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DLASD2(l)			       )			     DLASD2(l)

NAME
       DLASD2  -  merge the two sets of singular values together into a single
       sorted set

SYNOPSIS
       SUBROUTINE DLASD2( NL, NR, SQRE, K, D, Z,  ALPHA,  BETA,	 U,  LDU,  VT,
			  LDVT, DSIGMA, U2, LDU2, VT2, LDVT2, IDXP, IDX, IDXC,
			  IDXQ, COLTYP, INFO )

	   INTEGER	  INFO, K, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE

	   DOUBLE	  PRECISION ALPHA, BETA

	   INTEGER	  COLTYP( * ), IDX( * ), IDXC( * ), IDXP( * ), IDXQ( *
			  )

	   DOUBLE	  PRECISION  D(	 *  ),	DSIGMA(	 * ), U( LDU, * ), U2(
			  LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), Z( * )

PURPOSE
       DLASD2 merges the two sets of singular values together  into  a	single
       sorted set. Then it tries to deflate the size of the problem. There are
       two ways in which deflation can occur:  when two or more singular  val‐
       ues  are	 close	together  or if there is a tiny entry in the Z vector.
       For each such occurrence the order  of  the  related  secular  equation
       problem is reduced by one.

       DLASD2 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      (output) INTEGER
	      Contains the dimension of the non-deflated matrix, This  is  the
	      order of the related secular equation. 1 <= K <=N.

       D      (input/output) DOUBLE PRECISION array, dimension(N)
	      On  entry	 D contains the singular values of the two submatrices
	      to be combined.  On exit D contains the trailing	(N-K)  updated
	      singular values (those which were deflated) sorted into increas‐
	      ing order.

       ALPHA  (input) DOUBLE PRECISION
	      Contains the diagonal element associated with the added row.

       BETA   (input) DOUBLE PRECISION
	      Contains the off-diagonal element associated with the added row.

       U      (input/output) DOUBLE PRECISION array, dimension(LDU,N)
	      On entry U contains the left singular vectors of two submatrices
	      in  the  two  square blocks with corners at (1,1), (NL, NL), and
	      (NL+2, NL+2), (N,N).  On exit  U	contains  the  trailing	 (N-K)
	      updated left singular vectors (those which were deflated) in its
	      last N-K columns.

       LDU    (input) INTEGER
	      The leading dimension of the array U.  LDU >= N.

       Z      (output) DOUBLE PRECISION array, dimension(N)
	      On exit Z contains the updating row vector in the secular	 equa‐
	      tion.

	      DSIGMA (output) DOUBLE PRECISION array, dimension (N) Contains a
	      copy of the diagonal elements (K-1 singular values and one zero)
	      in the secular equation.

       U2     (output) DOUBLE PRECISION array, dimension(LDU2,N)
	      Contains	a  copy	 of  the first K-1 left singular vectors which
	      will be used by DLASD3 in a matrix multiply (DGEMM) to solve for
	      the  new left singular vectors. U2 is arranged into four blocks.
	      The first block contains a column with 1 at NL+1 and zero every‐
	      where  else;  the second block contains non-zero entries only at
	      and above NL; the third contains	non-zero  entries  only	 below
	      NL+1; and the fourth is dense.

       LDU2   (input) INTEGER
	      The leading dimension of the array U2.  LDU2 >= N.

       VT     (input/output) DOUBLE PRECISION array, dimension(LDVT,M)
	      On  entry	 VT' contains the right singular vectors of two subma‐
	      trices in the two square blocks with corners  at	(1,1),	(NL+1,
	      NL+1), and (NL+2, NL+2), (M,M).  On exit VT' contains the trail‐
	      ing (N-K) updated	 right	singular  vectors  (those  which  were
	      deflated)	 in  its  last N-K columns.  In case SQRE =1, the last
	      row of VT spans the right null space.

       LDVT   (input) INTEGER
	      The leading dimension of the array VT.  LDVT >= M.

       VT2    (output) DOUBLE PRECISION array, dimension(LDVT2,N)
	      VT2' contains a copy of the first K right singular vectors which
	      will be used by DLASD3 in a matrix multiply (DGEMM) to solve for
	      the new right singular  vectors.	VT2  is	 arranged  into	 three
	      blocks.  The  first block contains a row that corresponds to the
	      special 0 diagonal element in SIGMA; the second  block  contains
	      non-zeros	 only  at  and	before NL +1; the third block contains
	      non-zeros only at and after  NL +2.

       LDVT2  (input) INTEGER
	      The leading dimension of the array VT2.  LDVT2 >= M.

       IDXP   (workspace) INTEGER array, dimension(N)
	      This will contain the permutation used to place deflated	values
	      of D at the end of the array. On output IDXP(2:K)
	      points to the nondeflated D-values and IDXP(K+1:N) points to the
	      deflated singular values.

       IDX    (workspace) INTEGER array, dimension(N)
	      This will contain the permutation used to sort the contents of D
	      into ascending order.

       IDXC   (output) INTEGER array, dimension(N)
	      This will contain the permutation used to arrange the columns of
	      the deflated U matrix into three groups:	the first  group  con‐
	      tains non-zero entries only at and above NL, the second contains
	      non-zero entries only below NL+2, and the third is dense.

	      COLTYP  (workspace/output)  INTEGER   array,   dimension(N)   As
	      workspace,  this	will contain a label which will indicate which
	      of the following types a column in the U2 matrix or a row in the
	      VT2 matrix is:
	      1 : non-zero in the upper half only
	      2 : non-zero in the lower half only
	      3 : dense
	      4 : deflated

	      On exit, it is an array of dimension 4, with COLTYP(I) being the
	      dimension of the I-th type columns.

       IDXQ   (input) INTEGER array, dimension(N)
	      This contains the permutation which  separately  sorts  the  two
	      sub-problems  in	D  into ascending order.  Note that entries in
	      the first hlaf of this permutation must first be moved one posi‐
	      tion  backward;  and  entries in the second half must first have
	      NL+1 added to their values.

       INFO   (output) INTEGER
	      = 0:  successful exit.
	      < 0:  if INFO = -i, the i-th argument had an illegal value.

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			     DLASD2(l)
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