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DGELSS(1)	      LAPACK driver routine (version 3.2)	     DGELSS(1)

NAME
       DGELSS  -  computes  the	 minimum  norm solution to a real linear least
       squares problem

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

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

	   DOUBLE	  PRECISION RCOND

	   DOUBLE	  PRECISION  A( LDA, * ), B( LDB, * ), S( * ), WORK( *
			  )

PURPOSE
       DGELSS 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	singu‐
       lar 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) DOUBLE PRECISION 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 rowwise.

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

       B       (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
	       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)   DOUBLE   PRECISION   array,    dimension
       (MAX(1,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 bidiag‐
	       onal form did not converge to zero.

 LAPACK driver routine (version 3November 2008			     DGELSS(1)
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