ZGGRQF - compute a generalized RQ factorization of an M-
by-N matrix A and a P-by-N matrix B
SYNOPSIS
SUBROUTINE ZGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB,
WORK, LWORK, INFO )
INTEGER INFO, LDA, LDB, LWORK, M, N, P
COMPLEX*16 A( LDA, * ), B( LDB, * ), TAUA( * ),
TAUB( * ), WORK( * )
PURPOSE
ZGGRQF 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 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 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.
The number of columns of the matrices A and B. N
>= 0.
A (input/output) COMPLEX*16 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 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*16 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*16
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 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*16 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*16
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 ZUNMRQ.
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.
= 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 ZUNGRQ.
To use Q to update another matrix, use LAPACK subroutine
ZUNMRQ.
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 ZUNGQR.
To use Z to update another matrix, use LAPACK subroutine
ZUNMQR.
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