SRC/dggrqf.f(3)

NAME
SRC/dggrqf.f

SYNOPSIS

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

Function/Subroutine Documentation

  subroutine dggrqf (integer m, integer p, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) taua, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) taub, double precision, dimension( * ) work, integer lwork, integer info)

DGGRQF

Purpose:

    !>
    !> DGGRQF computes a generalized RQ factorization of an M-by-N matrix A
    !> and a P-by-N matrix B:
    !>
    !>             A = R*Q,        B = Z*T*Q,
    !>
    !> 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 M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
    !>                  N-M  M                           ( R21 ) N
    !>                                                      N
    !>
    !> where R12 or R21 is upper triangular, and
    !>
    !> if P >= N,  T = ( T11 ) N  ,   or if P < N,  T = ( T11  T12 ) P,
    !>                 (  0  ) P-N                         P   N-P
    !>                    N
    !>
    !> where T11 is upper triangular.
    !>
    !> In particular, if B is square and nonsingular, the GRQ factorization
    !> of A and B implicitly gives the RQ factorization of A*inv(B):
    !>
    !>              A*inv(B) = (R*inv(T))*Z**T
    !>
    !> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
    !> transpose of the matrix Z.
    !>

Parameters
    M

    !>          M is INTEGER
    !>          The number of rows of the matrix A.  M >= 0.
    !>

    P

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

    N

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

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA,N)
    !>          On entry, the M-by-N matrix A.
    !>          On exit, if M <= N, the upper triangle of the subarray
    !>          A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
    !>          if M > N, the elements on and above the (M-N)-th subdiagonal
    !>          contain the M-by-N upper trapezoidal matrix R; the remaining
    !>          elements, with the array TAUA, represent the orthogonal
    !>          matrix Q as a product of elementary reflectors (see Further
    !>          Details).
    !>

    LDA

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

    TAUA

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

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB,N)
    !>          On entry, the P-by-N matrix B.
    !>          On exit, the elements on and above the diagonal of the array
    !>          contain the min(P,N)-by-N upper trapezoidal matrix T (T is
    !>          upper triangular if P >= N); the elements below the diagonal,
    !>          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,P).
    !>

    TAUB

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

    WORK

    !>          WORK is DOUBLE PRECISION 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 RQ factorization
    !>          of an M-by-N matrix, NB2 is the optimal blocksize for the
    !>          QR factorization of a P-by-N matrix, and NB3 is the optimal
    !>          blocksize for a call of DORMRQ.
    !>
    !>          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 INF0= -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(m,n).
    !>
    !>  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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
    !>  A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
    !>  To form Q explicitly, use LAPACK subroutine DORGRQ.
    !>  To use Q to update another matrix, use LAPACK subroutine DORMRQ.
    !>
    !>  The matrix Z is represented as a product of elementary reflectors
    !>
    !>     Z = H(1) H(2) . . . H(k), where k = min(p,n).
    !>
    !>  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(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
    !>  and taub in TAUB(i).
    !>  To form Z explicitly, use LAPACK subroutine DORGQR.
    !>  To use Z to update another matrix, use LAPACK subroutine DORMQR.
    !>

Definition at line 212 of file dggrqf.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.