Fast convergence to higher multiplicity zeros

TL;DR

This paper demonstrates that the Newton-Anderson method, enhanced with Anderson acceleration, achieves superlinear convergence to multiple roots without prior knowledge of root multiplicity or extra computations.

Contribution

The paper introduces and analyzes a modified Newton method using Anderson acceleration that converges superlinearly to multiple roots without additional information or evaluations.

Findings

Achieves superlinear convergence to multiple roots.

Does not require prior knowledge of root multiplicity.

No extra function evaluations needed.

Abstract

In this paper, the Newton-Anderson method, which results from applying an extrapolation technique known as Anderson acceleration to Newton's method, is shown both analytically and numerically to provide superlinear convergence to non-simple roots of scalar equations. The method requires neither a priori knowledge of the multiplicities of the roots, nor computation of any additional function evaluations or derivatives.