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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 over-
	       written with its right singular vectors, stored rowwise.

       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 ele-
	       ments 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  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.

       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)
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