
SLASD3(l) ) SLASD3(l)
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 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 con
ceivably 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 NRbyNR square matrix.
= 1: the lower block is an NRby(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 nondeflated 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 nondeflated 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 nonzero entries only at and above (or before) NL
+1; the second contains nonzero 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 deflationadjusted
updating row vector.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = i, the ith argument had an illegal value.
> 0: if INFO = 1, an singular value did not converge
FURTHER DETAILS
Based on contributions by
Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA
LAPACK version 3.0 15 June 2000 SLASD3(l) 
