Numerische Mathematik

, Volume 27, Issue 3, pp 257–269

# A family of root finding methods

• Eldon Hansen
• Merrell Patrick
Article

## Summary

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.

## Keywords

Mathematical Method Simple Root Parameter Family Finding Method Euler Method

## References

1. 1.
Traub, J. F.: Iterative methods for the solution of equations. New York: Prentice-Hall 1964Google Scholar
2. 2.
Ostrowski, A. M.: Solution of equations and systems of equations. 3rd ed. New York-London: Academic Press 1973Google Scholar
3. 3.
Davies, M., Dawson, B.: On the global convergence of Halley's iteration formula. Num. Math.24, 133–135 (1975)Google Scholar
4. 4.
Parlett, B.: Laguerre's method applied to the matrix eigenvalue problem. Math. Comp.18, 464–485 (1964)Google Scholar
5. 5.
Kahan, W.: Where does Laguerre's method come from? Fourth Annual Princeton Conference on Information Sciences and Systems (1970)Google Scholar
6. 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. 7.
Laguerre, E. N.: Oeuvres de Laguerre, Vol. 1, pp. 87–103Google Scholar
8. 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. 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