CGELSX  -  routine  is deprecated and has been replaced by
       routine CGELSY


SYNOPSIS

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

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

           REAL           RCOND

           INTEGER        JPVT( * )

           REAL           RWORK( * )

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


PURPOSE

       This  routine  is deprecated and has been replaced by rou­
       tine CGELSY.  CGELSX computes the minimum-norm solution to
       a complex 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 unitary transformations from the right, arriv­
       ing 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) COMPLEX 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) COMPLEX 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) COMPLEX array, dimension
               (min(M,N) + max( 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


Man(1) output converted with man2html