#include "ESMF_LapackBlas.inc" !> \brief \b DLASD6 ! ! =========== DOCUMENTATION =========== ! ! Online html documentation available at ! http://www.netlib.org/lapack/explore-html/ ! !> \htmlonly !> Download DLASD6 + dependencies !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd6.f"> !> [TGZ]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd6.f"> !> [ZIP]</a> !> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd6.f"> !> [TXT]</a> !> \endhtmlonly ! ! Definition: ! =========== ! ! SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, ! IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, ! LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, ! IWORK, INFO ) ! ! .. Scalar Arguments .. ! INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL, ! $ NR, SQRE ! DOUBLE PRECISION ALPHA, BETA, C, S ! .. ! .. Array Arguments .. ! INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ), ! $ PERM( * ) ! DOUBLE PRECISION D( * ), DIFL( * ), DIFR( * ), ! $ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ), ! $ VF( * ), VL( * ), WORK( * ), Z( * ) ! .. ! ! !> \par Purpose: ! ============= !> !> \verbatim !> !> DLASD6 computes the SVD of an updated upper bidiagonal matrix B !> obtained by merging two smaller ones by appending a row. This !> routine is used only for the problem which requires all singular !> values and optionally singular vector matrices in factored form. !> B is an N-by-M matrix with N = NL + NR + 1 and M = N + SQRE. !> A related subroutine, DLASD1, handles the case in which all singular !> values and singular vectors of the bidiagonal matrix are desired. !> !> DLASD6 computes the SVD as follows: !> !> ( D1(in) 0 0 0 ) !> B = U(in) * ( Z1**T a Z2**T b ) * VT(in) !> ( 0 0 D2(in) 0 ) !> !> = U(out) * ( D(out) 0) * VT(out) !> !> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M !> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros !> elsewhere; and the entry b is empty if SQRE = 0. !> !> The singular values of B can be computed using D1, D2, the first !> components of all the right singular vectors of the lower block, and !> the last components of all the right singular vectors of the upper !> block. These components are stored and updated in VF and VL, !> respectively, in DLASD6. Hence U and VT are not explicitly !> referenced. !> !> The singular values are stored in D. The algorithm consists of two !> stages: !> !> The first stage consists of deflating the size of the problem !> when there are multiple singular values or if there is a zero !> in the Z vector. For each such occurence the dimension of the !> secular equation problem is reduced by one. This stage is !> performed by the routine DLASD7. !> !> The second stage consists of calculating the updated !> singular values. This is done by finding the roots of the !> secular equation via the routine DLASD4 (as called by DLASD8). !> This routine also updates VF and VL and computes the distances !> between the updated singular values and the old singular !> values. !> !> DLASD6 is called from DLASDA. !> \endverbatim ! ! Arguments: ! ========== ! !> \param[in] ICOMPQ !> \verbatim !> ICOMPQ is INTEGER !> Specifies whether singular vectors are to be computed in !> factored form: !> = 0: Compute singular values only. !> = 1: Compute singular vectors in factored form as well. !> \endverbatim !> !> \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 row dimension N = NL + NR + 1, !> and column dimension M = N + SQRE. !> \endverbatim !> !> \param[in,out] D !> \verbatim !> D is DOUBLE PRECISION array, dimension ( NL+NR+1 ). !> On entry D(1:NL,1:NL) contains the singular values of the !> upper block, and D(NL+2:N) contains the singular values !> of the lower block. On exit D(1:N) contains the singular !> values of the modified matrix. !> \endverbatim !