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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 )
INTEGER INFO, LDA, LDB, LWORK, M, N, P
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 M-by-M orthogonal matrix, and Z is an N-by-N
orthogonal matrix. R and T assume one of the forms:
if M <= N, or if M > N
R = ( 0 R12 ) M, R = ( R11 ) M-N
N-M M ( R21 ) N
N
where R12 or R21 is upper triangular, and
if P >= N, or if P < N
T = ( T11 ) N T = ( T11 T12 ) P
( 0 ) P-N P N-P
N
where T11 is an upper triangular matrix.
In particular, if B is square and nonsingular, the GRQ fac-
torization of A and B implicitly gives the RQ factorization
of matrix 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 subdiag-
onal 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 (see
Further Details).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B. On exit, the ele-
ments 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 orthog-
onal 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 (see
Further Details).
WORK (workspace) 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(1,N,M,P)*MAX(NB1,NB2,NB3), where NB1 is the
optimal blocksize for the RQ factorization of the
M-by-N matrix A. NB2 is the optimal blocksize for
the QR factorization of the P-by-N matrix B. NB3 is
the optimal blocksize for 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 - taub * v * v'
where taub 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 taub 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 - taua * v * v'
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:M) is stored on exit in
B(i+1:P,i), and taua in TAUB(i).
To form Z explicitly, use LAPACK subroutine DORGQR.
To use Z to update another matrix, use LAPACK subroutine
DORMQR.