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

SYNOPSIS
     SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT,  RCOND,
               RANK, WORK, INFO )

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

     REAL RCOND

     INTEGER JPVT( * )

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



     #include <sunperf.h>

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

PURPOSE
     SGELSX computes the minimum-norm solution to a  real  linear
     least squares problem:
         minimize || A * X - B ||
     using a complete orthogonal factorization 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 routine first computes a QR  factorization  with  column
     pivoting:
         A * P = Q * [ R11 R12 ]
                     [  0  R22 ]
     with R11 defined as  the  largest  leading  submatrix  whose
     estimated  condition number is less than 1/RCOND.  The order
     of R11, RANK, is the effective rank of A.

     Then, R22 is considered to be negligible, and R12 is annihi-
     lated by orthogonal transformations from the right, arriving
     at the complete orthogonal factorization:
        A * P = Q * [ T11 0 ] * Z
                    [  0  0 ]
     The minimum-norm solution is then
        X = P * Z' [ inv(T11)*Q1'*B ]
                   [        0       ]

     where Q1 consists of the first RANK columns of Q.


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 matrices B and X. NRHS >= 0.

     A         (input/output) REAL array, dimension (LDA,N)
               On entry, the M-by-N matrix A.   On  exit,  A  has
               been   overwritten  by  details  of  its  complete
               orthogonal factorization.

     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, 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,M,N).

     JPVT      (input/output) INTEGER array, dimension (N)
               On entry, if JPVT(i) .ne. 0, the i-th column of  A
               is  an  initial  column,  otherwise  it  is a free
               column.  Before the QR  factorization  of  A,  all
               initial  columns are permuted to the leading posi-
               tions; only the remaining free columns  are  moved
               as a result of column pivoting during the factori-
               zation.  On exit, if JPVT(i) = k,  then  the  i-th
               column of A*P was the k-th column of A.

     RCOND     (input) REAL
               RCOND is used to determine the effective  rank  of
               A,  which  is  defined as the order of the largest
               leading triangular submatrix R11 in the QR factor-
               ization with pivoting of A, whose estimated condi-
               tion number < 1/RCOND.

     RANK      (output) INTEGER
               The effective rank of A, i.e., the  order  of  the
               submatrix  R11.   This is the same as the order of
               the submatrix T11 in the complete orthogonal  fac-
               torization of A.

     WORK      (workspace) REAL array, dimension
               (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),

     INFO      (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille-
               gal value

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