Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3 - computes the minimum-norm solution to a real linear least squares problem

SYNOPSIS

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

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

           REAL           RCOND

           INTEGER        JPVT( * )

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

PURPOSE

       SGELSY 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.
        This routine is basically identical to the original xGELSX except
        three differences:
          o The call to the subroutine xGEQPF has been substituted by the
            the call to the subroutine xGEQP3. This subroutine is a Blas-3
            version of the QR factorization with column pivoting.
          o Matrix B (the right hand side) is updated with Blas-3.
          o The permutation of matrix B (the right hand side) is faster and
            more simple.

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

        B       (input/output) 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.

        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 permuted
                to the front of AP, otherwise column i is a free column.
                On exit, if JPVT(i) = k, then the i-th column of AP
                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 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/output) REAL array, dimension (MAX(1,LWORK))
                On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

        LWORK   (input) INTEGER
                The dimension of the array WORK.
                The unblocked strategy requires that:
                LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
                where MN = min( M, N ).
                The block algorithm requires that:
                LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
                where NB is an upper bound on the blocksize returned
                by ILAENV for the routines SGEQP3, STZRZF, STZRQF, SORMQR,
                and SORMRZ.
                If LWORK = -1, then a workspace query is assumed; the routine
                only calculates the optimal size of the WORK array, returns
                this value as the first entry of the WORK array, and no error
                message related to LWORK is issued by XERBLA.

        INFO    (output) INTEGER
                = 0: successful exit
                < 0: If INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS

        Based on contributions by
          A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
          E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
          G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

 LAPACK driver routine (version 3.3.1)      April 2011                            SGELSY(3lapack)