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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