Computing multiple roots of inexact polynomials

TL;DR

This paper introduces a combined algorithmic approach for accurately computing multiple roots of inexact polynomials, leveraging a numerical GCD finder and a regularized nonlinear least squares method, achieving high precision without multiprecision arithmetic.

Contribution

The novel combined method accurately computes multiple roots of inexact polynomials, including a new GCD-finder and a regularization approach, with high reliability and without multiprecision arithmetic.

Findings

High accuracy in computing multiple roots in practice

Effective handling of inexact polynomial coefficients

Proposed condition number for sensitivity analysis

Abstract

We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations. To fulfill the input requirement of Algorithm I, we develop a numerical GCD-finder containing a successive singular value updating and an iterative GCD refinement as the main engine of Algorithm II that calculates the multiplicity structure and the initial root approximation. While limitations of our algorithm exist in identifying the multiplicity structure in certain situations, the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic even if the coefficients are inexact. This is perhaps the first blackbox-type root-finder with such capabilities. To measure the sensitivity of the multiple roots, a structure-preserving condition number is proposed and error bounds are established. According to our computational experiments and error analysis, a polynomial being ill-conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved, and its multiple roots can be computed with high accuracy.