version 1.8, 2010/12/21 13:53:33
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version 1.18, 2017/06/17 11:06:26
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*> \brief \b DLASDA computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by sbdsdc. |
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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 DLASDA + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasda.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/dlasda.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/dlasda.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 DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, |
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* DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, |
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* PERM, GIVNUM, C, S, WORK, IWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE |
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* .. |
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* .. Array Arguments .. |
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* INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ), |
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* $ K( * ), PERM( LDGCOL, * ) |
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* DOUBLE PRECISION C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ), |
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* $ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ), |
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* $ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ), |
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* $ Z( LDU, * ) |
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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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*> Using a divide and conquer approach, DLASDA computes the singular |
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*> value decomposition (SVD) of a real upper bidiagonal N-by-M matrix |
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*> B with diagonal D and offdiagonal E, where M = N + SQRE. The |
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*> algorithm computes the singular values in the SVD B = U * S * VT. |
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*> The orthogonal matrices U and VT are optionally computed in |
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*> compact form. |
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*> |
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*> A related subroutine, DLASD0, computes the singular values and |
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*> the singular vectors in explicit form. |
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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] ICOMPQ |
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*> \verbatim |
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*> ICOMPQ is INTEGER |
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*> Specifies whether singular vectors are to be computed |
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*> in compact form, as follows |
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*> = 0: Compute singular values only. |
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*> = 1: Compute singular vectors of upper bidiagonal |
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*> matrix in compact form. |
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*> \endverbatim |
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*> |
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*> \param[in] SMLSIZ |
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*> \verbatim |
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*> SMLSIZ is INTEGER |
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*> The maximum size of the subproblems at the bottom of the |
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*> computation tree. |
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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 row dimension of the upper bidiagonal matrix. This is |
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*> also the dimension of the main diagonal array D. |
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*> \endverbatim |
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*> |
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*> \param[in] SQRE |
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*> \verbatim |
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*> SQRE is INTEGER |
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*> Specifies the column dimension of the bidiagonal matrix. |
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*> = 0: The bidiagonal matrix has column dimension M = N; |
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*> = 1: The bidiagonal matrix has column dimension M = N + 1. |
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*> \endverbatim |
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*> |
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*> \param[in,out] D |
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*> \verbatim |
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*> D is DOUBLE PRECISION array, dimension ( N ) |
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*> On entry D contains the main diagonal of the bidiagonal |
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*> matrix. On exit D, if INFO = 0, contains its singular values. |
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*> \endverbatim |
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*> |
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*> \param[in] E |
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*> \verbatim |
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*> E is DOUBLE PRECISION array, dimension ( M-1 ) |
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*> Contains the subdiagonal entries of the bidiagonal matrix. |
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*> On exit, E has been destroyed. |
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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, |
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*> dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced |
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*> if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left |
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*> singular vector matrices of all subproblems at the bottom |
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*> level. |
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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, LDU = > N. |
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*> The leading dimension of arrays U, VT, DIFL, DIFR, POLES, |
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*> GIVNUM, and Z. |
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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, |
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*> dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced |
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*> if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right |
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*> singular vector matrices of all subproblems at the bottom |
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*> level. |
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*> \endverbatim |
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*> |
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*> \param[out] K |
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*> \verbatim |
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*> K is INTEGER array, |
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*> dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. |
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*> If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th |
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*> secular equation on the computation tree. |
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*> \endverbatim |
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*> |
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*> \param[out] DIFL |
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*> \verbatim |
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*> DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ), |
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*> where NLVL = floor(log_2 (N/SMLSIZ))). |
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*> \endverbatim |
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*> |
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*> \param[out] DIFR |
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*> \verbatim |
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*> DIFR is DOUBLE PRECISION array, |
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*> dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and |
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*> dimension ( N ) if ICOMPQ = 0. |
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*> If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1) |
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*> record distances between singular values on the I-th |
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*> level and singular values on the (I -1)-th level, and |
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*> DIFR(1:N, 2 * I ) contains the normalizing factors for |
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*> the right singular vector matrix. See DLASD8 for details. |
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*> \endverbatim |
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*> |
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*> \param[out] Z |
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*> \verbatim |
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*> Z is DOUBLE PRECISION array, |
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*> dimension ( LDU, NLVL ) if ICOMPQ = 1 and |
