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# Stability and Accuracy Assessments Z. Bai and R. Li

In this section, we discuss the tools to assess the accuracy of computed eigenvalues and corresponding eigenvectors of the GNHEP of a regular matrix pair . We only assume the availability of residual vectors which are usually available upon the exit of a successful computation or cost marginal to compute afterwards. For the treatment of error estimation for the computed eigenvalues, eigenvectors, and deflating subspaces of dense GNHEPs, see Chapter 4 of the LAPACK Users' Guide [12].

The situation for general regular pairs is more complicated than the standard NHEP discussed in §7.13 (p. ), especially when is singular, in which case the characteristic polynomial no longer has degree , the dimension of the matrices and . Even when is mathematically nonsingular but nearly singular, problems arise when one tries to convert it to a standard eigenvalue problem for , which then could have huge eigenvalues and consequently cause numerical instability. To account for all possibilities, a homogeneous representation of an eigenvalue by a nonzero pair of numbers has been proposed:

When , such pairs represent infinite eigenvalues, and this occurs when is singular. Such representations are clearly not unique since represents the same ratio for any , and consequently the same eigenvalue. So really a pair is a representative from a class of pairs that give the same ratio.

With this new representation of an eigenvalue, the characteristic polynomial takes the form , which does have total degree of in and . (In fact the th term in its expansion is a multiple of .)

But how do we measure the difference of two eigenvalues, given the fact of non-uniqueness in their representations? We resort to the chordal metric for and ; their distance in chordal metric is defined as

We are now ready to address the issue of assessing the accuracy of computed approximations.

Subsections

Next: Residual Vectors. Up: Generalized Non-Hermitian Eigenvalue Problems Previous: Notes and References   Contents   Index
Susan Blackford 2000-11-20