Extrapolating on Taylor Series Solutions of Homotopies with Nearby Poles

TL;DR

The work addresses when extrapolation can speed up slowly converging Taylor-series solutions along polynomial-homotopy paths that reach isolated singularities. It develops a theoretical framework based on Fabry's ratio theorem and product decompositions to relate coefficient-ratio expansions to the proximity of nearby poles, showing that a pole near the disk of convergence can destroy extrapolation effectiveness, while a distant pole permits $1/n$-type expansions and successful acceleration (via Richardson, rho, and related methods). Computational experiments with the rho algorithm and homotopies of the form $x^2 - C (1 - t)(P - t)$ corroborate the theory, illustrating practical thresholds (e.g., $|P|$) and providing guidance on when extrapolation will be beneficial. Overall, the paper offers a concrete criterion for the use of extrapolation in accelerating slowly converging Taylor-series solution paths of polynomial homotopies and clarifies the role of nearby poles in determining its success.

Abstract

A polynomial homotopy is a family of polynomial systems in one parameter, which defines solution paths starting from known solutions and ending at solutions of a system that has to be solved. We consider paths leading to isolated singular solutions, to which the Taylor series converges logarithmically. Whether or not extrapolation algorithms manage to accelerate the slowly converging series depends on the proximity of poles close to the disk of convergence of the Taylor series.

Figures (1)

• Figure 1: One nearby pole $P$ near the disk with radius $1$.

Theorems & Definitions (5)

• Theorem 2.1: the ratio theorem of Fabry Fab1896
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Theorem 3.4