DGELSX  -  routine  is deprecated and has been replaced by
       routine DGELSY


SYNOPSIS

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

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

           DOUBLE         PRECISION RCOND

           INTEGER        JPVT( * )

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


PURPOSE

       This routine is deprecated and has been replaced  by  rou­
       tine DGELSY.  DGELSX 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.

               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) DOUBLE PRECISION 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)   DOUBLE   PRECISION   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) DOUBLE PRECISION
               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) DOUBLE PRECISION array, dimension
               (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),

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


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