N right singular vectors to be computed in VT and
|
|
* no left singular vectors to be computed
|
|
*
|
|
IF( LWORK.GE.M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IR = 1
|
|
IF( LWORK.GE.WRKBL+LDA*M ) THEN
|
|
*
|
|
* WORK(IR) is LDA by M
|
|
*
|
|
LDWRKR = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IR) is M by M
|
|
*
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Copy L to WORK(IR), zeroing out above it
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ),
|
|
$ LDWRKR )
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
|
|
$ WORK( IR+LDWRKR ), LDWRKR )
|
|
*
|
|
* Generate Q in VT
|
|
* (Workspace: need M*M + M + N, prefer M*M + M + N*NB)
|
|
*
|
|
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IR)
|
|
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IR)
|
|
* (Workspace: need M*M + 4*M-1,
|
|
* prefer M*M+3*M+(M-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of L in WORK(IR)
|
|
* (Workspace: need M*M + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
|
|
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IR) by
|
|
* Q in VT, storing result in A
|
|
* (Workspace: need M*M)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IR ),
|
|
$ LDWRKR, VT, LDVT, ZERO, A, LDA )
|
|
*
|
|
* Copy right singular vectors of A from A to VT
|
|
*
|
|
CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (Workspace: need M + N, prefer M + N*NB)
|
|
*
|
|
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Zero out above L in A
|
|
*
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
|
|
$ LDA )
|
|
*
|
|
* Bidiagonalize L in A
|
|
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right bidiagonalizing vectors in A by Q
|
|
* in VT
|
|
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
|
|
*
|
|
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing right
|
|
* singular vectors of A in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, N, 0, 0, S, WORK( IE ), VT,
|
|
$ LDVT, DUM, 1, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUO ) THEN
|
|
*
|
|
* Path 8t(N much larger than M, JOBU='O', JOBVT='A')
|
|
* N right singular vectors to be computed in VT and
|
|
* M left singular vectors to be overwritten on A
|
|
*
|
|
IF( LWORK.GE.2*M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+2*LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by M and WORK(IR) is LDA by M
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = LDA
|
|
ELSE IF( LWORK.GE.WRKBL+( LDA + M )*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by M and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = LDA
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = M
|
|
ELSE
|
|
*
|
|
* WORK(IU) is M by M and WORK(IR) is M by M
|
|
*
|
|
LDWRKU = M
|
|
IR = IU + LDWRKU*M
|
|
LDWRKR = M
|
|
END IF
|
|
ITAU = IR + LDWRKR*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (Workspace: need 2*M*M + M + N, prefer 2*M*M + M + N*NB)
|
|
*
|
|
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IU), zeroing out above it
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
|
|
$ WORK( IU+LDWRKU ), LDWRKU )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IU), copying result to
|
|
* WORK(IR)
|
|
* (Workspace: need 2*M*M + 4*M,
|
|
* prefer 2*M*M+3*M+2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( IR ), LDWRKR )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IU)
|
|
* (Workspace: need 2*M*M + 4*M-1,
|
|
* prefer 2*M*M+3*M+(M-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in WORK(IR)
|
|
* (Workspace: need 2*M*M + 4*M, prefer 2*M*M + 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
|
|
$ WORK( ITAUQ ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of L in WORK(IR) and computing
|
|
* right singular vectors of L in WORK(IU)
|
|
* (Workspace: need 2*M*M + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
|
|
$ WORK( IU ), LDWRKU, WORK( IR ),
|
|
$ LDWRKR, DUM, 1, WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IU) by
|
|
* Q in VT, storing result in A
|
|
* (Workspace: need M*M)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
|