> !> \param[in,out] VF !> \verbatim !> VF is DOUBLE PRECISION array, dimension ( M ) !> On entry, VF(1:NL+1) contains the first components of all !> right singular vectors of the upper block; and VF(NL+2:M) !> contains the first components of all right singular vectors !> of the lower block. On exit, VF contains the first components !> of all right singular vectors of the bidiagonal matrix. !> \endverbatim !> !> \param[in,out] VL !> \verbatim !> VL is DOUBLE PRECISION array, dimension ( M ) !> On entry, VL(1:NL+1) contains the last components of all !> right singular vectors of the upper block; and VL(NL+2:M) !> contains the last components of all right singular vectors of !> the lower block. On exit, VL contains the last components of !> all right singular vectors of the bidiagonal matrix. !> \endverbatim !> !> \param[in,out] ALPHA !> \verbatim !> ALPHA is DOUBLE PRECISION !> Contains the diagonal element associated with the added row. !> \endverbatim !> !> \param[in,out] BETA !> \verbatim !> BETA is DOUBLE PRECISION !> Contains the off-diagonal element associated with the added !> row. !> \endverbatim !> !> \param[out] IDXQ !> \verbatim !> IDXQ is INTEGER array, dimension ( N ) !> This contains the permutation which will reintegrate the !> subproblem just solved back into sorted order, i.e. !> D( IDXQ( I = 1, N ) ) will be in ascending order. !> \endverbatim !> !> \param[out] PERM !> \verbatim !> PERM is INTEGER array, dimension ( N ) !> The permutations (from deflation and sorting) to be applied !> to each block. Not referenced if ICOMPQ = 0. !> \endverbatim !> !> \param[out] GIVPTR !> \verbatim !> GIVPTR is INTEGER !> The number of Givens rotations which took place in this !> subproblem. Not referenced if ICOMPQ = 0. !> \endverbatim !> !> \param[out] GIVCOL !> \verbatim !> GIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) !> Each pair of numbers indicates a pair of columns to take place !> in a Givens rotation. Not referenced if ICOMPQ = 0. !> \endverbatim !> !> \param[in] LDGCOL !> \verbatim !> LDGCOL is INTEGER !> leading dimension of GIVCOL, must be at least N. !> \endverbatim !> !> \param[out] GIVNUM !> \verbatim !> GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) !> Each number indicates the C or S value to be used in the !> corresponding Givens rotation. Not referenced if ICOMPQ = 0. !> \endverbatim !> !> \param[in] LDGNUM !> \verbatim !> LDGNUM is INTEGER !> The leading dimension of GIVNUM and POLES, must be at least N. !> \endverbatim !> !> \param[out] POLES !> \verbatim !> POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) !> On exit, POLES(1,*) is an array containing the new singular !> values obtained from solving the secular equation, and !> POLES(2,*) is an array containing the poles in the secular !> equation. Not referenced if ICOMPQ = 0. !> \endverbatim !> !> \param[out] DIFL !> \verbatim !> DIFL is DOUBLE PRECISION array, dimension ( N ) !> On exit, DIFL(I) is the distance between I-th updated !> (undeflated) singular value and the I-th (undeflated) old !> singular value. !> \endverbatim !> !> \param[out] DIFR !> \verbatim !> DIFR is DOUBLE PRECISION array, !> dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and !> dimension ( N ) if ICOMPQ = 0. !> On exit, DIFR(I, 1) is the distance between I-th updated !> (undeflated) singular value and the I+1-th (undeflated) old !> singular value. !> !> If ICOMPQ = 1, DIFR(1:K,2) is an array containing the !> normalizing factors for the right singular vector matrix. !> !> See DLASD8 for details on DIFL and DIFR. !> \endverbatim !> !> \param[out] Z !> \verbatim !> Z is DOUBLE PRECISION array, dimension ( M ) !> The first elements of this array contain the components !> of the deflation-adjusted updating row vector. !