Computing multiple roots of inexact polynomials

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 it 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 exist 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.