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*> dimension ( N ) if ICOMPQ = 0. |
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*> The first K elements of Z(1, I) contain the components of |
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*> the deflation-adjusted updating row vector for subproblems |
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*> on the I-th level. |
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*> \endverbatim |
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*> |
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*> \param[out] POLES |
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*> \verbatim |
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*> POLES is DOUBLE PRECISION array, |
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*> dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced |
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*> if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and |
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*> POLES(1, 2*I) contain the new and old singular values |
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*> involved in the secular equations on the I-th level. |
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*> \endverbatim |
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*> |
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*> \param[out] GIVPTR |
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*> \verbatim |
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*> GIVPTR is INTEGER array, |
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*> dimension ( N ) if ICOMPQ = 1, and not referenced if |
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*> ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records |
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*> the number of Givens rotations performed on the I-th |
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*> problem on the computation tree. |
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*> \endverbatim |
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*> |
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*> \param[out] GIVCOL |
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*> \verbatim |
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*> GIVCOL is INTEGER array, |
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*> dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not |
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*> referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, |
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*> GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations |
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*> of Givens rotations performed on the I-th level on the |
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*> computation tree. |
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*> \endverbatim |
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*> |
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*> \param[in] LDGCOL |
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*> \verbatim |
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*> LDGCOL is INTEGER, LDGCOL = > N. |
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*> The leading dimension of arrays GIVCOL and PERM. |
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*> \endverbatim |
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*> |
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*> \param[out] PERM |
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*> \verbatim |
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*> PERM is INTEGER array, |
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*> dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced |
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*> if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records |
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*> permutations done on the I-th level of the computation tree. |
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*> \endverbatim |
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*> |
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*> \param[out] GIVNUM |
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*> \verbatim |
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*> GIVNUM is DOUBLE PRECISION array, |
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*> dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not |
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*> referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, |
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*> GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S- |
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*> values of Givens rotations performed on the I-th level on |
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*> the computation tree. |
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*> \endverbatim |
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*> |
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*> \param[out] C |
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*> \verbatim |
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*> C is DOUBLE PRECISION array, |
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*> dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. |
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*> If ICOMPQ = 1 and the I-th subproblem is not square, on exit, |
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*> C( I ) contains the C-value of a Givens rotation related to |
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*> the right null space of the I-th subproblem. |
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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 ( N ) if |
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*> ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 |
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*> and the I-th subproblem is not square, on exit, S( I ) |
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*> contains the S-value of a Givens rotation related to |
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*> the right null space of the I-th subproblem. |
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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 |
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*> (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)). |
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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. |
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*> Dimension must be at least (7 * 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: if INFO = 1, a singular value did not converge |
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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 December 2016 |
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* |
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*> \ingroup OTHERauxiliary |
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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 DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, |
SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, |
$ DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, |
$ DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL, |
$ PERM, GIVNUM, C, S, WORK, IWORK, INFO ) |
$ PERM, GIVNUM, C, S, WORK, IWORK, INFO ) |
* |
* |
* -- LAPACK auxiliary routine (version 3.2.2) -- |
* -- LAPACK auxiliary routine (version 3.7.0) -- |
* -- 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..-- |
* June 2010 |
* December 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE |
INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE |
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$ Z( LDU, * ) |
$ Z( LDU, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* Using a divide and conquer approach, DLASDA computes the singular |
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* value decomposition (SVD) of a real upper bidiagonal N-by-M matrix |
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* B with diagonal D and offdiagonal E, where M = N + SQRE. The |
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* algorithm computes the singular values in the SVD B = U * S * VT. |
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* The orthogonal matrices U and VT are optionally computed in |
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* compact form. |
|
* |
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* A related subroutine, DLASD0, computes the singular values and |
|
* the singular vectors in explicit form. |
|
* |
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* Arguments |
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* ========= |
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* |
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* ICOMPQ (input) INTEGER |
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* Specifies whether singular vectors are to be computed |
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* in compact form, as follows |
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* = 0: Compute singular values only. |
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* = 1: Compute singular vectors of upper bidiagonal |
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* matrix in compact form. |
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* |
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* SMLSIZ (input) INTEGER |
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* The maximum size of the subproblems at the bottom of the |
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* computation tree. |
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* |
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* N (input) INTEGER |
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* The row dimension of the upper bidiagonal matrix. This is |
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* also the dimension of the main diagonal array D. |
