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 values are close together or if there is a tiny entry in the Z vec-
       tor.  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 singu-
	      lar values (those which were deflated) sorted into increasing 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 equation.

	      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 everywhere
	      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 submatrices 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 trailing (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  contains
	      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 fol-
	      lowing 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 position 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)