dggrqf(3P)

NAME

dggrqf - compute a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B

SYNOPSIS

SUBROUTINE DGGRQF(
M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO )

void dggrqf(long int m, long int p, long int n, double ∗da,
long int lda, double ∗taua, double ∗db, long int ldb, double ∗taub, long int ∗info)

INTEGER INFO, LDA, LDB, LWORK, M, N, P

DOUBLE PRECISION A( LDA, ∗ ), B( LDB, ∗ ), TAUA( ∗ ), TAUB( ∗ ), WORK( ∗ )

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’

where inv(B) denotes the inverse of the matrix B, and Z’ denotes the transpose of the matrix Z.

ARGUMENTS

M (input) INTEGER
The number of rows of the matrix A. M >= 0.

P (input) INTEGER
The number of rows of the matrix B. P >= 0.

N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.

A (input/output) 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 (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).

TAUA (output) DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors which represent the orthogonal matrix Q (see Further Details). B (input/output) 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 (input) INTEGER The leading dimension of the array B. LDB >= max(1,P).

TAUB (output) DOUBLE PRECISION array, dimension (min(P,N))
The scalar factors of the elementary reflectors which represent the orthogonal matrix Z (see Further Details). WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) 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.

INFO (output) INTEGER
= 0: successful exit
< 0: if INF0= -i, the i-th argument had an illegal value.

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’

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’

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.