Resolution of singularities by rational functions. (English) Zbl 1535.65049
Summary: Results on the rational approximation of functions containing singularities are presented. We build further on the “lightning method,” recently proposed by
A. Gopal and L. N. Trefethen [SIAM J. Numer. Anal. 57, No. 5, 2074–2094 (2019; Zbl 1431.65223)],
based on exponentially clustering poles close to the singularities. Our results are obtained by augmenting the lightning approximation set with either a low-degree polynomial basis or partial fractions with poles clustering toward infinity in order to obtain a robust approximation of the smooth behavior of the function. This leads to a significant increase in the achievable accuracy as well as the convergence rate of the numerical scheme. For the approximation of \(x^\alpha\) on \([0,1]\), the optimal convergence rate as shown by
H. Stahl [Bull. Am. Math. Soc., New Ser. 28, No. 1, 116–122 (1993; Zbl 0768.41015)]
is now achieved simply by least-squares fitting.
MSC:
| 65F20 | Numerical solutions to overdetermined systems, pseudoinverses |
| 41A20 | Approximation by rational functions |
| 65E05 | General theory of numerical methods in complex analysis (potential theory, etc.) |
| 65D05 | Numerical interpolation |
Keywords:
rational functions; approximation theory; complex analysis; least-squares; ill-conditioningSoftware:
Lightning Laplace SolverReferences:
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