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CLALSA(l)			       )			     CLALSA(l)

NAME
       CLALSA  - i an itermediate step in solving the least squares problem by
       computing the SVD of the coefficient matrix in compact form (The singu‐
       lar vectors are computed as products of simple orthorgonal matrices.)

SYNOPSIS
       SUBROUTINE CLALSA( ICOMPQ,  SMLSIZ,  N, NRHS, B, LDB, BX, LDBX, U, LDU,
			  VT, K, DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
			  PERM, GIVNUM, C, S, RWORK, IWORK, INFO )

	   INTEGER	  ICOMPQ,  INFO, LDB, LDBX, LDGCOL, LDU, N, NRHS, SML‐
			  SIZ

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

	   REAL		  C( * ), DIFL( LDU, * ), DIFR( LDU, * ), GIVNUM( LDU,
			  * ), POLES( LDU, * ), RWORK( * ), S( * ), U( LDU,  *
			  ), VT( LDU, * ), Z( LDU, * )

	   COMPLEX	  B( LDB, * ), BX( LDBX, * )

PURPOSE
       CLALSA  is  an itermediate step in solving the least squares problem by
       computing the SVD of the coefficient matrix in compact form (The singu‐
       lar  vectors are computed as products of simple orthorgonal matrices.).
       If ICOMPQ = 0, CLALSA applies the inverse of the left  singular	vector
       matrix  of  an  upper  bidiagonal matrix to the right hand side; and if
       ICOMPQ = 1, CLALSA applies the right  singular  vector  matrix  to  the
       right hand side. The singular vector matrices were generated in compact
       form by CLALSA.

ARGUMENTS
       ICOMPQ (input) INTEGER Specifies whether the left or the right singular
       vector matrix is involved.  = 0: Left singular vector matrix
       = 1: Right singular vector matrix

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

       N      (input) INTEGER
	      The row and column dimensions of the upper bidiagonal matrix.

       NRHS   (input) INTEGER
	      The number of columns of B and BX. NRHS must be at least 1.

       B      (input) COMPLEX array, dimension ( LDB, NRHS )
	      On input, B contains the right hand sides of the	least  squares
	      problem  in rows 1 through M. On output, B contains the solution
	      X in rows 1 through N.

       LDB    (input) INTEGER
	      The leading dimension of B in the calling subprogram.  LDB  must
	      be at least max(1,MAX( M, N ) ).

       BX     (output) COMPLEX array, dimension ( LDBX, NRHS )
	      On  exit, the result of applying the left or right singular vec‐
	      tor matrix to B.

       LDBX   (input) INTEGER
	      The leading dimension of BX.

       U      (input) REAL array, dimension ( LDU, SMLSIZ ).
	      On entry, 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     (input) REAL array, dimension ( LDU, SMLSIZ+1 ).
	      On entry, VT' contains the right singular vector matrices of all
	      subproblems at the bottom level.

       K      (input) INTEGER array, dimension ( N ).

       DIFL   (input) REAL array, dimension ( LDU, NLVL ).
	      where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.

       DIFR   (input) REAL array, dimension ( LDU, 2 * NLVL ).
	      On  entry,  DIFL(*,  I)  and  DIFR(*, 2 * I -1) record distances
	      between singular values on the I-th level and singular values on
	      the  (I  -1)-th level, and DIFR(*, 2 * I) record the normalizing
	      factors of the right singular vectors matrices of subproblems on
	      I-th level.

       Z      (input) REAL array, dimension ( LDU, NLVL ).
	      On  entry,  Z(1,	I)  contains  the components of the deflation-
	      adjusted updating row vector for subproblems on the I-th level.

       POLES  (input) REAL array, dimension ( LDU, 2 * NLVL ).
	      On entry, POLES(*, 2 * I -1: 2 * I) contains  the	 new  and  old
	      singular	values	involved  in the secular equations on the I-th
	      level.

	      GIVPTR (input) INTEGER  array,  dimension	 (  N  ).   On	entry,
	      GIVPTR(  I ) records the number of Givens rotations performed on
	      the I-th problem on the computation tree.

	      GIVCOL (input) INTEGER array, dimension ( LDGCOL, 2  *  NLVL  ).
	      On  entry,  for  each I, GIVCOL(*, 2 * I - 1: 2 * I) records 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   (input) INTEGER array, dimension ( LDGCOL, NLVL ).
	      On entry, PERM(*, I) records permutations done on the I-th level
	      of the computation tree.

	      GIVNUM  (input)  REAL  array,  dimension	( LDU, 2 * NLVL ).  On
	      entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and  S-	values
	      of  Givens rotations performed on the I-th level on the computa‐
	      tion tree.

       C      (input) REAL array, dimension ( N ).
	      On entry, if the I-th subproblem is not square, C( I )  contains
	      the C-value of a Givens rotation related to the right null space
	      of the I-th subproblem.

       S      (input) REAL array, dimension ( N ).
	      On entry, if the I-th subproblem is not square, S( I )  contains
	      the S-value of a Givens rotation related to the right null space
	      of the I-th subproblem.

       RWORK  (workspace) REAL array, dimension at least
	      max ( N, (SMLSZ+1)*NRHS*3 ).

       IWORK  (workspace) INTEGER array.
	      The dimension must be at least 3 * N

       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 Ren-Cang Li, Computer Science Division, University of
	    California at Berkeley, USA
	  Osni Marques, LBNL/NERSC, USA

LAPACK version 3.0		 15 June 2000			     CLALSA(l)
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