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Stable polefinding and rational least-squares fitting via eigenvalues.

Shinji Ito1,2, Yuji Nakatsukasa3

  • 11Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656 Japan.

Numerische Mathematik
|July 6, 2018
PubMed
Summary
This summary is machine-generated.

This study introduces a novel, one-step algorithm for finding poles of meromorphic functions. It offers guaranteed numerical stability and automatically determines the rational approximant type, outperforming existing methods.

Keywords:
65D05 Numerical analysis, Interpolation65D15 Numerical analysis, Algorithms for functional approximation

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Area of Science:

  • Numerical Analysis
  • Complex Analysis
  • Scientific Computing

Background:

  • Evaluating meromorphic functions at sample points is common when explicit forms are unknown.
  • Traditional pole-finding involves two steps: rational interpolation and root finding, often lacking numerical stability analysis.
  • Existing algorithms require users to specify rational interpolant types and have unexplored numerical stability.

Purpose of the Study:

  • To develop a numerically stable, one-step algorithm for finding poles of meromorphic functions.
  • To create an algorithm that automatically determines the appropriate rational approximant type.
  • To provide guaranteed numerical stability in rational interpolation and approximation.

Main Methods:

  • A generalized eigenvalue problem is used to find poles directly from sampled function values.
  • Implicit polynomial basis change via QR factorization ensures numerical stability.
  • Oversampling combined with least-squares fitting is employed for robust approximation.

Main Results:

  • The algorithm automatically identifies the rational approximant type, simplifying user input.
  • Poles are found in a single step using a generalized eigenvalue problem.
  • Guaranteed numerical stability is achieved, with small errors at sample points, a first for rational interpolation.

Conclusions:

  • The new algorithm offers significant improvements in accuracy and stability over existing methods.
  • It provides a robust and user-friendly approach to pole-finding for meromorphic functions.
  • This work establishes a benchmark for numerically stable rational approximation algorithms.