SRC/sggqrf.f(3)

NAME
SRC/sggqrf.f

SYNOPSIS

  Functions/Subroutines
subroutine sggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)
SGGQRF

Function/Subroutine Documentation

  subroutine sggqrf (integer n, integer m, integer p, real, dimension( lda, * ) a, integer lda, real, dimension( * ) taua, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) taub, real, dimension( * ) work, integer lwork, integer info)

SGGQRF

Purpose:

    !>
    !> SGGQRF computes a generalized QR factorization of an N-by-M matrix A
    !> and an N-by-P matrix B:
    !>
    !>             A = Q*R,        B = Q*T*Z,
    !>
    !> where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
    !> matrix, and R and T assume one of the forms:
    !>
    !> if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
    !>                 (  0  ) N-M                         N   M-N
    !>                    M
    !>
    !> where R11 is upper triangular, and
    !>
    !> if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
    !>                  P-N  N                           ( T21 ) P
    !>                                                      P
    !>
    !> where T12 or T21 is upper triangular.
    !>
    !> In particular, if B is square and nonsingular, the GQR factorization
    !> of A and B implicitly gives the QR factorization of inv(B)*A:
    !>
    !>              inv(B)*A = Z**T*(inv(T)*R)
    !>
    !> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
    !> transpose of the matrix Z.
    !>

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.  M >= 0.
    !>

    P

    !>          P is INTEGER
    !>          The number of columns of the matrix B.  P >= 0.
    !>

    A

    !>          A is REAL array, dimension (LDA,M)
    !>          On entry, the N-by-M matrix A.
    !>          On exit, the elements on and above the diagonal of the array
    !>          contain the min(N,M)-by-M upper trapezoidal matrix R (R is
    !>          upper triangular if N >= M); the elements below the diagonal,
    !>          with the array TAUA, represent the orthogonal matrix Q as a
    !>          product of min(N,M) elementary reflectors (see Further
    !>          Details).
    !>

    LDA

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

    TAUA

    !>          TAUA is REAL array, dimension (min(N,M))
    !>          The scalar factors of the elementary reflectors which
    !>          represent the orthogonal matrix Q (see Further Details).
    !>

    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; the remaining
    !>          elements, with the array TAUB, represent the orthogonal
    !>          matrix Z as a product of elementary reflectors (see Further
    !>          Details).
    !>

    LDB

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

    TAUB

    !>          TAUB is REAL array, dimension (min(N,P))
    !>          The scalar factors of the elementary reflectors which
    !>          represent the orthogonal matrix Z (see Further Details).
    !>

    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 >= max(N,M,P)*max(NB1,NB2,NB3),
    !>          where NB1 is the optimal blocksize for the QR factorization
    !>          of an N-by-M matrix, NB2 is the optimal blocksize for the
    !>          RQ factorization of an N-by-P matrix, and NB3 is the optimal
    !>          blocksize for a call of SORMQR.
    !>
    !>          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.

Further Details:

    !>
    !>  The matrix Q is represented as a product of elementary reflectors
    !>
    !>     Q = H(1) H(2) . . . H(k), where k = min(n,m).
    !>
    !>  Each H(i) has the form
    !>
    !>     H(i) = I - taua * v * v**T
    !>
    !>  where taua is a real scalar, and v is a real vector with
    !>  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
    !>  and taua in TAUA(i).
    !>  To form Q explicitly, use LAPACK subroutine SORGQR.
    !>  To use Q to update another matrix, use LAPACK subroutine SORMQR.
    !>
    !>  The matrix Z is represented as a product of elementary reflectors
    !>
    !>     Z = H(1) H(2) . . . H(k), where k = min(n,p).
    !>
    !>  Each H(i) has the form
    !>
    !>     H(i) = I - taub * v * v**T
    !>
    !>  where taub is a real scalar, and v is a real vector with
    !>  v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
    !>  B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
    !>  To form Z explicitly, use LAPACK subroutine SORGRQ.
    !>  To use Z to update another matrix, use LAPACK subroutine SORMRQ.
    !>

Definition at line 213 of file sggqrf.f.

Author

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