SYNOPSIS
 SUBROUTINE DLASD3(
 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( * )
 DOUBLE PRECISION D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ), U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), Z( * )
PURPOSE
DLASD3 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 DLASD4 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.
DLASD3 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 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) DOUBLE PRECISION array, dimension(K)
 On exit the square roots of the roots of the secular equation, in ascending order.
 Q (workspace) DOUBLE PRECISION array,
 dimension at least (LDQ,K).
 LDQ (input) INTEGER
 The leading dimension of the array Q. LDQ >= K. DSIGMA (input) DOUBLE PRECISION 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 (output) DOUBLE PRECISION 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/output) DOUBLE PRECISION 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 (output) DOUBLE PRECISION 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/output) DOUBLE PRECISION 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 DLASD4 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) DOUBLE PRECISION 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 byMing Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA