--- rpl/lapack/lapack/dlasd2.f 2011/07/22 07:38:07 1.8 +++ rpl/lapack/lapack/dlasd2.f 2011/11/21 20:42:58 1.9 @@ -1,11 +1,278 @@ +*> \brief \b DLASD2 +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download DLASD2 + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT, +* LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, +* IDXC, IDXQ, COLTYP, INFO ) +* +* .. Scalar Arguments .. +* INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE +* DOUBLE PRECISION ALPHA, BETA +* .. +* .. Array Arguments .. +* INTEGER COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ), +* $ IDXQ( * ) +* DOUBLE PRECISION D( * ), DSIGMA( * ), U( LDU, * ), +* $ U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), +* $ Z( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> DLASD2 merges the two sets of singular values together into a single +*> sorted set. Then it tries to deflate the size of the problem. +*> There are two ways in which deflation can occur: when two or more +*> singular values are close together or if there is a tiny entry in the +*> Z vector. For each such occurrence the order of the related secular +*> equation problem is reduced by one. +*> +*> DLASD2 is called from DLASD1. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] NL +*> \verbatim +*> NL is INTEGER +*> The row dimension of the upper block. NL >= 1. +*> \endverbatim +*> +*> \param[in] NR +*> \verbatim +*> NR is INTEGER +*> The row dimension of the lower block. NR >= 1. +*> \endverbatim +*> +*> \param[in] SQRE +*> \verbatim +*> SQRE is INTEGER +*> = 0: the lower block is an NR-by-NR square matrix. +*> = 1: the lower block is an NR-by-(NR+1) rectangular matrix. +*> +*> The bidiagonal matrix has N = NL + NR + 1 rows and +*> M = N + SQRE >= N columns. +*> \endverbatim +*> +*> \param[out] K +*> \verbatim +*> K is INTEGER +*> Contains the dimension of the non-deflated matrix, +*> This is the order of the related secular equation. 1 <= K <=N. +*> \endverbatim +*> +*> \param[in,out] D +*> \verbatim +*> D is DOUBLE PRECISION array, dimension(N) +*> On entry D contains the singular values of the two submatrices +*> to be combined. On exit D contains the trailing (N-K) updated +*> singular values (those which were deflated) sorted into +*> increasing order. +*> \endverbatim +*> +*> \param[out] Z +*> \verbatim +*> Z is DOUBLE PRECISION array, dimension(N) +*> On exit Z contains the updating row vector in the secular +*> equation. +*> \endverbatim +*> +*> \param[in] ALPHA +*> \verbatim +*> ALPHA is DOUBLE PRECISION +*> Contains the diagonal element associated with the added row. +*> \endverbatim +*> +*> \param[in] BETA +*> \verbatim +*> BETA is DOUBLE PRECISION +*> Contains the off-diagonal element associated with the added +*> row. +*> \endverbatim +*> +*> \param[in,out] U +*> \verbatim +*> U is DOUBLE PRECISION array, dimension(LDU,N) +*> On entry U contains the left singular vectors of two +*> submatrices in the two square blocks with corners at (1,1), +*> (NL, NL), and (NL+2, NL+2), (N,N). +*> On exit U contains the trailing (N-K) updated left singular +*> vectors (those which were deflated) in its last N-K columns. +*> \endverbatim +*> +*> \param[in] LDU +*> \verbatim +*> LDU is INTEGER +*> The leading dimension of the array U. LDU >= N. +*> \endverbatim +*> +*> \param[in,out] VT +*> \verbatim +*> VT is DOUBLE PRECISION array, dimension(LDVT,M) +*> On entry VT**T contains the right singular vectors of two +*> submatrices in the two square blocks with corners at (1,1), +*> (NL+1, NL+1), and (NL+2, NL+2), (M,M). +*> On exit VT**T contains the trailing (N-K) updated right singular +*> vectors (those which were deflated) in its last N-K columns. +*> In case SQRE =1, the last row of VT spans the right null +*> space. +*> \endverbatim +*> +*> \param[in] LDVT +*> \verbatim +*> LDVT is INTEGER +*> The leading dimension of the array VT. LDVT >= M. +*> \endverbatim +*> +*> \param[out] DSIGMA +*> \verbatim +*> DSIGMA is DOUBLE PRECISION array, dimension (N) +*> Contains a copy of the diagonal elements (K-1 singular values +*> and one zero) in the secular equation. +*> \endverbatim +*> +*> \param[out] U2 +*> \verbatim +*> U2 is DOUBLE PRECISION array, dimension(LDU2,N) +*> Contains a copy of the first K-1 left singular vectors which +*> will be used by DLASD3 in a matrix multiply (DGEMM) to solve +*> for the new left singular vectors. U2 is arranged into four +*> blocks. The first block contains a column with 1 at NL+1 and +*> zero everywhere else; the second block contains non-zero +*> entries only at and above NL; the third contains non-zero +*> entries only below