DLASD7 - merge the two sets of singular values together into a single sorted set

SUBROUTINE DLASD7(

ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL, VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ, PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, C, S, INFO )

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

DOUBLE PRECISION ALPHA, BETA, C, S

INTEGER GIVCOL( LDGCOL, * ), IDX( * ), IDXP( * ), IDXQ( * ), PERM( * )

DOUBLE PRECISION D( * ), DSIGMA( * ), GIVNUM( LDGNUM, * ), VF( * ), VFW( * ), VL( * ), VLW( * ), Z( * ), ZW( * )

DLASD7 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.

DLASD7 is called from DLASD6.

ICOMPQ (input) INTEGER

Specifies whether singular vectors are to be computed in compact form, as follows:

= 0: Compute singular values only.

= 1: Compute singular vectors of upper bidiagonal matrix in compact form.

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 ( M )

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

ZW (workspace) DOUBLE PRECISION array, dimension ( M )

Workspace for Z.

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.

VFW (workspace) DOUBLE PRECISION array, dimension ( M )

Workspace for VF.

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.

VLW (workspace) DOUBLE PRECISION array, dimension ( M )

Workspace for VL.

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.

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.

IDX (workspace) INTEGER array, dimension ( N )

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

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.

IDXQ (input) 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 half 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.

PERM (output) INTEGER array, dimension ( N )

The permutations (from deflation and sorting) to be applied to each singular 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 The 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, must be at least 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.

INFO (output) INTEGER

= 0: successful exit.

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

Based on contributions by

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