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DLASDA(l)			       )			     DLASDA(l)

NAME
       DLASDA  -  a  divide and conquer approach, DLASDA computes the singular
       value decomposition (SVD) of a real upper bidiagonal  N-by-M  matrix  B
       with diagonal D and offdiagonal E, where M = N + SQRE

SYNOPSIS
       SUBROUTINE DLASDA( ICOMPQ,  SMLSIZ, N, SQRE, D, E, U, LDU, VT, K, DIFL,
			  DIFR,	 Z,  POLES,  GIVPTR,  GIVCOL,  LDGCOL,	 PERM,
			  GIVNUM, C, S, WORK, IWORK, INFO )

	   INTEGER	  ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE

	   INTEGER	  GIVCOL(  LDGCOL,  * ), GIVPTR( * ), IWORK( * ), K( *
			  ), PERM( LDGCOL, * )

	   DOUBLE	  PRECISION C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU,
			  * ), E( * ), GIVNUM( LDU, * ), POLES( LDU, * ), S( *
			  ), U( LDU, * ), VT( LDU, * ), WORK( * ), Z( LDU, * )

PURPOSE
       Using a divide and conquer approach, DLASDA computes the singular value
       decomposition  (SVD)  of	 a  real upper bidiagonal N-by-M matrix B with
       diagonal D and offdiagonal E, where M = N + SQRE.  The  algorithm  com‐
       putes  the  singular  values in the SVD B = U * S * VT.	The orthogonal
       matrices U and VT are optionally computed in compact form.

       A related subroutine, DLASD0, computes the singular values and the sin‐
       gular vectors in explicit form.

ARGUMENTS
       ICOMPQ  (input)	INTEGER	 Specifies  whether singular vectors are to be
       computed in compact form, as follows = 0: Compute singular values only.
       = 1: Compute singular vectors of upper  bidiagonal  matrix  in  compact
       form.

       SMLSIZ  (input) INTEGER The maximum size of the subproblems at the bot‐
       tom of the computation tree.

       N      (input) INTEGER
	      The row dimension of the upper bidiagonal matrix. This  is  also
	      the dimension of the main diagonal array D.

       SQRE   (input) INTEGER
	      Specifies	 the  column dimension of the bidiagonal matrix.  = 0:
	      The bidiagonal matrix has column dimension M = N;
	      = 1: The bidiagonal matrix has column dimension M = N + 1.

       D      (input/output) DOUBLE PRECISION array, dimension ( N )
	      On entry D contains the main diagonal of the bidiagonal  matrix.
	      On exit D, if INFO = 0, contains its singular values.

       E      (input) DOUBLE PRECISION array, dimension ( M-1 )
	      Contains	the  subdiagonal entries of the bidiagonal matrix.  On
	      exit, E has been destroyed.

       U      (output) DOUBLE PRECISION array,
	      dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not	referenced  if
	      ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left singular
	      vector matrices of all subproblems at the bottom level.

       LDU    (input) INTEGER, LDU = > N.
	      The leading dimension  of	 arrays	 U,  VT,  DIFL,	 DIFR,	POLES,
	      GIVNUM, and Z.

       VT     (output) DOUBLE PRECISION array,
	      dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced if
	      ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right  sin‐
	      gular vector matrices of all subproblems at the bottom level.

       K      (output) INTEGER array,
	      dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.  If
	      ICOMPQ = 1, on exit, K(I) is the dimension of the	 I-th  secular
	      equation on the computation tree.

       DIFL   (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ),
	      where NLVL = floor(log_2 (N/SMLSIZ))).

       DIFR   (output) DOUBLE PRECISION array,
	      dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and dimension ( N ) if
	      ICOMPQ = 0.  If ICOMPQ = 1, on exit, DIFL(1:N, I) and  DIFR(1:N,
	      2	 * I - 1) record distances between singular values on the I-th
	      level and singular values on the (I -1)-th level, and  DIFR(1:N,
	      2	 * I ) contains the normalizing factors for the right singular
	      vector matrix. See DLASD8 for details.

       Z      (output) DOUBLE PRECISION array,
	      dimension ( LDU, NLVL ) if ICOMPQ = 1 and dimension  (  N	 )  if
	      ICOMPQ  = 0.  The first K elements of Z(1, I) contain the compo‐
	      nents of the deflation-adjusted updating row vector for subprob‐
	      lems on the I-th level.

       POLES  (output) DOUBLE PRECISION array,
	      dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced if
	      ICOMPQ = 0. If ICOMPQ = 1,  on  exit,  POLES(1,  2*I  -  1)  and
	      POLES(1,	2*I) contain  the new and old singular values involved
	      in the secular equations on the I-th level.

	      GIVPTR (output) INTEGER array, dimension ( N ) if	 ICOMPQ	 =  1,
	      and  not	referenced  if	ICOMPQ	=  0.  If ICOMPQ = 1, on exit,
	      GIVPTR( I ) records the number of Givens rotations performed  on
	      the I-th problem on the computation tree.

	      GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 * NLVL ) if
	      ICOMPQ = 1, and not referenced if ICOMPQ = 0. If ICOMPQ = 1,  on
	      exit, for each I, GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record
	      the locations of Givens rotations performed on the I-th level on
	      the computation tree.

	      LDGCOL  (input) INTEGER, LDGCOL = > N.  The leading dimension of
	      arrays GIVCOL and PERM.

       PERM   (output) INTEGER array,
	      dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced  if
	      ICOMPQ  = 0. If ICOMPQ = 1, on exit, PERM(1, I) records permuta‐
	      tions done on the I-th level of the computation tree.

	      GIVNUM (output) DOUBLE PRECISION array, dimension (  LDU,	  2  *
	      NLVL  )  if  ICOMPQ  =  1,  and not referenced if ICOMPQ = 0. If
	      ICOMPQ = 1, on exit, for	each  I,  GIVNUM(1,  2	*I  -  1)  and
	      GIVNUM(1,	 2 *I) record the C- and S- values of Givens rotations
	      performed on the I-th level on the computation tree.

       C      (output) DOUBLE PRECISION array,
	      dimension ( N ) if ICOMPQ = 1, and dimension 1 if	 ICOMPQ	 =  0.
	      If ICOMPQ = 1 and the I-th subproblem is not square, on exit, C(
	      I ) contains the C-value of a Givens  rotation  related  to  the
	      right null space of the I-th subproblem.

       S      (output) DOUBLE PRECISION array, dimension ( N ) if
	      ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1 and the
	      I-th subproblem is not square, on exit, S( I ) contains  the  S-
	      value  of	 a  Givens rotation related to the right null space of
	      the I-th subproblem.

       WORK   (workspace) DOUBLE PRECISION array, dimension
	      (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).

       IWORK  (workspace) INTEGER array.
	      Dimension must be at least (7 * N).

       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			     DLASDA(l)
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