|
$ LDWRKU, VT, LDVT, ZERO, A, LDA )
|
|
*
|
|
* Copy right singular vectors of A from A to VT
|
|
*
|
|
CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Copy left singular vectors of A from WORK(IR) to A
|
|
*
|
|
CALL DLACPY( 'F', M, M, WORK( IR ), LDWRKR, A,
|
|
$ LDA )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (Workspace: need M + N, prefer M + N*NB)
|
|
*
|
|
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Zero out above L in A
|
|
*
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
|
|
$ LDA )
|
|
*
|
|
* Bidiagonalize L in A
|
|
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right bidiagonalizing vectors in A by Q
|
|
* in VT
|
|
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
|
|
*
|
|
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in A
|
|
* (Workspace: need 4*M, prefer 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in A and computing right
|
|
* singular vectors of A in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
|
|
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
ELSE IF( WNTUAS ) THEN
|
|
*
|
|
* Path 9t(N much larger than M, JOBU='S' or 'A',
|
|
* JOBVT='A')
|
|
* N right singular vectors to be computed in VT and
|
|
* M left singular vectors to be computed in U
|
|
*
|
|
IF( LWORK.GE.M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
|
|
*
|
|
* Sufficient workspace for a fast algorithm
|
|
*
|
|
IU = 1
|
|
IF( LWORK.GE.WRKBL+LDA*M ) THEN
|
|
*
|
|
* WORK(IU) is LDA by M
|
|
*
|
|
LDWRKU = LDA
|
|
ELSE
|
|
*
|
|
* WORK(IU) is M by M
|
|
*
|
|
LDWRKU = M
|
|
END IF
|
|
ITAU = IU + LDWRKU*M
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (Workspace: need M*M + M + N, prefer M*M + M + N*NB)
|
|
*
|
|
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to WORK(IU), zeroing out above it
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
|
|
$ LDWRKU )
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
|
|
$ WORK( IU+LDWRKU ), LDWRKU )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IU), copying result to U
|
|
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
|
|
$ WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, U,
|
|
$ LDU )
|
|
*
|
|
* Generate right bidiagonalizing vectors in WORK(IU)
|
|
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + (M-1)*NB)
|
|
*
|
|
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
|
|
$ WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in U
|
|
* (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of L in U and computing right
|
|
* singular vectors of L in WORK(IU)
|
|
* (Workspace: need M*M + BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
|
|
$ WORK( IU ), LDWRKU, U, LDU, DUM, 1,
|
|
$ WORK( IWORK ), INFO )
|
|
*
|
|
* Multiply right singular vectors of L in WORK(IU) by
|
|
* Q in VT, storing result in A
|
|
* (Workspace: need M*M)
|
|
*
|
|
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
|
|
$ LDWRKU, VT, LDVT, ZERO, A, LDA )
|
|
*
|
|
* Copy right singular vectors of A from A to VT
|
|
*
|
|
CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
ELSE
|
|
*
|
|
* Insufficient workspace for a fast algorithm
|
|
*
|
|
ITAU = 1
|
|
IWORK = ITAU + M
|
|
*
|
|
* Compute A=L*Q, copying result to VT
|
|
* (Workspace: need 2*M, prefer M + M*NB)
|
|
*
|
|
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
*
|
|
* Generate Q in VT
|
|
* (Workspace: need M + N, prefer M + N*NB)
|
|
*
|
|
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Copy L to U, zeroing out above it
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
|
|
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
|
|
$ LDU )
|
|
IE = ITAU
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in U
|
|
* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
|
|
*
|
|
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Multiply right bidiagonalizing vectors in U by Q
|
|
* in VT
|
|
* (Workspace: need 3*M + N, prefer 3*M + N*NB)
|
|
*
|
|
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
|
|
$ WORK( ITAUP ), VT, LDVT,
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
*
|
|
* Generate left bidiagonalizing vectors in U
|
|
* (Workspace: need 4*M, prefer 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration, computing left
|
|
* singular vectors of A in U and computing right
|
|
* singular vectors of A in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
|
|
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
|
|
$ INFO )
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
ELSE
|
|
*
|
|
* N .LT. MNTHR
|
|
*
|
|
* Path 10t(N greater than M, but not much larger)
|
|
* Reduce to bidiagonal form without LQ decomposition
|
|
*
|
|
IE = 1
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize A
|
|
* (Workspace: need 3*M + N, prefer 3*M + (M + N)*NB)
|
|
*
|
|
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
|
|
$ IERR )
|
|
IF( WNTUAS ) THEN
|
|
*
|
|
* If left singular vectors desired in U, copy result to U
|
|
* and generate left bidiagonalizing vectors in U
|
|
* (Workspace: need 4*M-1, prefer 3*M + (M-1)*NB)
|
|
*
|
|
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
|
|
CALL DORGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTVAS ) THEN
|
|
*
|
|
* If right singular vectors desired in VT, copy result to
|
|
* VT and generate right bidiagonalizing vectors in VT
|
|
* (Workspace: need 3*M + NRVT, prefer 3*M + NRVT*NB)
|
|
*
|
|
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
|
|
IF( WNTVA )
|
|
$ NRVT = N
|
|
IF( WNTVS )
|
|
$ NRVT = M
|
|
CALL DORGBR( 'P', NRVT, N, M, VT, LDVT, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTUO ) THEN
|
|
*
|
|
* If left singular vectors desired in A, generate left
|
|
* bidiagonalizing vectors in A
|
|
* (Workspace: need 4*M-1, prefer 3*M + (M-1)*NB)
|
|
*
|
|
CALL DORGBR( 'Q', M, M, N, A, LDA, WORK( ITAUQ ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IF( WNTVO ) THEN
|
|
*
|
|
* If right singular vectors desired in A, generate right
|
|
* bidiagonalizing vectors in A
|
|
* (Workspace: need 4*M, prefer 3*M + M*NB)
|
|
*
|
|
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
|
|
END IF
|
|
IWORK = IE + M
|
|
IF( WNTUAS .OR. WNTUO )
|
|
$ NRU = M
|
|
IF( WNTUN )
|
|
$ NRU = 0
|
|
IF( WNTVAS .OR. WNTVO )
|
|
$ NCVT = N
|
|
IF( WNTVN )
|
|
$ NCVT = 0
|
|
IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in U and computing right singular
|
|
* vectors in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), VT,
|
|
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ), INFO )
|
|
ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in U and computing right singular
|
|
* vectors in A
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), A, LDA,
|
|
$ U, LDU, DUM, 1, WORK( IWORK ), INFO )
|
|
ELSE
|
|
*
|
|
* Perform bidiagonal QR iteration, if desired, computing
|
|
* left singular vectors in A and computing right singular
|
|
* vectors in VT
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), VT,
|
|
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ), INFO )
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
END IF
|
|
*
|
|
* If DBDSQR failed to converge, copy unconverged superdiagonals
|
|
* to WORK( 2:MINMN )
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
IF( IE.GT.2 ) THEN
|
|
DO 50 I = 1, MINMN - 1
|
|
WORK( I+1 ) = WORK( I+IE-1 )
|
|
50 CONTINUE
|
|
END IF
|
|
IF( IE.LT.2 ) THEN
|
|
DO 60 I = MINMN - 1, 1, -1
|
|
WORK( I+1 ) = WORK( I+IE-1 )
|
|
60 CONTINUE
|
|
END IF
|
|
END IF
|
|
*
|
|
* Undo scaling if necessary
|
|
*
|
|
IF( ISCL.EQ.1 ) THEN
|
|
IF( ANRM.GT.BIGNUM )
|
|
$ CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
|
|
$ IERR )
|
|
IF( INFO.NE.0 .AND. ANRM.GT.BIGNUM )
|
|
$ CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN-1, 1, WORK( 2 ),
|
|
$ MINMN, IERR )
|
|
IF( ANRM.LT.SMLNUM )
|
|
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
|
|
$ IERR )
|
|
IF( INFO.NE.0 .AND. ANRM.LT.SMLNUM )
|
|
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN-1, 1, WORK( 2 ),
|
|
$ MINMN, IERR )
|
|
END IF
|
|
*
|
|
* Return optimal workspace in WORK(1)
|
|
*
|
|
WORK( 1 ) = MAXWRK
|
|
*
|
|
RETURN
|
|
*
|
|
* End of DGESVD
|
|
*
|
|
END
|