Chebyshev Expansions for Solutions of Linear Differential Equations

TL;DR

This paper explores Chebyshev expansions for solving linear differential equations, analyzing existing algorithms and proposing a faster method for computing solutions using recurrence relations.

Contribution

It introduces a novel interpretation of recurrence relations as fractions of linear operators, enabling improved algorithmic efficiency for high-order solutions.

Findings

New interpretation of recurrence equations as operator fractions

Complexity analysis of existing algorithms

Development of a faster algorithm for large orders

Abstract

A Chebyshev expansion is a series in the basis of Chebyshev polynomials of the first kind. When such a series solves a linear differential equation, its coefficients satisfy a linear recurrence equation. We interpret this equation as the numerator of a fraction of linear recurrence operators. This interpretation lets us give a simple view of previous algorithms, analyze their complexity, and design a faster one for large orders.