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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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## 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

## Copyright information

© Springer-Verlag 1977

## Authors and Affiliations

• Eldon Hansen
• 1
• Merrell Patrick
• 2
1. 1.Lockheed Palo Alto Research LaboratoryPalo AltoUSA
2. 2.Computer Science DepartmentDuke UniversityDurhamUSA