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

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.