SLALSD(l)								 )								 SLALSD(l)

NAME

       SLALSD  -  use the singular value decomposition of A to solve the least squares problem of finding X to minimize the Euclidean norm of each
       column of A*X-B, where A is N-by-N upper bidiagonal, and X and B are N-by-NRHS

SYNOPSIS

       SUBROUTINE SLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK, WORK, IWORK, INFO )

	   CHARACTER	  UPLO

	   INTEGER	  INFO, LDB, N, NRHS, RANK, SMLSIZ

	   REAL 	  RCOND

	   INTEGER	  IWORK( * )

	   REAL 	  B( LDB, * ), D( * ), E( * ), WORK( * )

PURPOSE

       SLALSD uses the singular value decomposition of A to solve the least squares problem of finding X to minimize the Euclidean  norm  of  each
       column  of  A*X-B,  where  A  is N-by-N upper bidiagonal, and X and B are N-by-NRHS. The solution X overwrites B.  The singular values of A
       smaller than RCOND times the largest singular value are treated as zero in solving the least squares problem; in this case a  minimum  norm
       solution is returned.  The actual singular values are returned in D in ascending order.

       This  code  makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on
       those binary machines without guard digits which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.  It could conceivably fail	on
       hexadecimal or decimal machines without guard digits, but we know of none.

ARGUMENTS

       UPLO   (input) CHARACTER*1
	      = 'U': D and E define an upper bidiagonal matrix.
	      = 'L': D and E define a  lower bidiagonal matrix.

	      SMLSIZ (input) INTEGER The maximum size of the subproblems at the bottom of the computation tree.

       N      (input) INTEGER
	      The dimension of the  bidiagonal matrix.	N >= 0.

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

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

       E      (input) REAL array, dimension (N-1)
	      Contains the super-diagonal entries of the bidiagonal matrix.  On exit, E has been destroyed.

       B      (input/output) REAL array, dimension (LDB,NRHS)
	      On input, B contains the right hand sides of the least squares problem. On output, B contains the solution X.

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

       RCOND  (input) REAL
	      The  singular  values  of  A  less  than or equal to RCOND times the largest singular value are treated as zero in solving the least
	      squares problem. If RCOND is negative, machine precision is used instead.  For example, if diag(S)*X=B were the least squares  prob-
	      lem,  where  diag(S)  is	a  diagonal  matrix  of  singular values, the solution would be X(i) = B(i) / S(i) if S(i) is greater than
	      RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to RCOND*max(S).

       RANK   (output) INTEGER
	      The number of singular values of A greater than RCOND times the largest singular value.

       WORK   (workspace) REAL array, dimension at least
	      (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2), where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).

       IWORK  (workspace) INTEGER array, dimension at least
	      (3*N*NLVL + 11*N)

       INFO   (output) INTEGER
	      = 0:  successful exit.
	      < 0:  if INFO = -i, the i-th argument had an illegal value.
	      > 0:  The algorithm failed to compute an singular value while working on the submatrix lying in rows and columns INFO/(N+1)  through
	      MOD(INFO,N+1).

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 							 SLALSD(l)