*> \brief \b SLASD6 computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by sbdsdc. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLASD6 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLASD6( 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 * REAL ALPHA, BETA, C, S * .. * .. Array Arguments .. * INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ), * $ PERM( * ) * REAL D( * ), DIFL( * ), DIFR( * ), * $ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ), * $ VF( * ), VL( * ), WORK( * ), Z( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLASD6 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, SLASD1, handles the case in which all singular *> values and singular vectors of the bidiagonal matrix are desired. *> *> SLASD6 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 SLASD6. 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 occurrence the dimension of the *> secular equation problem is reduced by one. This stage is *> performed by the routine SLASD7. *> *> The second stage consists of calculating the updated *> singular values. This is done by finding the roots of the *> secular equation via the routine SLASD4 (as called by SLASD8). *> This routine also updates VF and VL and computes the distances *> between the updated singular values and the old singular *> values. *> *> SLASD6 is called from SLASDA. *> \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 REAL 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 REAL 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 REAL 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 REAL *> Contains the diagonal element associated with the added row. *> \endverbatim *> *> \param[in,out] BETA *> \verbatim *> BETA is REAL *> Contains the off-diagonal element associated with the added *> row. *> \endverbatim *> *> \param[in,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 REAL 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 REAL 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 REAL 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 REAL array, *> dimension ( LDDIFR, 2 ) if ICOMPQ = 1 and *> dimension ( K ) if ICOMPQ = 0. *> On exit, DIFR(I,1) = D(I) - DSIGMA(I+1), DIFR(K,1) is not *> defined and will not be referenced. *> *> If ICOMPQ = 1, DIFR(1:K,2) is an array containing the *> normalizing factors for the right singular vector matrix. *> *> See SLASD8 for details on DIFL and DIFR. *> \endverbatim *> *> \param[out] Z *> \verbatim *> Z is REAL 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 REAL *> 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 REAL *> 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 REAL 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 June 2016 * *> \ingroup OTHERauxiliary * *> \par Contributors: * ================== *> *> Ming Gu and Huan Ren, Computer Science Division, University of *> California at Berkeley, USA *> * ===================================================================== SUBROUTINE SLASD6( 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 3.6.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2016 * * .. Scalar Arguments .. INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL, $ NR, SQRE REAL ALPHA, BETA, C, S * .. * .. Array Arguments .. INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ), $ PERM( * ) REAL D( * ), DIFL( * ), DIFR( * ), $ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ), $ VF( * ), VL( * ), WORK( * ), Z( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ONE, ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. * .. Local Scalars .. INTEGER I, IDX, IDXC, IDXP, ISIGMA, IVFW, IVLW, IW, M, $ N, N1, N2 REAL ORGNRM * .. * .. External Subroutines .. EXTERNAL SCOPY, SLAMRG, SLASCL, SLASD7, SLASD8, 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( 'SLASD6', -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 SLASD7 and SLASD8. * 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 SLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO ) ALPHA = ALPHA / ORGNRM BETA = BETA / ORGNRM * * Sort and Deflate singular values. * CALL SLASD7( 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 SLASD8( ICOMPQ, K, D, Z, VF, VL, DIFL, DIFR, LDGNUM, $ WORK( ISIGMA ), WORK( IW ), INFO ) * * Report the possible convergence failure. * IF( INFO.NE.0 ) THEN RETURN END IF * * Save the poles if ICOMPQ = 1. * IF( ICOMPQ.EQ.1 ) THEN CALL SCOPY( K, D, 1, POLES( 1, 1 ), 1 ) CALL SCOPY( K, WORK( ISIGMA ), 1, POLES( 1, 2 ), 1 ) END IF * * Unscale. * CALL SLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO ) * * Prepare the IDXQ sorting permutation. * N1 = K N2 = N - K CALL SLAMRG( N1, N2, D, 1, -1, IDXQ ) * RETURN * * End of SLASD6 * END