slasd6(3)

NAME

SLASD6 - compute the SVD of an updated upper bidiagonal

matrix B obtained by merging two smaller ones by appending a row

SYNOPSIS

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 )
    INTEGER          GIVPTR,   ICOMPQ,  INFO,  K,  LDGCOL,
LDGNUM, NL, NR, SQRE
    REAL           ALPHA, BETA, C, S
    INTEGER        GIVCOL( LDGCOL, * ), IDXQ( * ),  IWORK(
* ), PERM( * )
    REAL            D(  * ), DIFL( * ), DIFR( * ), GIVNUM(
LDGNUM, * ), POLES( LDGNUM, * ), VF( * ), VL( * ), WORK( * ),  Z(
* )

PURPOSE

SLASD6 computes the SVD of an updated upper bidiagonal ma

trix B obtained by merging two smaller ones by appending a row.

This routine is used only for the problem which requires all sin

gular 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' 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 SLASD6. Hence U and VT are not explicit

ly referenced.

The singular values are stored in D. The algorithm con

sists 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 dimen

sion 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 sin

gular
values.

SLASD6 is called from SLASDA.

ARGUMENTS

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) 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.

VF (input/output) 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 compo

nents of all right singular vectors of the bidiagonal matrix.

VL (input/output) 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 compo

nents of all right singular vectors of the bidiagonal matrix.

ALPHA (input) REAL

Contains the diagonal element associated with the

added row.

BETA (input) REAL

Contains the off-diagonal element associated with

the added row.

IDXQ (output) INTEGER array, dimension ( N )

This contains the permutation which will reinte

grate 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 rota

tions 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) REAL array, dimension ( LDGNUM, 2 )

Each number indicates the C or S value to be used in the corre

sponding Givens rotation. Not referenced if ICOMPQ = 0.

LDGNUM (input) INTEGER The leading dimension of

GIVNUM and POLES, must be at least N.

POLES (output) 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 equa

tion. Not referenced if ICOMPQ = 0.

DIFL (output) REAL array, dimension ( N )

On exit, DIFL(I) is the distance between I-th up

dated (undeflated) singular value and the I-th (undeflated) old

singular value.

DIFR (output) REAL 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 (unde

flated) old singular value.

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.

Z (output) REAL array, dimension ( M )

The first elements of this array contain the compo

nents 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) REAL

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) REAL

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) REAL 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 ille

gal value.
> 0: if INFO = 1, an singular value did not con

verge

FURTHER DETAILS

Based on contributions by

Ming Gu and Huan Ren, Computer Science Division, Uni

versity of
California at Berkeley, USA

LAPACK version 3.0 15 June 2000

SLASD6(3)