version 1.1, 2010/01/26 15:22:46
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version 1.15, 2015/11/26 11:44:15
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*> \brief \b DGESDD |
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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 DGESDD + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgesdd.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/dgesdd.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/dgesdd.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 DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, |
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* LWORK, IWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER JOBZ |
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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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* INTEGER IWORK( * ) |
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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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*> DGESDD computes the singular value decomposition (SVD) of a real |
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*> M-by-N matrix A, optionally computing the left and right singular |
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*> vectors. If singular vectors are desired, it uses a |
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*> divide-and-conquer algorithm. |
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*> |
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*> 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 VT = V**T, not V. |
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*> |
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*> The divide and conquer algorithm makes very mild assumptions about |
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*> floating point arithmetic. It will work on machines with a guard |
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*> digit in add/subtract, or on those binary machines without guard |
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*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or |
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*> Cray-2. It could conceivably fail on hexadecimal or decimal machines |
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*> without guard digits, but we know of none. |
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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] JOBZ |
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*> \verbatim |
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*> JOBZ 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 and all N rows of V**T are |
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*> returned in the arrays U and VT; |
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*> = 'S': the first min(M,N) columns of U and the first |
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*> min(M,N) rows of V**T are returned in the arrays U |
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*> and VT; |
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*> = 'O': If M >= N, the first N columns of U are overwritten |
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*> on the array A and all rows of V**T are returned in |
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*> the array VT; |
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*> otherwise, all columns of U are returned in the |
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*> array U and the first M rows of V**T are overwritten |
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*> in the array A; |
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*> = 'N': no columns of U or rows of V**T are computed. |
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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 JOBZ = 'O', A is overwritten with the first N columns |
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*> of U (the left singular vectors, stored |
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*> columnwise) if M >= N; |
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*> A is overwritten with the first M rows |
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*> of V**T (the right singular vectors, stored |
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*> rowwise) otherwise. |
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*> if JOBZ .ne. 'O', the contents of A 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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*> UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N; |
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*> UCOL = min(M,N) if JOBZ = 'S'. |
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*> If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M |
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*> orthogonal matrix U; |
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*> if JOBZ = '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 JOBZ = 'O' and M >= N, or JOBZ = 'N', 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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*> JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, 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 JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the |
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*> N-by-N orthogonal matrix V**T; |
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*> if JOBZ = '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 JOBZ = 'O' and M < N, or JOBZ = 'N', 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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*> JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N; |
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*> if JOBZ = '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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*> \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. LWORK >= 1. |
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*> If JOBZ = 'N', |
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*> LWORK >= 3*min(M,N) + max(max(M,N),7*min(M,N)). |
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*> If JOBZ = 'O', |
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*> LWORK >= 3*min(M,N) + |
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*> max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)). |
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*> If JOBZ = 'S' or 'A' |
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*> LWORK >= min(M,N)*(7+4*min(M,N)) |
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*> For good performance, LWORK should generally be larger. |
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*> If LWORK = -1 but other input arguments are legal, WORK(1) |
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*> returns the optimal LWORK. |
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*> \endverbatim |
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*> |
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*> \param[out] IWORK |
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*> \verbatim |
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*> IWORK is INTEGER array, dimension (8*min(M,N)) |
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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: DBDSDC did not converge, updating process failed. |
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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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*> \date November 2015 |
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* |
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*> \ingroup doubleGEsing |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> Ming Gu and Huan Ren, Computer Science Division, University of |
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*> California at Berkeley, USA |
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*> |
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* ===================================================================== |
SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, |
SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, |
$ LWORK, IWORK, INFO ) |
$ LWORK, IWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2.1) -- |
* -- LAPACK driver routine (version 3.6.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* March 2009 |
* November 2015 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBZ |
CHARACTER JOBZ |
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$ VT( LDVT, * ), WORK( * ) |
$ VT( LDVT, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
|
* |
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* DGESDD computes the singular value decomposition (SVD) of a real |
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* M-by-N matrix A, optionally computing the left and right singular |
|
* vectors. If singular vectors are desired, it uses a |
|
* divide-and-conquer algorithm. |
|
* |
