Aberth's method for the parallel iterative finding of polynomial zeros

Abstract

Aberth's method for the iterative finding of polynomial zeros using simultaneous interacting guesses is seen to be the systematic use of Newton's method under Parallel Anticipatory Implicit Deflation (PAID), changing the single polynomial problem into that for a system of rational functions. It offers the prospect of parallel execution, discourages coincident convergences, and subjects all iterates to the same algorithm, without the uneven roundoff error accumulation due to deflation typical in serial iterative methods. Yet it converges slowly to multiple zeros, and consumes far more total computing power than serial methods. Factors hindering convergence include problem symmetry and clustering of iterates, alleviated by asymmetric initial guesses and an immediate updating strategy, as seen through the speedup in the experimental APL program PAIDAX.