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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- Javier Segura
- Affiliation: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, 39005-Santander, Spain
- MR Author ID: 627158
- Email: email@example.com
- 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.
- © Copyright 2016 American Mathematical Society
- Journal: Math. Comp. 86 (2017), 865-879
- MSC (2010): Primary 65H05; Secondary 33B20, 33E05
- DOI: https://doi.org/10.1090/mcom/3119
- MathSciNet review: 3584552