Let where and Then

We shall develop a crude but effective assessment of accuracy based upon this
estimate. Far more sophisticated analysis is available for the
symmetric problem in [36] and in [40] for
the non-symmetric case.
It is easily shown that

In the Hermitian case a small residual implies an accurate answer. However, in the non-Hermitian case, a small does not necessarily imply that the Ritz pair is an accurate approximation to an eigenpair of The following theorem indicates what accuracy might be expected of a Ritz value as an approximation to an eigenvalue of

**Theorem 14349**

Suppose that is a simple eigenvalue of nearest the eigenvalue of Denote the left and right eigenvectors for by and , respectively, each of unit length. Then

*Proof: *
See Wilkinson [48, p. 68].

The number is
the cosine of the angle between and and it
determines the conditioning of . Thus, if and are
nearly orthogonal, the eigenvalue is highly sensitive to
perturbations in but if they
are nearly parallel
then is insensitive to perturbations.
For Hermitian matrices so that and will be an excellent approximation to .However, if the left and right eigenvectors are nearly orthogonal, then even
if , where is machine
precision, may contain few digits, if any, of accuracy. Roughly, as
a rule of thumb, if
and then the leading *t*-*d* decimal
digits of will agree with those of .

The question of how close the Ritz vector is to is complicated by the fact that an eigenvector is not a unique quantity. Any scaling of an eigenvector by a nonzero complex number is also an eigenvector. For this reason, it is better to estimate the positive angle between an eigenvector and its approximation.

**Theorem 14386**

Suppose that is a Schur form for and let be a simple eigenvalue of nearest the eigenvalue of with corresponding eigenvectors and respectively. If is the positive angle between and then

*Proof: *
See 5 in [5].

The definition of the quantity *sep* in this theorem is

This is a more refined indicator of eigenvector sensitivity
that
accounts for non-normality as well as clustering of eigenvalues.
Varah [46] shows that

The conclusion we must draw is that the eigenvalues of a non-symmetric matrix may be very sensitive to perturbations such as those introduced by round-off error. This sensitivity is intricately tied to the departure from normality of the given matrix. The classic example of a matrix with an extremely ill-conditioned eigensystem is where is bi-diagonal with the number on the diagonal and all ones on the super-diagonal. The eigenvalues of this perturbed Jordan matrix satisfy . This is quite contrary to the behavior of eigenvalues of normal matrices. If the matrix is near to a matrix with such an ill-conditioned eigensystem, then we can say little about the accuracy of the computed eigenvalues and eigenvectors. The best we can say is that we have computed the exact eigenvalues and eigenvectors of a nearby matrix (or matrix pencil).

We must be content with a stopping criterion that assures small backward error. This strategy is used in ARPACK, where the Ritz pair is considered converged when

is satisfied where is machine precision. d. Since , this implies that ( 4.6.2) is satisfied with and this test is invariant to scaling of through multiplication by a nonzero scalar.The backward error is defined as the smallest, in norm, perturbation such that the Ritz pair is an eigenpair for .The recent study [25] presents a thorough discussion of the many issues involved in determining stopping criteria for the non-symmetric eigenvalue problem. In ARPACK we are more stringent than just asking for a small backward error relative to . We instead ask for small backward error relative to the projected matrix .