SGELSD(l)								 )								 SGELSD(l)

NAME

       SGELSD - compute the minimum-norm solution to a real linear least squares problem

SYNOPSIS

       SUBROUTINE SGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, IWORK, INFO )

	   INTEGER	  INFO, LDA, LDB, LWORK, M, N, NRHS, RANK

	   REAL 	  RCOND

	   INTEGER	  IWORK( * )

	   REAL 	  A( LDA, * ), B( LDB, * ), S( * ), WORK( * )

PURPOSE

       SGELSD computes the minimum-norm solution to a real linear least squares problem:     minimize 2-norm(| b - A*x |)
       using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient.

       Several	right  hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS
       right hand side matrix B and the N-by-NRHS solution matrix X.

       The problem is solved in three steps:
       (1) Reduce the coefficient matrix A to bidiagonal form with
	   Householder transformations, reducing the original problem
	   into a "bidiagonal least squares problem" (BLS)
       (2) Solve the BLS using a divide and conquer approach.
       (3) Apply back all the Householder tranformations to solve
	   the original least squares problem.

       The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

       The divide and conquer algorithm 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 X-MP, Cray Y-MP, Cray C-90, or Cray-2. It
       could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

ARGUMENTS

       M       (input) INTEGER
	       The number of rows of A. M >= 0.

       N       (input) INTEGER
	       The number of columns of A. N >= 0.

       NRHS    (input) INTEGER
	       The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.

       A       (input) REAL array, dimension (LDA,N)
	       On entry, the M-by-N matrix A.  On exit, A has been destroyed.

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

       B       (input/output) REAL array, dimension (LDB,NRHS)
	       On entry, the M-by-NRHS right hand side matrix B.  On exit, B is overwritten by the N-by-NRHS solution matrix X.  If  m	>=  n  and
	       RANK  = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that
	       column.

       LDB     (input) INTEGER
	       The leading dimension of the array B. LDB >= max(1,max(M,N)).

       S       (output) REAL array, dimension (min(M,N))
	       The singular values of A in decreasing order.  The condition number of A in the 2-norm = S(1)/S(min(m,n)).

       RCOND   (input) REAL
	       RCOND is used to determine the effective rank of A.  Singular values S(i) <= RCOND*S(1) are treated as zero.  If RCOND < 0, machine
	       precision is used instead.

       RANK    (output) INTEGER
	       The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1).

       WORK    (workspace/output) REAL array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The  dimension of the array WORK. LWORK must be at least 1.  The exact minimum amount of workspace needed depends on M, N and NRHS.
	       As long as LWORK is at least 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N  or  12*M  +
	       2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is less than N, the code will execute correctly.  SMLSIZ is returned by ILAENV
	       and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and NLVL  =  MAX(	0,
	       INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) For good performance, LWORK should generally be larger.

	       If  LWORK  =  -1,  then	a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this
	       value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

       IWORK   (workspace) INTEGER array, dimension (LIWORK)
	       LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, where MINMN = MIN( M,N ).

       INFO    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value.
	       > 0:  the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of  an  intermediate	bidiagonal
	       form did not converge to zero.

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 							 SGELSD(l)