dggrqf(3)

NAME

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 (integerM, integerP, integerN, double precision, dimension( lda, * )A, integerLDA, double precision, dimension( * )TAUA, double precision, dimension( ldb, * )B, integerLDB, double precision, dimension( * )TAUB, double precision, dimension( * )WORK, integerLWORK, integerINFO)

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

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

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

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 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.

Date:

: November 2011

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 214 of file dggrqf.f.

Author

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