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DLASD3(1)	    LAPACK auxiliary routine (version 3.2)	     DLASD3(1)

NAME
       DLASD3  -  finds all the square roots of the roots of the secular equa‐
       tion, 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 updating prob‐
	      lem.  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 non-deflated 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	singu‐
	      lar 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 non-deflated right sin‐
	      gular 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 mul‐
	      tiplies 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 auxiliary routine (versioNovember 2008			     DLASD3(1)
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