A polynomially accelerated fixed-point iteration for vector problems

TL;DR

This work introduces a constant-memory three-point polynomial accelerator (TPA) for fixed-point iterations that exploits residual dynamics to estimate the dominant contraction factor $m$ and constructs a regularised three-point update to cancel the leading error mode. The derivation ties to Aitken's $\Delta^2$ process and Anderson acceleration, showing that TPA reduces to known extrapolation limits under special choices while preserving a lightweight, three-vector memory footprint. Numerical experiments on linear systems with clustered spectra, nonlinear $\tanh$ fixed points, and a discretised 2D Poisson equation demonstrate substantial reductions in map evaluations compared with Picard, SOR, and shallow/deep Anderson schemes, with performance improving as problem difficulty increases. The results suggest practical benefits for accelerating contractive fixed-point iterations with minimal overhead, along with caveats for non-normal Jacobians and a roadmap for adaptive parameters and higher-degree extensions.

Abstract

Fixed-point procedures frequently slow down because an error mode decays much more slowly than the others, leaving the base iteration with a persistent residual plateau. To counter this obstruction we formulate a three-point polynomial accelerator (TPA) that fits inside existing fixed-point algorithms with negligible modification and computational cost. TPA first infers the dominant contraction factor directly from the residual dynamics and then assembles a regularised three-point update from the last three iterates. We show that across a suite of tests: a linear system with clustered eigenvalues, a nonlinear tanh mapping, and a discretised Poisson equation, TPA attains a prescribed tolerance in markedly fewer map evaluations than Picard iteration, weighted Jacobi/SOR, and shallow Anderson schemes while preserving a minimal memory and arithmetic footprint.

Figures (3)

• Figure 1: Linear system with clustered spectrum. Residual histories showing that TPA damps the dominant modes faster than the reference methods.
• Figure 2: Nonlinear $\tanh$ fixed point. Residual histories showing the smooth convergence of TPA relative to the baselines.
• Figure 3: Discretised 2D Poisson equation. Residual histories highlighting the faster damping achieved by TPA.

Theorems & Definitions (6)

• Remark 1: Jacobian interpretation of Assumption \ref{['ass:A1']}
• Lemma 1: Error polynomial
• proof
• Proposition 1: Optimal quadratic coefficients
• Lemma 2: Residual recursion
• proof