SYNOPSIS

SUBROUTINE DLASD2(

NL, NR, SQRE, K, D, Z, ALPHA, BETA, U, LDU, VT, LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, IDXC, IDXQ, COLTYP, INFO )

    

INTEGER INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE

    

DOUBLE PRECISION ALPHA, BETA

    

INTEGER COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ), IDXQ( * )

    

DOUBLE PRECISION D( * ), DSIGMA( * ), U( LDU, * ), U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), Z( * )

PURPOSE

DLASD2 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more singular values are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.
DLASD2 is called from DLASD1.

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 N = NL + NR + 1 rows and M = N + SQRE >= N columns.

K (output) INTEGER

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

D (input/output) DOUBLE PRECISION array, dimension(N)

On entry D contains the singular values of the two submatrices to be combined. On exit D contains the trailing (N-K) updated singular values (those which were deflated) sorted into increasing order.

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

On exit Z contains the updating row vector in the secular equation.

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 contains the left singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL, NL), and (NL+2, NL+2), (N,N). On exit U contains the trailing (N-K) updated left singular vectors (those which were deflated) in its last N-K columns.

LDU (input) INTEGER

The leading dimension of the array U. LDU >= N.

VT (input/output) DOUBLE PRECISION array, dimension(LDVT,M)

On entry VT' contains the right singular vectors of two submatrices in the two square blocks with corners at (1,1), (NL+1, NL+1), and (NL+2, NL+2), (M,M). On exit VT' contains the trailing (N-K) updated right singular vectors (those which were deflated) in its last N-K columns. In case SQRE =1, the last row of VT spans the right null space.

LDVT (input) INTEGER

The leading dimension of the array VT. LDVT >= M. DSIGMA (output) DOUBLE PRECISION array, dimension (N) Contains a copy of the diagonal elements (K-1 singular values and one zero) in the secular equation.

U2 (output) DOUBLE PRECISION array, dimension(LDU2,N)

Contains a copy of the first K-1 left singular vectors which will be used by DLASD3 in a matrix multiply (DGEMM) to solve for the new left singular vectors. U2 is arranged into four blocks. The first block contains a column with 1 at NL+1 and zero everywhere else; the second block contains non-zero entries only at and above NL; the third contains non-zero entries only below NL+1; and the fourth is dense.

LDU2 (input) INTEGER

The leading dimension of the array U2. LDU2 >= N.

VT2 (output) DOUBLE PRECISION array, dimension(LDVT2,N)

VT2' contains a copy of the first K right singular vectors which will be used by DLASD3 in a matrix multiply (DGEMM) to solve for the new right singular vectors. VT2 is arranged into three blocks. The first block contains a row that corresponds to the special 0 diagonal element in SIGMA; the second block contains non-zeros only at and before NL +1; the third block contains non-zeros only at and after NL +2.

LDVT2 (input) INTEGER

The leading dimension of the array VT2. LDVT2 >= M.

IDXP (workspace) INTEGER array dimension(N)

This will contain the permutation used to place deflated values of D at the end of the array. On output IDXP(2:K)
points to the nondeflated D-values and IDXP(K+1:N) points to the deflated singular values.

IDX (workspace) INTEGER array dimension(N)

This will contain the permutation used to sort the contents of D into ascending order.

IDXC (output) INTEGER array dimension(N)

This will contain the permutation used to arrange the columns of the deflated U matrix into three groups: the first group contains non-zero entries only at and above NL, the second contains non-zero entries only below NL+2, and the third is dense.

IDXQ (input/output) INTEGER array dimension(N)

This contains the permutation which separately sorts the two sub-problems in D into ascending order. Note that entries in the first hlaf of this permutation must first be moved one position backward; and entries in the second half must first have NL+1 added to their values. COLTYP (workspace/output) INTEGER array dimension(N) As workspace, this will contain a label which will indicate which of the following types a column in the U2 matrix or a row in the VT2 matrix is:
1 : non-zero in the upper half only
2 : non-zero in the lower half only
3 : dense
4 : deflated On exit, it is an array of dimension 4, with COLTYP(I) being the dimension of the I-th type columns.

INFO (output) INTEGER

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

FURTHER DETAILS

Based on contributions by

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