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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( * )



     #include <sunperf.h>

     void sgelss(int m, int n, int  nrhs,  float  *sa,  int  lda,
               float  *sb,  int  ldb,  float *s, float rcond, int
               *rank,
                 int *info) ;

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 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 solu-
               tion matrix X.  If m >= n and RANK = n, the  resi-
               dual  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  performance,  LWORK
               should generally be larger.

     INFO      (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille-
               gal 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  con-
               verge to zero.

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