table of contents
/home/abuild/rpmbuild/BUILD/lapack-3.12.0/SRC/zggrqf.f(3) | Library Functions Manual | /home/abuild/rpmbuild/BUILD/lapack-3.12.0/SRC/zggrqf.f(3) |
NAME¶
/home/abuild/rpmbuild/BUILD/lapack-3.12.0/SRC/zggrqf.f
SYNOPSIS¶
Functions/Subroutines¶
subroutine ZGGRQF (m, p, n, a, lda, taua, b, ldb, taub,
work, lwork, info)
ZGGRQF
Function/Subroutine Documentation¶
subroutine ZGGRQF (integer m, integer p, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) taua, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) taub, complex*16, dimension( * ) work, integer lwork, integer info)¶
ZGGRQF
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 factorization of A*inv(B): !> !> A*inv(B) = (R*inv(T))*Z**H !> !> where inv(B) denotes the inverse of the matrix B, and Z**H denotes the !> conjugate transpose of the matrix Z. !>
Parameters
M
!> M is INTEGER !> The number of rows of the matrix A. M >= 0. !>
P
!> P is INTEGER !> The number of rows of the matrix B. P >= 0. !>
N
!> N is INTEGER !> The number of columns of the matrices A and B. N >= 0. !>
A
!> A is 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 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
!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !>
TAUA
!> TAUA is COMPLEX*16 array, dimension (min(M,N)) !> The scalar factors of the elementary reflectors which !> represent the unitary matrix Q (see Further Details). !>
B
!> B is 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
!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,P). !>
TAUB
!> TAUB is COMPLEX*16 array, dimension (min(P,N)) !> The scalar factors of the elementary reflectors which !> represent the unitary matrix Z (see Further Details). !>
WORK
!> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
LWORK
!> LWORK is 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. !>
INFO
!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO=-i, the i-th argument had an illegal value. !>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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**H !> !> 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**H !> !> 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. !>
Definition at line 212 of file zggrqf.f.
Author¶
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