slasd3(3)

NAME

SLASD3 - find all the square roots of the roots of the

secular equation, as defined by the values in D and Z

SYNOPSIS

SUBROUTINE  SLASD3( NL, NR, SQRE, K, D, Q, LDQ, DSIGMA, U,
LDU, U2, LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, INFO )
    INTEGER        INFO, K, LDQ, LDU, LDU2,  LDVT,  LDVT2,
NL, NR, SQRE
    INTEGER        CTOT( * ), IDXC( * )
    REAL            D(  *  ), DSIGMA( * ), Q( LDQ, * ), U(
LDU, * ), U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), Z( * )

PURPOSE

SLASD3 finds all the square roots of the roots of the sec

ular equation, as defined by the values in D and Z. It makes the

appropriate calls to SLASD4 and then updates the singular vectors

by matrix multiplication.

This code makes very mild assumptions about floating point

arithmetic. It will work on machines with a guard digit in

add/subtract, or on those binary machines without guard digits

which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.

It could conceivably fail on hexadecimal or decimal machines

without guard digits, but we know of none.

SLASD3 is called from SLASD1.

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 (input) INTEGER

The size of the secular equation, 1 =< K = < N.

D (output) REAL array, dimension(K)

On exit the square roots of the roots of the secu

lar equation, in ascending order.

Q (workspace) REAL array,

dimension at least (LDQ,K).

LDQ (input) INTEGER

The leading dimension of the array Q. LDQ >= K.

DSIGMA (input) REAL array, dimension(K) The first K

elements of this array contain the old roots of the deflated up

dating problem. These are the poles of the secular equation.

U (input) REAL array, dimension (LDU, N)

The last N - K columns of this matrix contain the

deflated left singular vectors.

LDU (input) INTEGER

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

U2 (input) REAL array, dimension (LDU2, N)

The first K columns of this matrix contain the non

deflated left singular vectors for the split problem.

LDU2 (input) INTEGER

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

VT (input) REAL array, dimension (LDVT, M)

The last M - K columns of VT' contain the deflated

right singular vectors.

LDVT (input) INTEGER

The leading dimension of the array VT. LDVT >= N.

VT2 (input) REAL array, dimension (LDVT2, N)

The first K columns of VT2' contain the non-deflat

ed right singular vectors for the split problem.

LDVT2 (input) INTEGER

The leading dimension of the array VT2. LDVT2 >=

N.

IDXC (input) INTEGER array, dimension ( N )

The permutation used to arrange the columns of U

(and rows of VT) into three groups: the first group contains

non-zero entries only at and above (or before) NL +1; the second

contains non-zero entries only at and below (or after) NL+2; and

the third is dense. The first column of U and the row of VT are

treated separately, however.

The rows of the singular vectors found by SLASD4

must be likewise permuted before the matrix multiplies can take

place.

CTOT (input) INTEGER array, dimension ( 4 )

A count of the total number of the various types of

columns in U (or rows in VT), as described in IDXC. The fourth

column type is any column which has been deflated.

Z (input) REAL array, dimension (K)

The first K elements of this array contain the com

ponents of the deflation-adjusted updating row vector.

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

SLASD3(3)