The function may be called by the names: f08cvc, nag_lapackeig_zgerqf or nag_zgerqf.
f08cvc forms the factorization of an arbitrary rectangular real matrix. If , the factorization is given by
where is an lower triangular matrix and is an unitary matrix. If the factorization is given by
where is an upper trapezoidal matrix and is again an unitary matrix. In the case where the factorization can be expressed as
where consists of the first rows of and the remaining rows.
The matrix is not formed explicitly, but is represented as a product of elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with in this representation (see Section 9).
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
2: – IntegerInput
On entry: , the number of rows of the matrix .
3: – IntegerInput
On entry: , the number of columns of the matrix .
4: – ComplexInput/Output
Note: the dimension, dim, of the array a
must be at least
where appears in this document, it refers to the array element
On entry: the matrix .
On exit: if , the upper triangle of the subarray contains the upper triangular matrix .
If , the elements on and above the th subdiagonal contain the upper trapezoidal matrix ; the remaining elements, with the array tau, represent the unitary matrix as a product of elementary reflectors (see Section 3.4.6 in the F08 Chapter Introduction).
5: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
if , .
6: – ComplexOutput
Note: the dimension, dim, of the array tau
must be at least
On exit: the scalar factors of the elementary reflectors.
7: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
On entry, .
On entry, . Constraint: .
On entry, and .
On entry, and .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
The computed factorization is the exact factorization of a nearby matrix , where
and is the machine precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08cvc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is approximately if , or if .
To form the unitary matrix f08cvc may be followed by a call to f08cwc
nag_lapackeig_zungrq(order, n, n, minmn, a, pda, tau, &fail)
but note that the first dimension of the array a must be at least n, which may be larger than was required by f08cvc. When , it is often only the first rows of that are required and they may be formed
by the call:
nag_lapackeig_zungrq(order, m, n, m, a, pda, tau, c, pdc, &fail)
To apply to an arbitrary real rectangular matrix , f08cvc may be followed by a call to f08cxc
. For example:
nag_lapackeig_zunmrq(Nag_LeftSide, Nag_ConjTrans, n, p, minmn, a, pda, tau, c, pdc, &fail)