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 .
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
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:
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
distance in chordal metric is defined as