A polynomially accelerated fixed-point iteration for vector problems

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.