The Mathematical Institute, University of Oxford, Eprints Archive

Numerical algorithms based on analytic function values at roots of unity

Austin, Anthony P. and Kravanja, Peter and Trefethen, Lloyd N. (2013) Numerical algorithms based on analytic function values at roots of unity. Technical Report. SINUM. (Submitted)

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Abstract

Let $f(z)$ be an analytic or meromorphic function in the closed unit disk sampled at the $n$th 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 the possibilities and emphasize a basic 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.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1733
Deposited By: Lotti Ekert
Deposited On:01 Aug 2013 09:07
Last Modified:29 May 2015 19:25

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