Chebyshev Subdivision and Reduction Methods for Solving Multivariable Systems of Equations

TL;DR

This paper introduces a new Chebyshev-based algorithm for efficiently finding all isolated real zeros of multivariable systems within a bounded domain, demonstrating superior speed and accuracy through theoretical analysis and numerical experiments.

Contribution

The paper presents a novel Chebyshev subdivision and reduction algorithm with proven quadratic convergence and improved performance over existing methods for solving multivariable systems.

Findings

The algorithm achieves R-quadratic convergence near simple zeros.

Numerical experiments show the method is faster and more accurate than standard approaches.

The implementation is publicly available on GitHub.

Abstract

We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in $\mathbb{R}^n$. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has R-quadratic convergence locally near simple zeros of the system. We also analyze the temporal complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to three that the method is both fast and accurate on a wide range of problems. The algorithm should also work well in higher dimensions. Our tests show that the algorithm outperforms other standard methods on this problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available on GitHub.