DGGRQF(3F)DGGRQF(3F)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
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.
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DGGRQF(3F)DGGRQF(3F)
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
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DGGRQF(3F)DGGRQF(3F)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.
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