As for the nonsymmetric eigenvalue problem we provide both asymptotic and global error bounds. If the problem is ill-conditioned, the asymptotic bounds may only hold for extremely small values of . Therefore, we also provide similar global error bounds, which are valid for all perturbations that satisfy an upper bound on . The global error bounds in Table 4.8 are guaranteed to hold for all , where for an individual eigenvector pair, and for a cluster of eigenvalues or a deflating subspace pair. We let in Table 4.8 .
Global error bounds for the generalized nonsymmetric eigenvalue problem assuming .
For ill-conditioned problems the restriction on may also be small. See also the discussion of singular problems in section 4.11.1.4 . As for the nonsymmetric eigenvalue problem we have global error bounds for eigenvalues which are true for all E and F . We let the columns of and be the computed left and right eigenvectors associated with the computed generalized eigenvalue pairs . Moreover, we assume that and are normalized such that and , i.e., we overwrite with , with and with .