Bounds for individual eigenvalues and eigenvectors are provided by the driver xGGEVX (subsection 2.3.5.2) or the computational routine xTGSNA (subsection 2.4.8). Bounds for cluster of eigenvalues and their associated pair of deflating subspaces are provided by the driver xGGESX (subsection 2.3.5.2) or the computational routine xTGSEN (subsection 2.4.8).
We let be the i^{th} computed eigenvalue pair and the i^{th} exact eigenvalue pair.^{4.2}Let and be the corresponding computed right and left eigenvectors, and x_{i} and y_{i} the exact right and left eigenvectors (so that and ). As in the standard nonsymmetric eigenvalue problem, we also want to bound the error in the average of a cluster of eigenvalues, corresponding to a subset of the integers from 1 to n. However, since there are both finite and infinite eigenvalues, we need a proper definition for the average of the eigenvalues for . Here we let denote the average of the selected eigenvalues^{4.3}: , and similarly for . We also let and denote the exact pair of left and right deflating subspaces associated with the cluster of selected eigenvalues. Similarly, and are the corresponding computed pair of left and right deflating subspaces.
The algorithms for the generalized nonsymmetric eigenproblem are normwise backward stable; the computed eigenvalues, eigenvectors and deflating subspaces are the exact ones of slightly perturbed matrices A + E and B +F, where . The code fragment in the previous subsection approximates by , where , and the values ABNRM and BBNRM returned by xGGEVX are the 1-norm of the matrices A and B, respectively.
xGGEVX (or xTGSNA) returns reciprocal condition numbers for each eigenvalue pair and corresponding left and right eigenvectors and : s_{i} and . s_{i} is a reciprocal condition number for the computed eigenpair , and is referred to as RCONDE(i) by xGGEVX. is a reciprocal condition number for the left and right eigenvectors and , and is referred to as RCONDV(i) by xGGEVX (see subsection 4.11.1.3 for definitions). Similarly, xGGESX (or xTGSEN) returns condition numbers for eigenvalue clusters and deflating subspaces corresponding to a subset of the eigenvalues. These are and , the reciprocal values of the left and right projection norms p and q, and estimates of the separation between two matrix pairs defined by and (see subsection 4.11.1.3 for definitions). xGGESX reports and in RCONDE(1) and RCONDE(2) (PL and PR in xTGSEN)), respectively, and estimates of and in RCONDV(1) and RCONDV(2) (DIF(1) and DIF(2) in xTGSEN), respectively.
As for the nonsymmetric eigenvalue problem we provide both asymptotic and global error bounds. The asymptotic approximate error bounds for eigenvalues, averages of eigenvalues, eigenvectors, and deflating subspaces provided in Table 4.7 are true only for sufficiently small .
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
We let in Table 4.8. If is small, then the computed pair of left and right deflating subspaces (or computed left and right eigenvectors) are small perturbations of the exact pair of deflating subspaces (or the true left and right eigenvectors). The error bounds conform with the corresponding bounds for the nonsymmetric eigenproblem (see subsection 4.8.1.1 ).
For ill-conditioned problems the restriction on may also be small. Indeed, a small value of shows that the cluster of eigenvalues (in the (1,1)-blocks of (A, B)) is ill-conditioned in the sense that small perturbations of (A, B) may imply that one eigenvalue in the cluster moves and coalesces with another eigenvalue (outside the cluster). Accordingly, this also means that the associated (left and right) deflating subspaces are sensitive to small perturbations, since the size of the perturbed subspaces may change for small perturbations of (A, B). 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.
Let (A, B) be a diagonalizable matrix pair. 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
.
Then all eigenvalues
of (A, B) with
lie in the union of n regions
(``spheres'')
Notation Conversion For easy of reference, the following table summarizes the notation used in mathematical expression of the error bounds in tables 4.7 and 4.8 and in the corresponding driver and computational routines.
Mathematical | Driver Routines | Computational Routines | ||
notation | name | parameter | name | parameter |
s_{i} | xGGEVX | RCONDE(i) | xTGSNA | S(i) |
xGGEVX | RCONDV(i) | xTGSNA | DIF(i) | |
xGGESX | RCONDE(1) | xTGSEN | PL | |
xGGESX | RCONDE(2) | xTGSEN | PR | |
xGGESX | RCONDV(1) | xTGSEN | DIF(1) | |
xGGESX | RCONDV(2) | xTGSEN | DIF(2) |
The quantities l_{i}, r_{i}, and used in Table 4.8 (for the global error bounds of the computed eigenvalue pair and the left and right eigenvectors and ) can be obtained by calling xTGSEN with .