DLASD1 - compute the SVD of an upper bidiagonal N-by-M matrix B,

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( * )

DLASD1 computes the SVD of an upper bidiagonal N-by-M matrix 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 singular vectors directly using the updated singular values. The singular vectors for the current problem are multiplied with the singular vectors from the overall problem.

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 singular 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, dimension(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, dimension(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 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