slasda(3)

NAME

SLASDA - a divide and conquer approach, SLASDA computes

the singular value decomposition (SVD) of a real upper bidiagonal

N-by-M matrix B with diagonal D and offdiagonal E, where M = N +

SQRE

SYNOPSIS

SUBROUTINE  SLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU,
VT, K, DIFL,  DIFR,  Z,  POLES,  GIVPTR,  GIVCOL,  LDGCOL,  PERM,
GIVNUM, C, S, WORK, IWORK, INFO )
    INTEGER         ICOMPQ,  INFO, LDGCOL, LDU, N, SMLSIZ,
SQRE
    INTEGER        GIVCOL(  LDGCOL,  *  ),  GIVPTR(  *  ),
IWORK( * ), K( * ), PERM( LDGCOL, * )
    REAL            C(  * ), D( * ), DIFL( LDU, * ), DIFR(
LDU, * ), E( * ), GIVNUM( LDU, * ), POLES( LDU, * ), S( *  ),  U(
LDU, * ), VT( LDU, * ), WORK( * ), Z( LDU, * )

PURPOSE

Using a divide and conquer approach, SLASDA computes the

singular value decomposition (SVD) of a real upper bidiagonal N

by-M matrix B with diagonal D and offdiagonal E, where M = N +

SQRE. The algorithm computes the singular values in the SVD B = U

* S * VT. The orthogonal matrices U and VT are optionally com

puted in compact form.

A related subroutine, SLASD0, computes the singular values

and the singular vectors in explicit form.

ARGUMENTS

ICOMPQ (input) INTEGER Specifies whether singular vectors

are to be computed in compact form, as follows = 0: Compute sin

gular values only.
= 1: Compute singular vectors of upper bidiagonal matrix

in compact form.

SMLSIZ (input) INTEGER The maximum size of the subproblems

at the bottom of the computation tree.

N (input) INTEGER

The row dimension of the upper bidiagonal matrix.

This is also the dimension of the main diagonal array D.

SQRE (input) INTEGER

Specifies the column dimension of the bidiagonal

matrix. = 0: The bidiagonal matrix has column dimension M = N;
= 1: The bidiagonal matrix has column dimension M =

N + 1.

D (input/output) REAL array, dimension ( N )

On entry D contains the main diagonal of the bidi

agonal matrix. On exit D, if INFO = 0, contains its singular val

ues.

E (input) REAL array, dimension ( M-1 )

Contains the subdiagonal entries of the bidiagonal

matrix. On exit, E has been destroyed.

U (output) REAL array,

dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not

referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the

left singular vector matrices of all subproblems at the bottom

level.

LDU (input) INTEGER, LDU = > N.

The leading dimension of arrays U, VT, DIFL, DIFR,

POLES, GIVNUM, and Z.

VT (output) REAL array,

dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not

referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains

the right singular vector matrices of all subproblems at the bot

tom level.

K (output) INTEGER array,

dimension ( N ) if ICOMPQ = 1 and dimension 1 if

ICOMPQ = 0. If ICOMPQ = 1, on exit, K(I) is the dimension of the

I-th secular equation on the computation tree.

DIFL (output) REAL array, dimension ( LDU, NLVL ),

where NLVL = floor(log_2 (N/SMLSIZ))).

DIFR (output) REAL array,

dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and di

mension ( N ) if ICOMPQ = 0. If ICOMPQ = 1, on exit, DIFL(1:N,

I) and DIFR(1:N, 2 * I - 1) record distances between singular

values on the I-th level and singular values on the (I -1)-th

level, and DIFR(1:N, 2 * I ) contains the normalizing factors for

the right singular vector matrix. See SLASD8 for details.

Z (output) REAL array,

dimension ( LDU, NLVL ) if ICOMPQ = 1 and dimension

( N ) if ICOMPQ = 0. The first K elements of Z(1, I) contain the

components of the deflation-adjusted updating row vector for sub

problems on the I-th level.

POLES (output) REAL array,

dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not

referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I

1) and POLES(1, 2*I) contain the new and old singular values in

volved in the secular equations on the I-th level.

GIVPTR (output) INTEGER array, dimension ( N ) if

ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1, on

exit, GIVPTR( I ) records the number of Givens rotations per

formed on the I-th problem on the computation tree.

GIVCOL (output) INTEGER array, dimension ( LDGCOL,

2 * NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If

ICOMPQ = 1, on exit, for each I, GIVCOL(1, 2 *I - 1) and GIV

COL(1, 2 *I) record the locations of Givens rotations performed

on the I-th level on the computation tree.

LDGCOL (input) INTEGER, LDGCOL = > N. The leading

dimension of arrays GIVCOL and PERM.

PERM (output) INTEGER array,

dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not

referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I)

records permutations done on the I-th level of the computation

tree.

GIVNUM (output) REAL array, dimension ( LDU, 2 *

NLVL ) if ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ

= 1, on exit, for each I, GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I)

record the C- and S- values of Givens rotations performed on the

I-th level on the computation tree.

C (output) REAL array,

dimension ( N ) if ICOMPQ = 1, and dimension 1 if

ICOMPQ = 0. If ICOMPQ = 1 and the I-th subproblem is not square,

on exit, C( I ) contains the C-value of a Givens rotation related

to the right null space of the I-th subproblem.

S (output) REAL array, dimension ( N ) if

ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOM

PQ = 1 and the I-th subproblem is not square, on exit, S( I )

contains the S-value of a Givens rotation related to the right

null space of the I-th subproblem.

WORK (workspace) REAL array, dimension

(6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).

IWORK (workspace) INTEGER array.

Dimension must be at least (7 * N).

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

SLASDA(3)