ESMF_DLASD6 Subroutine

subroutine ESMF_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)

\brief \b ESMF_DLASD6 \htmlonly Download ESMF_DLASD6 + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_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.

ESMF_DLASD6 computes the SVD as follows:

          ( D1(in)    0    0       0 )

B = U(in) * ( Z1T a Z2T b ) * VT(in) ( 0 0 D2(in) 0 )

= U(out) * ( D(out) 0) * VT(out)

where ZT = (Z1T a Z2T b) = uT 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 ESMF_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 ESMF_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 ESMF_DLASD4 (as called by ESMF_DLASD8).
  This routine also updates VF and VL and computes the distances
  between the updated singular values and the old singular
  values.

ESMF_DLASD6 is called from ESMF_DLASDA. \endverbatim \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 ESMF_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 \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

Arguments