SLASD1(l)								 )								 SLASD1(l)

NAME

       SLASD1 - compute the SVD of an upper bidiagonal N-by-M matrix B,

SYNOPSIS

       SUBROUTINE SLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT, IDXQ, IWORK, WORK, INFO )

	   INTEGER	  INFO, LDU, LDVT, NL, NR, SQRE

	   REAL 	  ALPHA, BETA

	   INTEGER	  IDXQ( * ), IWORK( * )

	   REAL 	  D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )

PURPOSE

       SLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, where N = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0.

       A related subroutine SLASD7 handles the case in which the singular values (and the singular vectors in factored form) are desired.

       SLASD1 computes the SVD as follows:

		     ( D1(in)  0    0	  0 )
	 B = U(in) * (	 Z1'   a   Z2'	  b ) * VT(in)
		     (	 0     0   D2(in) 0 )

	   = U(out) * ( D(out) 0) * VT(out)

       where  Z'  =  (Z1'  a Z2' b) = u' VT', and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros else-
       where; and the entry b is empty if SQRE = 0.

       The left singular vectors of the original matrix are stored in U, and the transpose of the right singular vectors are stored in VT, and the
       singular values are in D.  The algorithm consists of three stages:

	  The first stage consists of deflating the size of the problem
	  when there are multiple singular values or when there are zeros in
	  the Z vector.  For each such occurence the dimension of the
	  secular equation problem is reduced by one.  This stage is
	  performed by the routine SLASD2.

	  The second stage consists of calculating the updated
	  singular values. This is done by finding the square roots of the
	  roots of the secular equation via the routine SLASD4 (as called
	  by SLASD3). This routine also calculates the singular vectors of
	  the current problem.

	  The final stage consists of computing the updated singular vectors
	  directly using the updated singular values.  The singular vectors
	  for the current problem are multiplied with the singular vectors
	  from the overall problem.

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 row dimension N = NL + NR + 1, and column dimension M = N + SQRE.

       D      (input/output) REAL array,
	      dimension (N = NL+NR+1).	On entry D(1:NL,1:NL) contains the singular values of the
	      upper block; and D(NL+2:N) contains the singular values of
	      the lower block. On exit D(1:N) contains the singular values of the modified matrix.

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

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

       U      (input/output) REAL array, dimension(LDU,N)
	      On entry U(1:NL, 1:NL) contains the left singular vectors of
	      the  upper block; U(NL+2:N, NL+2:N) contains the left singular vectors of the lower block. On exit U contains the left singular vec-
	      tors of the bidiagonal matrix.

       LDU    (input) INTEGER
	      The leading dimension of the array U.  LDU >= max( 1, N ).

       VT     (input/output) REAL array, dimension(LDVT,M)
	      where M = N + SQRE.  On entry VT(1:NL+1, 1:NL+1)' contains the right singular
	      vectors of the upper block; VT(NL+2:M, NL+2:M)' contains the right singular vectors of the lower block. On  exit	VT'  contains  the
	      right singular vectors of the bidiagonal matrix.

       LDVT   (input) INTEGER
	      The leading dimension of the array VT.  LDVT >= max( 1, M ).

       IDXQ  (output) INTEGER array, dimension(N)
	     This  contains  the permutation which will reintegrate the subproblem just solved back into sorted order, i.e.  D( IDXQ( I = 1, N ) )
	     will be in ascending order.

       IWORK  (workspace) INTEGER array, dimension( 4 * N )

       WORK   (workspace) REAL array, dimension( 3*M**2 + 2*M )

       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 							 SLASD1(l)