The routine may be called by the names f08nff, nagf_lapackeig_dorghr or its LAPACK name dorghr.
f08nff is intended to be used following a call to f08nef, which reduces a real general matrix to upper Hessenberg form by an orthogonal similarity transformation: . f08nef represents the matrix as a product of elementary reflectors. Here and are values determined by f08nhf when balancing the matrix; if the matrix has not been balanced, and .
This routine 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: – IntegerInput
On entry: , the order of the matrix .
2: – IntegerInput
3: – IntegerInput
On entry: these must be the same arguments ilo and ihi, respectively, as supplied to f08nef.
if , ;
if , and .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: details of the vectors which define the elementary reflectors, as returned by f08nef.
On exit: the orthogonal matrix .
5: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08nff is called.
6: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array tau
must be at least
On entry: further details of the elementary reflectors, as returned by f08nef.
7: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
8: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08nff is called, unless , in which case a workspace query is assumed and the routine only calculates the optimal dimension of work (using the formula given below).
for optimal performance lwork should be at least , where is the block size.
9: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The computed matrix differs from an exactly orthogonal matrix by a matrix such that
where is the machine precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08nff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nff 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 routine. 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 .