dlasd2(3)

NAME

DLASD2 - merge the two sets of singular values together

into a single sorted set

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 in

to 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 in

to increasing order.

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 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 trail

ing (N-K) updated left singular vectors (those which were deflat

ed) in its last N-K columns.

LDU (input) INTEGER

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

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

On exit Z contains the updating row vector in the

secular equation.

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 vec

tors 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.

VT (input/output) DOUBLE PRECISION array, dimen

sion(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.

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 con

tains non-zeros only at and before NL +1; the third block con

tains 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 de

flated 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.

COLTYP (workspace/output) INTEGER array, dimen

sion(N) As workspace, this will contain a label which will indi

cate 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.

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 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.

INFO (output) INTEGER

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

gal value.

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

DLASD2(3)