On entry: the first dimension of the array q as declared in the (sub)program from which f08qff is called.
if , ;
if , .
7: – IntegerInput/Output
8: – IntegerInput/Output
On entry: ifst and ilst must specify the reordering of the diagonal elements or blocks of . The element or block with row index ifst is moved to row ilst by a sequence of exchanges between adjacent elements or blocks.
On exit: if ifst pointed to the second row of a block on entry, it is changed to point to the first row. ilst always points to the first row of the block in its final position (which may differ from its input value by ).
9: – Real (Kind=nag_wp) arrayWorkspace
10: – 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.
Two adjacent diagonal elements or blocks could not be successfully exchanged. This error can only occur if the exchange involves at least one block; it implies that the problem is very ill-conditioned, and that the eigenvalues of the two blocks are very close. On exit, may have been partially reordered, and ilst points to the first row of the current position of the block being moved; (if requested) is updated consistently with .
The computed matrix is exactly similar to a matrix , where
and is the machine precision.
Note that if a diagonal block is involved in the reordering, its off-diagonal elements are in general changed; the diagonal elements and the eigenvalues of the block are unchanged unless the block is sufficiently ill-conditioned, in which case they may be noticeably altered. It is possible for a block to break into two blocks, i.e., for a pair of complex eigenvalues to become purely real. The values of real eigenvalues however are never changed by the reordering.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f08qff 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 , and if , where .
The input matrix must be in canonical Schur form, as is the output matrix . This has the following structure.
If all the computed eigenvalues are real, is upper triangular and its diagonal elements are the eigenvalues.
If some of the computed eigenvalues form complex conjugate pairs, then has diagonal blocks. Each diagonal block has the form