CGGRQF(1) LAPACK routine (version 3.2) CGGRQF(1)NAME
CGGRQF - computes a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B
SYNOPSIS
SUBROUTINE CGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK,
INFO )
INTEGER INFO, LDA, LDB, LWORK, M, N, P
COMPLEX A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
WORK( * )
PURPOSE
CGGRQF 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 unitary matrix, Z is a P-by-P unitary 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
conjugate 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) COMPLEX 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, repre‐
sent the unitary 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) COMPLEX array, dimension (min(M,N))
The scalar factors of the elementary reflectors which represent
the unitary matrix Q (see Further Details). B
(input/output) COMPLEX array, dimension (LDB,N) On entry, the
P-by-N matrix B. On exit, the elements on and above the diago‐
nal 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 unitary 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) COMPLEX array, dimension (min(P,N))
The scalar factors of the elementary reflectors which represent
the unitary matrix Z (see Further Details). WORK
(workspace/output) COMPLEX array, dimension (MAX(1,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 factoriza‐
tion of a P-by-N matrix, and NB3 is the optimal blocksize for a
call of CUNMRQ. 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 INFO=-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 complex scalar, and v is a complex 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 CUNGRQ.
To use Q to update another matrix, use LAPACK subroutine CUNMRQ. 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 complex scalar, and v is a complex 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 CUNGQR.
To use Z to update another matrix, use LAPACK subroutine CUNMQR.
LAPACK routine (version 3.2) November 2008 CGGRQF(1)
