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A family of root finding methods

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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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Authors and Affiliations

  1. Lockheed Palo Alto Research Laboratory, 3251 Hanover Street, 94304, Palo Alto, California, USA

    Eldon Hansen

  2. Computer Science Department, Duke University, Durham, North Carolina, USA

    Merrell Patrick

Authors
  1. Eldon Hansen
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  2. Merrell Patrick
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Additional information

This research was supported by the National Science Foundation under grant number NSF-DCR-74-10042.

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Hansen, E., Patrick, M. A family of root finding methods. Numer. Math. 27, 257–269 (1976). https://doi.org/10.1007/BF01396176

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  • DOI: https://doi.org/10.1007/BF01396176

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