dgelsx(3F)

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

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
     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 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
             factorization 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 submatrix
             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
             complete 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