Computing multiple roots of inexact polynomials

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 regular nonlinear least squares problem on a pejorative manifold, and calculates multiple roots simultaneously. To fulfill the input requirement of Algorithm I, we employ 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. The combined method calculates multiple roots with high 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 sensitivity of the multiple roots, a structure-preserving condition number is proposed and error bounds are given. Extensive computational experiments and the error analysis confirm that 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 remarkable accuracy.