SGELSX  -  routine  is deprecated and has been replaced by
       routine SGELSY


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( * )


PURPOSE

       This  routine  is deprecated and has been replaced by rou­
       tine SGELSY.  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 anni­
       hilated  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.

               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  col­
               umn.   Before  the QR factorization of A, all ini­
               tial columns are permuted  to  the  leading  posi­
               tions;  only  the remaining free columns are moved
               as a result of column pivoting during the  factor­
               ization.   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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