SRC/sgelsy.f(3)

NAME
SRC/sgelsy.f

SYNOPSIS

  Functions/Subroutines
subroutine sgelsy (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, info)
SGELSY solves overdetermined or underdetermined systems for GE matrices

Function/Subroutine Documentation

  subroutine sgelsy (integer m, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, integer, dimension( * ) jpvt, real rcond, integer rank, real, dimension( * ) work, integer lwork, integer info)

SGELSY solves overdetermined or underdetermined systems for GE matrices

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.
    !>

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 = 0 or N = 0, B is not referenced.
    !>

    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 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

    !>          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.
    !>          If NRHS = 0, RANK = 0 on output.
    !>

    WORK

    !>          WORK is REAL array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          LWORK is 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

    !>          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.

Contributors:
    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

Definition at line 204 of file sgelsy.f.

Author

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