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 factor
ization 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.
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 trape
zoidal 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 Fur
ther Details). B (input/output) DOUBLE PRE
CISION 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 triangu
lar 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 Fur
ther 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.
If LWORK = -1, then a workspace query is assumed;
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 (output) INTEGER
= 0: successful exit
< 0: if INF0= -i, the i-th argument had an ille
gal 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.
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