NL+1; and the fourth is dense. +*> \endverbatim +*> +*> \param[in] LDU2 +*> \verbatim +*> LDU2 is INTEGER +*> The leading dimension of the array U2. LDU2 >= N. +*> \endverbatim +*> +*> \param[out] VT2 +*> \verbatim +*> VT2 is DOUBLE PRECISION array, dimension(LDVT2,N) +*> VT2**T contains a copy of the first K right singular vectors +*> which will be used by DLASD3 in a matrix multiply (DGEMM) to +*> solve for the new right singular vectors. VT2 is arranged into +*> three blocks. The first block contains a row that corresponds +*> to the special 0 diagonal element in SIGMA; the second block +*> contains non-zeros only at and before NL +1; the third block +*> contains non-zeros only at and after NL +2. +*> \endverbatim +*> +*> \param[in] LDVT2 +*> \verbatim +*> LDVT2 is INTEGER +*> The leading dimension of the array VT2. LDVT2 >= M. +*> \endverbatim +*> +*> \param[out] IDXP +*> \verbatim +*> IDXP is INTEGER array dimension(N) +*> This will contain the permutation used to place deflated +*> values of D at the end of the array. On output IDXP(2:K) +*> points to the nondeflated D-values and IDXP(K+1:N) +*> points to the deflated singular values. +*> \endverbatim +*> +*> \param[out] IDX +*> \verbatim +*> IDX is INTEGER array dimension(N) +*> This will contain the permutation used to sort the contents of +*> D into ascending order. +*> \endverbatim +*> +*> \param[out] IDXC +*> \verbatim +*> IDXC is INTEGER array dimension(N) +*> This will contain the permutation used to arrange the columns +*> of the deflated U matrix into three groups: the first group +*> contains non-zero entries only at and above NL, the second +*> contains non-zero entries only below NL+2, and the third is +*> dense. +*> \endverbatim +*> +*> \param[in,out] IDXQ +*> \verbatim +*> IDXQ is INTEGER array dimension(N) +*> This contains the permutation which separately sorts the two +*> sub-problems in D into ascending order. Note that entries in +*> the first hlaf of this permutation must first be moved one +*> position backward; and entries in the second half +*> must first have NL+1 added to their values. +*> \endverbatim +*> +*> \param[out] COLTYP +*> \verbatim +*> COLTYP is INTEGER array dimension(N) +*> As workspace, this will contain a label which will indicate +*> which of the following types a column in the U2 matrix or a +*> row in the VT2 matrix is: +*> 1 : non-zero in the upper half only +*> 2 : non-zero in the lower half only +*> 3 : dense +*> 4 : deflated +*> +*> On exit, it is an array of dimension 4, with COLTYP(I) being +*> the dimension of the I-th type columns. +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> = 0: successful exit. +*> < 0: if INFO = -i, the i-th argument had an illegal value. +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \date November 2011 +* +*> \ingroup auxOTHERauxiliary +* +*> \par Contributors: +* ================== +*> +*> Ming Gu and Huan Ren, Computer Science Division, University of +*> California at Berkeley, USA +*> +* ===================================================================== SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT, $ LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, $ IDXC, IDXQ, COLTYP, INFO ) * -* -- LAPACK auxiliary routine (version 3.2) -- +* -- LAPACK auxiliary routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* November 2006 +* November 2011 * * .. Scalar Arguments .. INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE @@ -19,152 +286,6 @@ $ Z( * ) * .. * -* Purpose -* ======= -* -* DLASD2 merges the two sets of singular values together into a single -* sorted set. Then it tries to deflate the size of the problem. -* There are two ways in which deflation can occur: when two or more -* singular values are close together or if there is a tiny entry in the -* Z vector. For each such occurrence the order of the related secular -* equation problem is reduced by one. -* -* DLASD2 is called from DLASD1. -* -* Arguments -* ========= -* -* NL (input) INTEGER -* The row dimension of the upper block. NL >= 1. -* -* NR (input) INTEGER -* The row dimension of the lower block. NR >= 1. -* -* SQRE (input) INTEGER -* = 0: the lower block is an NR-by-NR square matrix. -* = 1: the lower block is an NR-by-(NR+1) rectangular matrix. -* -* The bidiagonal matrix has N = NL + NR + 1 rows and -* M = N + SQRE >= N columns. -* -* K (output) INTEGER -* Contains the dimension of the non-deflated matrix, -* This is the order of the related secular equation. 