The function may be called by the names: f08nfc, nag_lapackeig_dorghr or nag_dorghr.
f08nfc is intended to be used following a call to f08nec, which reduces a real general matrix to upper Hessenberg form by an orthogonal similarity transformation: . f08nec represents the matrix as a product of elementary reflectors. Here and are values determined by f08nhc when balancing the matrix; if the matrix has not been balanced, and .
This function may be used to generate explicitly as a square matrix. has the structure:
where occupies rows and columns to .
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 order of the matrix .
3: – IntegerInput
4: – IntegerInput
On entry: these must be the same arguments ilo and ihi, respectively, as supplied to f08nec.
if , ;
if , and .
5: – doubleInput/Output
Note: the dimension, dim, of the array a
must be at least
On entry: details of the vectors which define the elementary reflectors, as returned by f08nec.
On exit: the orthogonal matrix .
6: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
7: – const doubleInput
Note: the dimension, dim, of the array tau
must be at least
On entry: further details of the elementary reflectors, as returned by f08nec.
8: – 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, . Constraint: .
On entry, and .
On entry, , and .
Constraint: if , ;
if , 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 matrix differs from an exactly orthogonal matrix by a matrix such that
where is the machine precision.
8Parallelism and Performance
f08nfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nfc 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 , where .