TABLE OF CONTENTS The Arnoldi iteration is two things: the basis of many of the iterative algorithms of numerical linear algebra and, more specifically, a technique for finding eigenvalues of nonhermitian matrices. Here we consider this second, specific role, and in the process, we describe a connection with polynomial approximation theory that is of broad importance.
The use of the Arnoldi iteration for computing eigenvalues proceeds as follows. The iteration is carried out as described in the last lecture (Algorithm 33.1). At each step n, or at occasional steps, the eigenvalues of the Hessenberg matrix H n are computed by standard methods such as the QR algorithm. (In practice this means a call to software such as provided by EISPACK, LAPACK, or MATLAB.) These are the "Arnoldi estimates" or "Ritz values" (33.10). Some of these numbers are typically observed to converge rapidly, often geometrically (i.e., linearly; see Exercise 25.2), and when they do, one may assume with reasonable confidence that the converged values are eigenvalues of A.
Since n ? m for a feasible computation, one cannot of course expect to compute all of the eigenvalues of A by this process. Which eigenvalues, then, does the Arnoldi iteration find? Typically, it finds extreme eigenvalues, that is, eigenvalues near the edge of the spectrum of A. Fortunately, these are precisely the eigenvalues of main interest in most applications.
For example, in problems of hydrodynamic stability, the aim is to determine whether...
