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 $\{A,B\}$. 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 $\{A,B\}$ is more complicated than the standard NHEP discussed in §7.13 (p. ), especially when $B$ is singular, in which case the characteristic polynomial $\det (A-\lambda B)$ no longer has degree $n$, the dimension of the matrices $A$ and $B$. Even when $B$ is mathematically nonsingular but nearly singular, problems arise when one tries to convert it to a standard eigenvalue problem for $B^{-1}A$, which then could have huge eigenvalues and consequently cause numerical instability. To account for all possibilities, a homogeneous representation of an eigenvalue $\lambda$ by a nonzero pair of numbers $(\alpha,\beta)$ has been proposed:

$$\lambda\equiv\alpha/\beta, \quad\quad \vert\alpha\vert^2+\vert\beta\vert^2>0.$$

When $\beta = 0$, such pairs represent infinite eigenvalues, and this occurs when $B$ is singular. Such representations are clearly not unique since $(\xi\alpha,\xi\beta)$ represents the same ratio for any $\xi\ne 0$, and consequently the same eigenvalue. So really a pair $(\alpha,\beta)$ 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 $\det(\beta A-\alpha B)$, which does have total degree of $n$ in $\alpha$ and $\beta$. (In fact the $i$th term in its expansion is a multiple of $\alpha^i\beta^{n-i}$.)

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 $\lambda = \alpha/\beta$ and $\wtd\lambda=\wtd\alpha/\wtd\beta$; their distance in chordal metric is defined as

$$\chi(\lambda,\wtd\lambda) = \chi((\alpha, \beta),(\wtd\alpha... ...1 + \vert\lambda\vert^2} \sqrt{1 + \vert\wtd\lambda\vert^2}}.$$

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