dlasd1(3)

NAME

DLASD1 - compute the SVD of an upper bidiagonal N-by-M ma

trix B,

SYNOPSIS

SUBROUTINE  DLASD1(  NL, NR, SQRE, D, ALPHA, BETA, U, LDU,
VT, LDVT, IDXQ, IWORK, WORK, INFO )
    INTEGER        INFO, LDU, LDVT, NL, NR, SQRE
    DOUBLE         PRECISION ALPHA, BETA
    INTEGER        IDXQ( * ), IWORK( * )
    DOUBLE         PRECISION D( * ), U( LDU, * ), VT(  LDVT, * ), WORK( * )

PURPOSE

DLASD1 computes the SVD of an upper bidiagonal N-by-M ma

trix B, where N = NL + NR + 1 and M = N + SQRE. DLASD1 is called

from DLASD0.

A related subroutine DLASD7 handles the case in which the

singular values (and the singular vectors in factored form) are

desired.

DLASD1 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 left singular vectors of the original matrix are

stored in U, and the transpose of the right singular vectors are

stored in VT, and the singular values are in D. The algorithm

consists of three stages:

The first stage consists of deflating the size of the

problem
when there are multiple singular values or when there

are zeros 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 DLASD2.

The second stage consists of calculating the updated
singular values. This is done by finding the square

roots of the
roots of the secular equation via the routine DLASD4

(as called
by DLASD3). This routine also calculates the singular

vectors of
the current problem.

The final stage consists of computing the updated sin

gular vectors
directly using the updated singular values. The singu

lar vectors
for the current problem are multiplied with the singu

lar vectors
from the overall problem.

ARGUMENTS

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 (N = 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 singu

lar values of the modified 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.

U (input/output) DOUBLE PRECISION array, dimen

sion(LDU,N)

On entry U(1:NL, 1:NL) contains the left singular

vectors of
the upper block; U(NL+2:N, NL+2:N) contains the

left singular vectors of the lower block. On exit U contains the

left singular vectors of the bidiagonal matrix.

LDU (input) INTEGER

The leading dimension of the array U. LDU >= max(

1, N ).

VT (input/output) DOUBLE PRECISION array, dimen

sion(LDVT,M)

where M = N + SQRE. On entry VT(1:NL+1, 1:NL+1)'

contains the right singular
vectors of the upper block; VT(NL+2:M, NL+2:M)'

contains the right singular vectors of the lower block. On exit

VT' contains the right singular vectors of the bidiagonal matrix.

LDVT (input) INTEGER

The leading dimension of the array VT. LDVT >=

max( 1, M ).

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.

IWORK (workspace) INTEGER array, dimension( 4 * N )

WORK (workspace) DOUBLE PRECISION array, dimension(

3*M**2 + 2*M )

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

DLASD1(3)