SRC/sggglm.f(3)

NAME
SRC/sggglm.f

SYNOPSIS

  Functions/Subroutines
subroutine sggglm (n, m, p, a, lda, b, ldb, d, x, y, work, lwork, info)
SGGGLM

Function/Subroutine Documentation

  subroutine sggglm (integer n, integer m, integer p, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) d, real, dimension( * ) x, real, dimension( * ) y, real, dimension( * ) work, integer lwork, integer info)

SGGGLM

Purpose:

    !>
    !> SGGGLM solves a general Gauss-Markov linear model (GLM) problem:
    !>
    !>         minimize || y ||_2   subject to   d = A*x + B*y
    !>             x
    !>
    !> where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
    !> given N-vector. It is assumed that M <= N <= M+P, and
    !>
    !>            rank(A) = M    and    rank( A B ) = N.
    !>
    !> Under these assumptions, the constrained equation is always
    !> consistent, and there is a unique solution x and a minimal 2-norm
    !> solution y, which is obtained using a generalized QR factorization
    !> of the matrices (A, B) given by
    !>
    !>    A = Q*(R),   B = Q*T*Z.
    !>          (0)
    !>
    !> In particular, if matrix B is square nonsingular, then the problem
    !> GLM is equivalent to the following weighted linear least squares
    !> problem
    !>
    !>              minimize || inv(B)*(d-A*x) ||_2
    !>                  x
    !>
    !> where inv(B) denotes the inverse of B.
    !>
    !> Callers of this subroutine should note that the singularity/rank-deficiency checks
    !> implemented in this subroutine are rudimentary. The STRTRS subroutine called by this
    !> subroutine only signals a failure due to singularity if the problem is exactly singular.
    !>
    !> It is conceivable for one (or more) of the factors involved in the generalized QR
    !> factorization of the pair (A, B) to be subnormally close to singularity without this
    !> subroutine signalling an error. The solutions computed for such almost-rank-deficient
    !> problems may be less accurate due to a loss of numerical precision.
    !>
    !>

Parameters
    N

    !>          N is INTEGER
    !>          The number of rows of the matrices A and B.  N >= 0.
    !>

    M

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

    P

    !>          P is INTEGER
    !>          The number of columns of the matrix B.  P >= N-M.
    !>

    A

    !>          A is REAL array, dimension (LDA,M)
    !>          On entry, the N-by-M matrix A.
    !>          On exit, the upper triangular part of the array A contains
    !>          the M-by-M upper triangular matrix R.
    !>

    LDA

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

    B

    !>          B is REAL array, dimension (LDB,P)
    !>          On entry, the N-by-P matrix B.
    !>          On exit, if N <= P, the upper triangle of the subarray
    !>          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
    !>          if N > P, the elements on and above the (N-P)th subdiagonal
    !>          contain the N-by-P upper trapezoidal matrix T.
    !>

    LDB

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

    D

    !>          D is REAL array, dimension (N)
    !>          On entry, D is the left hand side of the GLM equation.
    !>          On exit, D is destroyed.
    !>

    X

    !>          X is REAL array, dimension (M)
    !>

    Y

    !>          Y is REAL array, dimension (P)
    !>
    !>          On exit, X and Y are the solutions of the GLM problem.
    !>

    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,N+M+P).
    !>          For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB,
    !>          where NB is an upper bound for the optimal blocksizes for
    !>          SGEQRF, SGERQF, SORMQR and SORMRQ.
    !>
    !>          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.
    !>          = 1:  the upper triangular factor R associated with A in the
    !>                generalized QR factorization of the pair (A, B) is exactly
    !>                singular, so that rank(A) < M; the least squares
    !>                solution could not be computed.
    !>          = 2:  the bottom (N-M) by (N-M) part of the upper trapezoidal
    !>                factor T associated with B in the generalized QR
    !>                factorization of the pair (A, B) is exactly singular, so that
    !>                rank( A B ) < N; 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 193 of file sggglm.f.

Author

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