Self-accelerating root search and optimisation methods based on rational interpolation

TL;DR

This paper introduces new barycentric rational interpolation-based iteration methods that accelerate convergence in univariate root search and optimization, outperforming traditional methods in speed and efficiency.

Contribution

It develops novel derivative-free and first-derivative methods with higher convergence orders using barycentric rational interpolation, including frameworks for multivariate problems.

Findings

Derivative-free root search methods approach quadratic convergence.

First-derivative root search methods approach cubic convergence.

Full-memory optimization methods outperform secant method by 1.8 times in speed.

Abstract

Iteration methods based on barycentric rational interpolation are derived that exhibit accelerating orders of convergence. For univariate root search, the derivative-free methods approach quadratic convergence and the first-derivative methods approach cubic convergence. For univariate optimisation, the convergence order of the derivative-free methods approaches 1.62 and the order of the first-derivative methods approaches 2.42. Generally, performance advantages are found with respect to low-memory iteration methods. In optimisation problems where the objective function and gradient is calculated at each step, the full-memory iteration methods converge asymptotically 1.8 times faster than the secant method. Frameworks for multivariate root search and optimisation are also proposed, though without discovery of practical parameter choices.