> \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[out] C !> \verbatim !> C is DOUBLE PRECISION !> C contains garbage if SQRE =0 and the C-value of a Givens !> rotation related to the right null space if SQRE = 1. !> \endverbatim !> !> \param[out] S !> \verbatim !> S is DOUBLE PRECISION !> S contains garbage if SQRE =0 and the S-value of a Givens !> rotation related to the right null space if SQRE = 1. !> \endverbatim !> !> \param[out] WORK !> \verbatim !> WORK is DOUBLE PRECISION array, dimension ( 4 * M ) !> \endverbatim !> !> \param[out] IWORK !> \verbatim !> IWORK is INTEGER array, dimension ( 3 * N ) !> \endverbatim !> !> \param[out] INFO !> \verbatim !> INFO is INTEGER !> = 0: successful exit. !> < 0: if INFO = -i, the i-th argument had an illegal value. !> > 0: if INFO = 1, a singular value did not converge !> \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 DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, & & IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, & & LDGNUM, POLES, DIFL, DIFR, Z, K, C, S, WORK, & & IWORK, INFO ) ! ! -- 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 2011 ! ! .. Scalar Arguments .. INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL, & & NR, SQRE DOUBLE PRECISION ALPHA, BETA, C, S ! .. ! .. Array Arguments .. INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ), & & PERM( * ) DOUBLE PRECISION D( * ), DIFL( * ), DIFR( * ), & & GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ), & & VF( * ), VL( * ), WORK( * ), Z( * ) ! .. ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) ! .. ! .. Local Scalars .. INTEGER I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M, & & N, N1, N2 DOUBLE PRECISION ORGNRM ! .. ! .. External Subroutines .. EXTERNAL DCOPY, DLAMRG, DLASCL, DLASD7, DLASD8, XERBLA ! .. ! .. Intrinsic Functions .. INTRINSIC ABS, MAX ! .. ! .. Executable Statements .. ! ! Test the input parameters. ! INFO = 0 N = NL + NR + 1 M = N + SQRE ! IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN INFO = -1 ELSE IF( NL.LT.1 ) THEN INFO = -2 ELSE IF( NR.LT.1 ) THEN INFO = -3 ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN INFO = -4 ELSE IF( LDGCOL.LT.N ) THEN INFO = -14 ELSE IF( LDGNUM.LT.N ) THEN INFO = -16 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASD6', -INFO ) RETURN END IF ! ! The following values are for bookkeeping purposes only. They are ! integer pointers which indicate the portion of the workspace ! used by a particular array in DLASD7 and DLASD8. ! ISIGMA = 1 IW = ISIGMA + N IVFW = IW + M IVLW = IVFW + M ! IDX = 1 IDXC = IDX + N IDXP = IDXC + N ! ! Scale. ! ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) ) D( NL+1 ) = ZERO DO 10 I = 1, N IF( ABS( D( I ) ).GT.ORGNRM ) THEN ORGNRM = ABS( D( I ) ) END IF 10 CONTINUE CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO ) ALPHA = ALPHA / ORGNRM BETA = BETA / ORGNRM ! ! Sort and Deflate singular values. ! CALL DLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, WORK( IW ), VF, & & WORK( IVFW ), VL, WORK( IVLW ), ALPHA, BETA, & & WORK( ISIGMA ), IWORK( IDX ), IWORK( IDXP ), IDXQ, & & PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S, & & INFO ) ! ! Solve Secular Equation, compute DIFL, DIFR, and update VF, VL. ! CALL DLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM, & & WORK( ISIGMA ), WORK( IW ), INFO ) ! ! Handle error returned ! IF( INFO.NE.0 ) THEN CALL XERBLA( 'DLASD8', -INFO ) RETURN END IF ! ! Save the poles if ICOMPQ = 1. ! IF( ICOMPQ.EQ.1 ) THEN CALL DCOPY( K, D, 1, POLES( 1, 1 ), 1 ) CALL DCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 ) END IF ! ! Unscale. ! CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO ) ! ! Prepare the IDXQ sorting permutation. ! N1 = K N2 = N - K CALL DLAMRG( N1, N2, D, 1, -1, IDXQ ) ! RETURN ! ! End of DLASD6 ! END