SLASD3 - find all the square roots of the roots of the secular equation, as defined by the values in D and Z

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( * )

SLASD3 finds all the square roots of the roots of the secular 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.

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 secular 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 updating 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-deflated 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 components of the deflation-adjusted updating row vector.

INFO (output) INTEGER

= 0: successful exit.

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

> 0: if INFO = 1, an singular value did not converge

Based on contributions by

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