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* 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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* Note that the routine returns VT = V**T, not V. |
|
* |
|
* The divide and conquer algorithm makes very mild assumptions about |
|
* floating point arithmetic. It will work on machines with a guard |
|
* digit in add/subtract, or on those binary machines without guard |
|
* digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or |
|
* Cray-2. It could conceivably fail on hexadecimal or decimal machines |
|
* without guard digits, but we know of none. |
|
* |
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* Arguments |
|
* ========= |
|
* |
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* JOBZ (input) 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 and all N rows of V**T are |
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* returned in the arrays U and VT; |
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* = 'S': the first min(M,N) columns of U and the first |
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* min(M,N) rows of V**T are returned in the arrays U |
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* and VT; |
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* = 'O': If M >= N, the first N columns of U are overwritten |
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* on the array A and all rows of V**T are returned in |
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* the array VT; |
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* otherwise, all columns of U are returned in the |
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* array U and the first M rows of V**T are overwritten |
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* in the array A; |
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* = 'N': no columns of U or rows of V**T are computed. |
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* |
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* M (input) INTEGER |
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* The number of rows of the input matrix A. M >= 0. |
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* |
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* N (input) INTEGER |
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* The number of columns of the input matrix A. N >= 0. |
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* |
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* A (input/output) 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 JOBZ = 'O', A is overwritten with the first N columns |
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* of U (the left singular vectors, stored |
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* columnwise) if M >= N; |
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* A is overwritten with the first M rows |
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* of V**T (the right singular vectors, stored |
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* rowwise) otherwise. |
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* if JOBZ .ne. 'O', the contents of A are destroyed. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= max(1,M). |
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* |
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* S (output) 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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* |
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* U (output) DOUBLE PRECISION array, dimension (LDU,UCOL) |
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* UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N; |
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* UCOL = min(M,N) if JOBZ = 'S'. |
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* If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M |
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* orthogonal matrix U; |
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* if JOBZ = '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 JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced. |
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* |
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* LDU (input) INTEGER |
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* The leading dimension of the array U. LDU >= 1; if |
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* JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M. |
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* |
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* VT (output) DOUBLE PRECISION array, dimension (LDVT,N) |
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* If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the |
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* N-by-N orthogonal matrix V**T; |
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* if JOBZ = '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 JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced. |
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* |
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* LDVT (input) INTEGER |
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* The leading dimension of the array VT. LDVT >= 1; if |
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* JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N; |
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* if JOBZ = 'S', LDVT >= min(M,N). |
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* |
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* WORK (workspace/output) 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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* |
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* LWORK (input) INTEGER |
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* The dimension of the array WORK. LWORK >= 1. |
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* If JOBZ = 'N', |
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* LWORK >= 3*min(M,N) + max(max(M,N),7*min(M,N)). |
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* If JOBZ = 'O', |
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* LWORK >= 3*min(M,N) + |
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* max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)). |
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* If JOBZ = 'S' or 'A' |
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* LWORK >= 3*min(M,N) + |
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* max(max(M,N),4*min(M,N)*min(M,N)+4*min(M,N)). |
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* For good performance, LWORK should generally be larger. |
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* If LWORK = -1 but other input arguments are legal, WORK(1) |
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* returns the optimal LWORK. |
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* |
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* IWORK (workspace) INTEGER array, dimension (8*min(M,N)) |
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* |
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* INFO (output) 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: DBDSDC did not converge, updating process failed. |
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* |
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* Further Details |
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* =============== |
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* |
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* Based on contributions by |
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* Ming Gu and Huan Ren, Computer Science Division, University of |
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* California at Berkeley, USA |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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$ ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) ) |
$ ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) ) |
WRKBL = MAX( WRKBL, BDSPAC+3*N ) |
WRKBL = MAX( WRKBL, BDSPAC+3*N ) |
MAXWRK = WRKBL + N*N |
MAXWRK = WRKBL + N*N |
MINWRK = BDSPAC + N*N + 3*N |
MINWRK = BDSPAC + N*N + 2*N + M |
END IF |
END IF |
ELSE |
ELSE |
* |
* |
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NWORK = ITAU + N |
NWORK = ITAU + N |
* |
* |
* Compute A=Q*R, copying result to U |
* Compute A=Q*R, copying result to U |
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) |
* (Workspace: need N*N+N+M, prefer N*N+N+M*NB) |
* |
* |
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), |
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), |
$ LWORK-NWORK+1, IERR ) |
$ LWORK-NWORK+1, IERR ) |
CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) |
CALL DLACPY( 'L', M, N, A, LDA, U, LDU ) |
* |
* |
* Generate Q in U |
* Generate Q in U |
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB) |
* (Workspace: need N*N+N+M, prefer N*N+N+M*NB) |
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ), |
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ), |
$ WORK( NWORK ), LWORK-NWORK+1, IERR ) |
$ WORK( NWORK ), LWORK-NWORK+1, IERR ) |
* |
* |