Approximate Diagonalization

We describe a new method of computing functions of highly non-normal matrices, by using the concept of approximate diagonalization. We formulate a conjecture about its efficiency, and provide both theoretical and numerical evidence in support of the conjecture. We apply the method to compute arbitrary real powers of highly non-normal matrices. 1. Introduction. Let A be a non-normal n × n matrix and suppose that one wants to evaluate x = f (A)b or solve f (A)x = b for a large number of different analytic functions f rapidly, without caring too much about high accuracy. If A is diagonalizable, i.e. A := SDS −1 where D is diagonal, then one can solve the first problem by writing x := Sf (D)S −1 b, where f (D) is evaluated by applying the function f to the diagonal entries of D, which coincide with the eigenvalues of A. The second problem may be solved in a similar manner. This procedure may not be appropriate if A is highly non-normal, because the eigenvalues of A can be highly unstable under small perturbations, such as those associated with rounding errors in computation, and the matrix S may have an extremely large condition number κ(S) := S S −1. In the most extreme case, when A has a non-trivial Jordan form, the method breaks down entirely. In this paper we describe an approach which involves using an approximate di-agonalization of A. We emphasize that this does not mean that it is close to a true diagonalization, but rather that it has many of the features of a true diagonaliza-tion, and the amount of error associated with using it can be estimated. We start by describing the idea, and formulate a conjecture about its efficiency. Much of the remainder of the paper is devoted to providing theoretical and numerical evidence in support of the conjecture. In Section 5 we use the ideas developed to throw some light on the difficulties of computing fractional powers of matrices that are close to singular.