Convergence in iterative polynomial root-finding
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.
Unable to display preview. Download preview PDF.
- 1.Chen T.C.: Iterative zero-finding revisited, in W.I. Hogarth and B.J. Noye (Eds.) Computational Techniques and Applications:CTAC-89 (Proc. Computaional Techniques and Applications Conf., Brisbane, Australia, July 1989), New York. Hemisphere Pub. Corp. (1990) 583–590Google Scholar
- 2.Chen, T.C.: SCARFS, an efficient polynomial zero-finder system. APL 93 Conf. Proc. (APL Quote Quard 24 (1993) 47–54Google Scholar
- 3.Chen, T.C.: Symmetry and nonconvergence in iterative polynomial zero-finding. (To be published)Google Scholar
- 4.Ralston, A., Rabinowitz, P.: A First Course in Numerical Analysis. 2nd Ed., New York: McGraw-Hill, 1978Google Scholar
- 5.Atkinson, K.E.: An Introduction to Numerical Analysis. New York:Wiley, 1978Google Scholar