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* |
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* SQRE (input) INTEGER |
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* Specifies the column dimension of the bidiagonal matrix. |
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* = 0: The bidiagonal matrix has column dimension M = N; |
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* = 1: The bidiagonal matrix has column dimension M = N + 1. |
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* |
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* D (input/output) DOUBLE PRECISION array, dimension ( N ) |
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* On entry D contains the main diagonal of the bidiagonal |
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* matrix. On exit D, if INFO = 0, contains its singular values. |
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* |
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* E (input) DOUBLE PRECISION array, dimension ( M-1 ) |
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* Contains the subdiagonal entries of the bidiagonal matrix. |
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* On exit, E has been destroyed. |
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* |
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* U (output) DOUBLE PRECISION array, |
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* dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced |
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* if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left |
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* singular vector matrices of all subproblems at the bottom |
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* level. |
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* |
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* LDU (input) INTEGER, LDU = > N. |
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* The leading dimension of arrays U, VT, DIFL, DIFR, POLES, |
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* GIVNUM, and Z. |
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* |
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* VT (output) DOUBLE PRECISION array, |
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* dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced |
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* if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right |
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* singular vector matrices of all subproblems at the bottom |
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* level. |
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* |
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* K (output) INTEGER array, |
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* dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0. |
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* If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th |
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* secular equation on the computation tree. |
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* |
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* DIFL (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ), |
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* where NLVL = floor(log_2 (N/SMLSIZ))). |
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* |
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* DIFR (output) DOUBLE PRECISION array, |
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* dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and |
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* dimension ( N ) if ICOMPQ = 0. |
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* If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1) |
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* record distances between singular values on the I-th |
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* level and singular values on the (I -1)-th level, and |
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* DIFR(1:N, 2 * I ) contains the normalizing factors for |
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* the right singular vector matrix. See DLASD8 for details. |
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* |
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* Z (output) DOUBLE PRECISION array, |
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* dimension ( LDU, NLVL ) if ICOMPQ = 1 and |
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* dimension ( N ) if ICOMPQ = 0. |
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* The first K elements of Z(1, I) contain the components of |
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* the deflation-adjusted updating row vector for subproblems |
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* on the I-th level. |
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* |
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* POLES (output) DOUBLE PRECISION array, |
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* dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced |
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* if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and |
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* POLES(1, 2*I) contain the new and old singular values |
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* involved in the secular equations on the I-th level. |
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* |
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* GIVPTR (output) INTEGER array, |
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* dimension ( N ) if ICOMPQ = 1, and not referenced if |
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* ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records |
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* the number of Givens rotations performed on the I-th |
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* problem on the computation tree. |
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* |
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* GIVCOL (output) INTEGER array, |
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* dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not |
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* referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, |
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* GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations |
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* of Givens rotations performed on the I-th level on the |
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* computation tree. |
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* |
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* LDGCOL (input) INTEGER, LDGCOL = > N. |
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* The leading dimension of arrays GIVCOL and PERM. |
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* |
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* PERM (output) INTEGER array, |
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* dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced |
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* if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records |
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* permutations done on the I-th level of the computation tree. |
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* |
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* GIVNUM (output) DOUBLE PRECISION array, |
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* dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not |
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* referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I, |
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* GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S- |
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* values of Givens rotations performed on the I-th level on |
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* the computation tree. |
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* |
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* C (output) DOUBLE PRECISION array, |
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* dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. |
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* If ICOMPQ = 1 and the I-th subproblem is not square, on exit, |
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* C( I ) contains the C-value of a Givens rotation related to |
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* the right null space of the I-th subproblem. |
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* |
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* S (output) DOUBLE PRECISION array, dimension ( N ) if |
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* ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 |
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* and the I-th subproblem is not square, on exit, S( I ) |
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* contains the S-value of a Givens rotation related to |
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* the right null space of the I-th subproblem. |
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* |
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* WORK (workspace) DOUBLE PRECISION array, dimension |
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* (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)). |
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* |
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* IWORK (workspace) INTEGER array. |
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* Dimension must be at least (7 * 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: if INFO = 1, a singular value did not converge |
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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 .. |