TL;DR
This paper reviews the theory of optimal polynomial and rational Chebyshev approximations, focusing on Zolotarev's formula for the sign function, and discusses efficient application of rational approximations to large sparse matrices using advanced numerical methods.
Contribution
It provides a comprehensive review of approximation theory for matrices and explains how to efficiently implement rational approximations with modern computational techniques.
Findings
Detailed explanation of Chebyshev and Zolotarev approximations
Application of rational approximations to large sparse matrices
Use of multi-shift Krylov solvers for efficiency
Abstract
We review the theory of optimal polynomial and rational Chebyshev approximations, and Zolotarev's formula for the sign function over the range (\epsilon \leq |z| \leq1). We explain how rational approximations can be applied to large sparse matrices efficiently by making use of partial fraction expansions and multi-shift Krylov space solvers.
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