
DGGRQF(l) ) DGGRQF(l)
NAME
DGGRQF  compute a generalized RQ factorization of an MbyN matrix A and a PbyN 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 MbyN matrix A and a PbyN matrix
B:
A = R*Q, B = Z*T*Q,
where Q is an NbyN orthogonal matrix, Z is a PbyP orthogonal matrix, and R and T
assume one of the forms:
if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) MN,
NM 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 ) PN P NP
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 MbyN matrix A. On exit, if M <= N, the upper triangle of the sub
array A(1:M,NM+1:N) contains the MbyM upper triangular matrix R; if M > N, the
elements on and above the (MN)th subdiagonal contain the MbyN 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 Further Details). B (input/output) DOUBLE PRECISION array,
dimension (LDB,N) On entry, the PbyN matrix B. On exit, the elements on and
above the diagonal of the array contain the min(P,N)byN 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 reflec
tors (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 MbyN matrix, NB2 is the optimal blocksize for the QR fac
torization of a PbyN 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 (output) INTEGER
= 0: successful exit
< 0: if INF0= i, the ith 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(nk+i+1:n) = 0 and v(nk+i) = 1; v(1:nk+i1) is stored on exit in A(mk+i,1:nk+i1),
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:i1) = 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.
LAPACK version 3.0 15 June 2000 DGGRQF(l) 
