DLASD6 - compute the SVD of an updated upper bidiagonal matrix B obtained by merging two smaller ones by appending a row

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 )

INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL, NR, SQRE

DOUBLE PRECISION ALPHA, BETA, C, S

INTEGER GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ), PERM( * )

DOUBLE PRECISION D( * ), DIFL( * ), DIFR( * ), GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ), VF( * ), VL( * ), WORK( * ), Z( * )

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' a Z2' b ) * VT(in)

( 0 0 D2(in) 0 )

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

where Z' = (Z1' a Z2' b) = u' VT', 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.

ICOMPQ (input) 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.

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 row dimension N = NL + NR + 1, and column dimension M = N + SQRE.

D (input/output) 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.

VF (input/output) 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.

VL (input/output) 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.

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.

IDXQ (output) 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.

PERM (output) INTEGER array, dimension ( N )

The permutations (from deflation and sorting) to be applied to each block. Not referenced if ICOMPQ = 0.

GIVPTR (output) INTEGER The number of Givens rotations which took place in this subproblem. Not referenced if ICOMPQ = 0.

GIVCOL (output) 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.

LDGCOL (input) INTEGER leading dimension of GIVCOL, must be at least N.

GIVNUM (output) 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.

LDGNUM (input) INTEGER The leading dimension of GIVNUM and POLES, must be at least N.

POLES (output) 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.

DIFL (output) 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.

DIFR (output) 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.

Z (output) DOUBLE PRECISION array, dimension ( M )

The first elements of this array contain the components of the deflation-adjusted updating row vector.

K (output) INTEGER

Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1 <= K <=N.

C (output) 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.

S (output) 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.

WORK (workspace) DOUBLE PRECISION array, dimension ( 4 * M )

IWORK (workspace) INTEGER array, dimension ( 3 * N )

INFO (output) INTEGER

= 0: successful exit.

< 0: if INFO = -i, the i-th argument had an illegal value.

> 0: if INFO = 1, an singular value did not converge

Based on contributions by

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA