Numerische Mathematik

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

A family of root finding methods

  • Eldon Hansen
  • Merrell Patrick


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.


Mathematical Method Simple Root Parameter Family Finding Method Euler Method 
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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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