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Mathematics of Computation

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The Schwarzian-Newton method for solving nonlinear equations, with applications


Author: Javier Segura
Journal: Math. Comp.
MSC (2010): Primary 65H05; Secondary 33B20, 33E05
Published electronically: June 2, 2016
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Abstract: The Schwarzian-Newton method (SNM) can be defined as the minimal method for solving nonlinear equations $ f(x)=0$ which is exact for any function $ f$ with constant Schwarzian derivative. Exactness means that the method gives the exact root in one iteration for any starting value in a neighborhood of the root. This is a fourth order method which has Halley's method as limit when the Schwarzian derivative tends to zero. We obtain conditions for the convergence of the SNM in an interval and show how this method can be applied for a reliable and fast solution of some problems, like the inversion of cumulative distribution functions (gamma and beta distributions) and the inversion of elliptic integrals.


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Additional Information

Javier Segura
Affiliation: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, 39005-Santander, Spain
Email: javier.segura@unican.es

DOI: https://doi.org/10.1090/mcom/3119
Received by editor(s): February 5, 2015
Received by editor(s) in revised form: September 1, 2015, and September 22, 2015
Published electronically: June 2, 2016
Additional Notes: The author acknowledges financial support from Ministerio de Economía y Competitividad (project MTM2012-34787). The author thanks the anonymous referee for helpful comments.
Article copyright: © Copyright 2016 American Mathematical Society