Let f (z) be an analytic or meromorphic function in the closed unit disk sampled at the nth roots of unity. Based on these data, how can we approximately evaluate f (z) or f (m) (z) at a point z in the disk? How can we calculate the zeros or poles of f in the disk? These questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations, partial differential equations, and large-scale linear algebra. We analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of Cauchy integrals. We then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension. Finally we highlight the power of rational in comparison with polynomial approximations for some of these problems.
a b s t r a c tIn this paper Beyn's algorithm for solving nonlinear eigenvalue problems is given a new interpretation and a variant is designed in which the required information is extracted via the canonical polyadic decomposition of a Hankel tensor. A numerical example shows that the choice of the filter function is very important, particularly with respect to where it is positioned in the complex plane.
We present a stabilized superfast solver for nonsymmetric Toeplitz systems Tx = b. An explicit formula for T 1 is expressed in such a way that the matrixvector product T 1 b can be calculated via FFTs and Hadamard products. This inversion formula involves certain polynomials that can be computed by solving two linearized rational interpolation problems on the unit circle. The heart of our Toeplitz solver is a superfast algorithm to solve these interpolation problems. This algorithm is stabilized via pivoting, iterative improvement, downdating, and by giving \di cult" interpolation points an adequate treatment. We have implemented our algorithm in Fortran 90. Numerical examples illustrate the e ectiveness of our approach.
In the previous chapter we have presented an accurate algorithm, based on the theory of formal orthogonal polynomials, for computing zeros of analytic functions. More specifically, given an analytic function f and a Jordan curve 9' that does not pass through any zero of f, we have considered the problem of computing all the zeros zl,...,zn of f that lie inside 7, together with their respective multiplicities vl,.. •, ~n. Our principal means of obtaining information about the location of these zeros has been the symmetric bilinear form (','/, cf. Equation (1.12). This form can be evaluated via numerical integration along % This chapter continues the previous chapter. If f has one or several clusters of zeros, then the mapping from the ordinary moments associated with (., -~ to the zeros and their respective multiplicities is very ill-conditioned. We will show that the algorithm that we have presented in Chapter 1 can be used to calculate the centre of a cluster and its size, i.e., the arithmetic mean of the zeros that form a certain cluster and the total number of zeros in this cluster, respectively. This information enables one to zoom into a certain cluster: its zeros can be calculated separately from the other zeros of f. By shifting the origin in the complex plane to the centre of a certain cluster, its zeros become better relatively separated, which is appropriate in floating point arithmetic and reduces the ill-conditioning.In this chapter we will also attack our problem of computing all the zeros of f that lie inside V in an entirely different way, based on rational interpolation at roots of unity. We will show how the new approach complements the previous one and how it can be used effectively in case V is the unit circle. Note 2.0.1. Specifically for clusters of polynomial zeros, let us mention that Hribernig and Stetter [70] worked on detection and validation of clusters of zeros whereas Kirrinnis [84] studied Newton's iteration towards a cluster.2.1 How to obtain the centre of a cluster and its weight Suppose that the zeros of f that lie inside V can be grouped into m clusters. Let I1,..., Im be index sets that define these clusters, and let 2. Clusters of zeros of analytic functions
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