1 <= K <=N. -* -* D (input/output) DOUBLE PRECISION array, dimension(N) -* On entry D contains the singular values of the two submatrices -* to be combined. On exit D contains the trailing (N-K) updated -* singular values (those which were deflated) sorted into -* increasing order. -* -* Z (output) DOUBLE PRECISION array, dimension(N) -* On exit Z contains the updating row vector in the secular -* equation. -* -* ALPHA (input) DOUBLE PRECISION -* Contains the diagonal element associated with the added row. -* -* BETA (input) DOUBLE PRECISION -* Contains the off-diagonal element associated with the added -* row. -* -* U (input/output) DOUBLE PRECISION array, dimension(LDU,N) -* On entry U contains the left singular vectors of two -* submatrices in the two square blocks with corners at (1,1), -* (NL, NL), and (NL+2, NL+2), (N,N). -* On exit U contains the trailing (N-K) updated left singular -* vectors (those which were deflated) in its last N-K columns. -* -* LDU (input) INTEGER -* The leading dimension of the array U. LDU >= N. -* -* VT (input/output) DOUBLE PRECISION array, dimension(LDVT,M) -* On entry VT**T contains the right singular vectors of two -* submatrices in the two square blocks with corners at (1,1), -* (NL+1, NL+1), and (NL+2, NL+2), (M,M). -* On exit VT**T contains the trailing (N-K) updated right singular -* vectors (those which were deflated) in its last N-K columns. -* In case SQRE =1, the last row of VT spans the right null -* space. -* -* LDVT (input) INTEGER -* The leading dimension of the array VT. LDVT >= M. -* -* DSIGMA (output) DOUBLE PRECISION array, dimension (N) -* Contains a copy of the diagonal elements (K-1 singular values -* and one zero) in the secular equation. -* -* U2 (output) DOUBLE PRECISION array, dimension(LDU2,N) -* Contains a copy of the first K-1 left singular vectors which -* will be used by DLASD3 in a matrix multiply (DGEMM) to solve -* for the new left singular vectors. U2 is arranged into four -* blocks. The first block contains a column with 1 at NL+1 and -* zero everywhere else; the second block contains non-zero -* entries only at and above NL; the third contains non-zero -* entries only below NL+1; and the fourth is dense. -* -* LDU2 (input) INTEGER -* The leading dimension of the array U2. LDU2 >= N. -* -* VT2 (output) DOUBLE PRECISION array, dimension(LDVT2,N) -* VT2**T contains a copy of the first K right singular vectors -* which will be used by DLASD3 in a matrix multiply (DGEMM) to -* solve for the new right singular vectors. VT2 is arranged into -* three blocks. The first block contains a row that corresponds -* to the special 0 diagonal element in SIGMA; the second block -* contains non-zeros only at and before NL +1; the third block -* contains non-zeros only at and after NL +2. -* -* LDVT2 (input) INTEGER -* The leading dimension of the array VT2. LDVT2 >= M. -* -* IDXP (workspace) INTEGER array dimension(N) -* This will contain the permutation used to place deflated -* values of D at the end of the array. On output IDXP(2:K) -* points to the nondeflated D-values and IDXP(K+1:N) -* points to the deflated singular values. -* -* IDX (workspace) INTEGER array dimension(N) -* This will contain the permutation used to sort the contents of -* D into ascending order. -* -* IDXC (output) INTEGER array dimension(N) -* This will contain the permutation used to arrange the columns -* of the deflated U matrix into three groups: the first group -* contains non-zero entries only at and above NL, the second -* contains non-zero entries only below NL+2, and the third is -* dense. -* -* IDXQ (input/output) INTEGER array dimension(N) -* This contains the permutation which separately sorts the two -* sub-problems in D into ascending order. Note that entries in -* the first hlaf of this permutation must first be moved one -* position backward; and entries in the second half -* must first have NL+1 added to their values. -* -* COLTYP (workspace/output) INTEGER array dimension(N) -* As workspace, this will contain a label which will indicate -* which of the following types a column in the U2 matrix or a -* row in the VT2 matrix is: -* 1 : non-zero in the upper half only -* 2 : non-zero in the lower half only -* 3 : dense -* 4 : deflated -* -* On exit, it is an array of dimension 4, with COLTYP(I) being -* the dimension of the I-th type columns. -* -* INFO (output) INTEGER -* = 0: successful exit. -* < 0: if INFO = -i, the i-th argument had an illegal value. -* -* Further Details -* =============== -* -* Based on contributions by -* Ming Gu and Huan Ren, Computer Science Division, University of -* California at Berkeley, USA -* * ===================================================================== * * .. Parameters ..