On entry: the first dimension of the array q as declared in the (sub)program from which f08hef is called.
if or , ;
if , .
11: – Real (Kind=nag_wp) arrayWorkspace
12: – 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 tridiagonal matrix is exactly similar to a nearby matrix , where
is a modestly increasing function of , and is the machine precision.
The elements of themselves may be sensitive to small perturbations in or to rounding errors in the computation, but this does not affect the stability of the eigenvalues and eigenvectors.
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.
f08hef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08hef 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 if with additional operations if .
This example computes all the eigenvalues and eigenvectors of the matrix , where
Here is symmetric and is treated as a band matrix. The program first calls f08hef to reduce to tridiagonal form , and to form the orthogonal matrix ; the results are then passed to f08jef which computes the eigenvalues and eigenvectors of .