DLASD3(l)								 )								 DLASD3(l)

NAME

       DLASD3 - find all the square roots of the roots of the secular equation, as defined by the values in D and Z

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 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) 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  updat-
	      ing problem.  These are the poles of the secular equation.

       U      (input) 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) DOUBLE PRECISION 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) 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) DOUBLE PRECISION 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 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 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

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 							 DLASD3(l)