Convergence in iterative polynomial root-finding

Abstract

For target α of the Nth-degree polynomial P (z), ¦δ^*/δ¦≡ ¦(z ^* −α)/(z −α)¦=O [σδ¦^q−1] < 1 if q > 1 and ¦σδ¦ ≪ 1, regardless of ¦δ¦ itself. Even if α is not a zero but the centroid of a cluster, the recomputed multiplicity estimate m (z) could lead to a component zero. In global iterations, popular methods proved inadequate, yet for symmetric clusters the CLAM formula z ^*=z −(NP/P′) (1 −Q ^m/n)/(1 −Q), where Q=[N (1 −PP″/P′ ²) −1]/(N/m −1), converges in principle to an m-fold zero in one iteration, using any finite guess outside the cluster centroid. Equipped with countermeasures against rebounds caused by local clusters, the formula has never been found to fail for general polynomials, and with an initial guess based on zeros of a symmetric cluster, usually converge in a few iterations.