A method computing multiple roots of inexact polynomials

Abstract

We present a combination of two novel algorithms that accurately calculate multiple roots of general polynomials. For a given multiplicity structure and initial root estimates, Algorithm I transforms the singular root-finding into a nonsingular least squares problem on a pejorative manifold, and calculates multiple roots simultaneously. To fulfill the input requirement of Algorithm I, we design a numerical GCD-finder, including a partial singular value decomposition and an iterative refinement, as the main engine for Algorithm II that calculates the multiplicity structure and the initial root approximation. The combined method calculates multiple roots with high forward accuracy 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 true sensitivity of the multiple roots, a pejorative condition number is proposed and error bounds are given. Extensive computational experiments are presented. The error analysis and numerical results confirm that a polynomial being ill-conditioned in conventional sense can be well conditioned pejoratively. In those cases, the multiple roots can be computed with remarkable accuracy.