Numerische Mathematik

, Volume 27, Issue 3, pp 257–269 | Cite as

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.

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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

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