An implementation of the matrix method using Chebyshev grid

TL;DR

This paper investigates adapting the matrix method for black hole quasinormal modes to Chebyshev grids, analyzing its precision, computational cost, and robustness compared to the original method.

Contribution

It introduces a Chebyshev grid implementation of the matrix method and evaluates its effectiveness and efficiency relative to the traditional approach.

Findings

Chebyshev grid implementation suppresses Runge's phenomenon

Increased precision does not always justify higher computational cost

Original matrix method remains robust for moderate grid sizes

Abstract

In this work, we explore the properties of the matrix method for black hole quasinormal modes on the nonuniform grid. In particular, the method is implemented to be adapted to the Chebyshev grid, aimed at effectively suppressing Runge's phenomenon. It is found that while such an implementation is favorable from a mathematical point of view, in practice, the increase in precision does not necessarily compensate for the penalty in computational time. On the other hand, the original matrix method, though subject to Runge's phenomenon, is shown to be reasonably robust and suffices for most applications with a moderate grid number. In terms of computational time and obtained significant figures, we carried out an analysis regarding the trade-off between the two aspects. The implications of the present study are also addressed.