DGELSX(3)             LAPACK driver routine (version 3.1)            DGELSX(3)



NAME
       DGELSX - i deprecated and has been replaced by routine DGELSY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       This routine is deprecated and has been replaced by routine DGELSY.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  con-
       dition  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  annihilated  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) 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 column.  Before the  QR  factor-
               ization  of  A, all initial columns are permuted to the leading
               positions; only the remaining  free  columns  are  moved  as  a
               result  of  column pivoting during the factorization.  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  subma-
               trix  R11  in  the  QR  factorization with pivoting of A, whose
               estimated condition 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 com-
               plete orthogonal factorization of A.

       WORK    (workspace) DOUBLE PRECISION 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 illegal value



 LAPACK driver routine (version 3.N1o)vember 2006                       DGELSX(3)