NAME

       sgelsx.f -

SYNOPSIS

   Functions/Subroutines
       subroutine sgelsx (M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, INFO)
            SGELSX solves overdetermined or underdetermined systems for GE matrices

Function/Subroutine Documentation

   subroutine sgelsx (integerM, integerN, integerNRHS, real, dimension( lda, * )A, integerLDA,
       real, dimension( ldb, * )B, integerLDB, integer, dimension( * )JPVT, realRCOND,
       integerRANK, real, dimension( * )WORK, integerINFO)
        SGELSX solves overdetermined or underdetermined systems for GE matrices

       Purpose:

            This routine is deprecated and has been replaced by routine SGELSY.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 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**T [ inv(T11)*Q1**T*B ]
                            [        0         ]
            where Q1 consists of the first RANK columns of Q.

       Parameters:
           M

                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.

           N

                     N is INTEGER
                     The number of columns of the matrix A.  N >= 0.

           NRHS

                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of
                     columns of matrices B and X. NRHS >= 0.

           A

                     A is 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

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,M).

           B

                     B is 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

                     LDB is INTEGER
                     The leading dimension of the array B. LDB >= max(1,M,N).

           JPVT

                     JPVT is 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

                     RCOND is 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 factorization with pivoting of A,
                     whose estimated condition number < 1/RCOND.

           RANK

                     RANK is 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

                     WORK is REAL array, dimension
                                 (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),

           INFO

                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2011

       Definition at line 178 of file sgelsx.f.

Author

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