ZGGRQF(3F) ZGGRQF(3F)
ZGGRQF - compute a generalized RQ factorization of an M-by-N matrix A and
a P-by-N matrix B
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(
* )
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 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.
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.
Page 1
ZGGRQF(3F) ZGGRQF(3F)
N (input) INTEGER
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 (MN)-th
subdiagonal contain the M-by-N upper trapezoidal 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 Mby-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.
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
Page 2
ZGGRQF(3F) ZGGRQF(3F)
H(i) = I - taua * v * v'
where taua is a complex scalar, and v is a complex vector with v(nk+i+1:n)
= 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in A(mk+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.
ZGGRQF(3F) ZGGRQF(3F)
ZGGRQF - compute a generalized RQ factorization of an M-by-N matrix A and
a P-by-N matrix B
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(
* )
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 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.
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.
Page 1
ZGGRQF(3F) ZGGRQF(3F)
N (input) INTEGER
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 (MN)-th
subdiagonal contain the M-by-N upper trapezoidal 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 Mby-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.
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
Page 2
ZGGRQF(3F) ZGGRQF(3F)
H(i) = I - taua * v * v'
where taua is a complex scalar, and v is a complex vector with v(nk+i+1:n)
= 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in A(mk+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.
PPPPaaaaggggeeee 3333 [ Back ]
|
