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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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
- Mathematical Method
- Simple Root
- Parameter Family
- Finding Method
- Euler Method