Matrix method and the suppression of Runge's phenomenon

TL;DR

This paper explores how the matrix method can be adapted to suppress Runge's phenomenon in polynomial interpolation, especially in the context of black hole quasinormal modes, by choosing boundary conditions and domain schemes.

Contribution

It introduces boundary condition strategies and domain schemes within the matrix method to effectively suppress Runge's phenomenon in polynomial interpolation for black hole physics.

Findings

Appropriate boundary conditions suppress oscillations in polynomial interpolation.

Delimited expansion domains improve results for discontinuous potentials.

The methods relate closely to Chebyshev grid principles.

Abstract

Higher-degree polynomial interpolations carried out on uniformly distributed nodes are often plagued by {\it overfitting}, known as Runge's phenomenon. This work investigates Runge's phenomenon and its suppression in various versions of the matrix method for black hole quasinormal modes. It is shown that an appropriate choice of boundary conditions gives rise to desirable suppression of oscillations associated with the increasing Lebesgue constant. For the case of discontinuous effective potentials, where the application of the above boundary condition is not feasible, the recently proposed scheme with delimited expansion domain also leads to satisfactory results. The onset of Runge's phenomenon and its effective suppression are demonstrated by evaluating the relevant waveforms. Furthermore, we argue that both scenarios are either closely related to or practical imitations of the Chebyshev grid. The implications of the present study are also addressed.