A one parameter family of iteration functions for finding roots is derived. The family includes the Laguerre, Halley, Ostrowski and Euler methods and, as a limiting case, Newton's method. All the methods of the family are cubically convergent for a simple root (except Newton's which is quadratically convergent). The superior behavior of Laguerre's method, when starting from a pointz for which |z| is large, is explained. It is shown that other methods of the family are superior if |z| is not large. It is also shown that a continuum of methods for the family exhibit global and monotonic convergence to roots of polynomials (and certain other functions) if all the roots are real.
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- 1.Traub, J. F.: Iterative methods for the solution of equations. New York: Prentice-Hall 1964Google Scholar
- 2.Ostrowski, A. M.: Solution of equations and systems of equations. 3rd ed. New York-London: Academic Press 1973Google Scholar
- 3.Davies, M., Dawson, B.: On the global convergence of Halley's iteration formula. Num. Math.24, 133–135 (1975)Google Scholar
- 4.Parlett, B.: Laguerre's method applied to the matrix eigenvalue problem. Math. Comp.18, 464–485 (1964)Google Scholar
- 5.Kahan, W.: Where does Laguerre's method come from? Fourth Annual Princeton Conference on Information Sciences and Systems (1970)Google Scholar
- 6.Bodewig, E.: Sur la méthode Laguerre pour l'approximation des racines de certaines équations algébriques et sur la critique d'Hermite. Indag. Math.8, 570–580 (1946)Google Scholar
- 7.Laguerre, E. N.: Oeuvres de Laguerre, Vol. 1, pp. 87–103Google Scholar
- 8.Dordević, L. N.: An iterative solution of algebraic equations with a parameter to accelerate convergence. Univ. Beograd. Publ. Elektrotechn. Fak. Ser. Mat. Fiz., # 449, pp. 179–182 (1973)Google Scholar
- 9.Tihonov, O. N.: On the rapid computation of the largest zeros of a polynomial [Russian]. Zap. Leningrad. Gorn. In-ta48, No. 3, 36–41 (1968)Google Scholar