SGELSS(l)								 )								 SGELSS(l)

NAME

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

SYNOPSIS

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

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

	   REAL 	  RCOND

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

PURPOSE

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

ARGUMENTS

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

       N       (input) INTEGER
	       The number of columns of the matrix 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/output) REAL array, dimension (LDA,N)
	       On entry, the M-by-N matrix A.  On exit, the first min(m,n) rows of A are overwritten with its right singular vectors, stored  row-
	       wise.

       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 >= 1, and also: LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS  )	For  good  perfor-
	       mance, 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.

       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.

LAPACK version 3.0						   15 June 2000 							 SGELSS(l)