Title: Numerical Algorithms Based on Analytic Function Values at Roots of Unity
Authors: Austin, Anthony P. ×
Kravanja, Peter
Trefethen, Lloyd N. #
Issue Date: 2014
Publisher: Society for Industrial and Applied Mathematics
Series Title: SIAM Journal on Numerical Analysis vol:52 issue:4 pages:1795-1821
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 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.
ISSN: 0036-1429
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Numerical Analysis and Applied Mathematics Section
× corresponding author
# (joint) last author

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