Stability and Accuracy Assessments

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

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.