SRC/sgels.f(3)

NAME
SRC/sgels.f

SYNOPSIS

  Functions/Subroutines
subroutine sgels (trans, m, n, nrhs, a, lda, b, ldb, work, lwork, info)
SGELS solves overdetermined or underdetermined systems for GE matrices

Function/Subroutine Documentation

  subroutine sgels (character trans, integer m, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) work, integer lwork, integer info)

SGELS solves overdetermined or underdetermined systems for GE matrices

Purpose:

    !>
    !> SGELS solves overdetermined or underdetermined real linear systems
    !> involving an M-by-N matrix A, or its transpose, using a QR or LQ
    !> factorization of A.
    !>
    !> It is assumed that A has full rank, and only a rudimentary protection
    !> against rank-deficient matrices is provided. This subroutine only detects
    !> exact rank-deficiency, where a diagonal element of the triangular factor
    !> of A is exactly zero.
    !>
    !> It is conceivable for one (or more) of the diagonal elements of the triangular
    !> factor of A to be subnormally tiny numbers without this subroutine signalling
    !> an error. The solutions computed for such almost-rank-deficient matrices may
    !> be less accurate due to a loss of numerical precision.
    !>
    !> The following options are provided:
    !>
    !> 1. If TRANS = 'N' and m >= n:  find the least squares solution of
    !>    an overdetermined system, i.e., solve the least squares problem
    !>                 minimize || B - A*X ||.
    !>
    !> 2. If TRANS = 'N' and m < n:  find the minimum norm solution of
    !>    an underdetermined system A * X = B.
    !>
    !> 3. If TRANS = 'T' and m >= n:  find the minimum norm solution of
    !>    an underdetermined system A**T * X = B.
    !>
    !> 4. If TRANS = 'T' and m < n:  find the least squares solution of
    !>    an overdetermined system, i.e., solve the least squares problem
    !>                 minimize || B - A**T * X ||.
    !>
    !> 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.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N': the linear system involves A;
    !>          = 'T': the linear system involves A**T.
    !>

    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 the matrices B and X. NRHS >=0.
    !>

    A

    !>          A is REAL array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix A.
    !>          On exit,
    !>            if M >= N, A is overwritten by details of its QR
    !>                       factorization as returned by SGEQRF;
    !>            if M <  N, A is overwritten by details of its LQ
    !>                       factorization as returned by SGELQF.
    !>

    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 matrix B of right hand side vectors, stored
    !>          columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
    !>          if TRANS = 'T'.
    !>          On exit, if INFO = 0, B is overwritten by the solution
    !>          vectors, stored columnwise:
    !>          if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
    !>          squares solution vectors; the residual sum of squares for the
    !>          solution in each column is given by the sum of squares of
    !>          elements N+1 to M in that column;
    !>          if TRANS = 'N' and m < n, rows 1 to N of B contain the
    !>          minimum norm solution vectors;
    !>          if TRANS = 'T' and m >= n, rows 1 to M of B contain the
    !>          minimum norm solution vectors;
    !>          if TRANS = 'T' and m < n, rows 1 to M of B contain the
    !>          least squares solution vectors; the residual sum of squares
    !>          for the solution in each column is given by the sum of
    !>          squares of elements M+1 to N in that column.
    !>

    LDB

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

    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.
    !>          LWORK >= max( 1, MN + max( MN, NRHS ) ).
    !>          For optimal performance,
    !>          LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
    !>          where MN = min(M,N) and NB is the optimum block size.
    !>
    !>          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
    !>          > 0:  if INFO =  i, the i-th diagonal element of the
    !>                triangular factor of A is exactly zero, so that A does not have
    !>                full rank; the least squares solution could not be
    !>                computed.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 191 of file sgels.f.

Author

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