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

Author: Javier Segura
Journal: Math. Comp. 86 (2017), 865-879
MSC (2010): Primary 65H05; Secondary 33B20, 33E05
Published electronically: June 2, 2016
MathSciNet review: 3584552
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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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  • [1] S. Amat, S. Busquier, and J. M. Gutiérrez, Geometric constructions of iterative functions to solve nonlinear equations, J. Comput. Appl. Math. 157 (2003), no. 1, 197-205. MR 1996476 (2004k:65077),
  • [2] John P. Boyd, Numerical, perturbative and Chebyshev inversion of the incomplete elliptic integral of the second kind, Appl. Math. Comput. 218 (2012), no. 13, 7005-7013. MR 2880288,
  • [3] George H. Brown Jr., On Halley's variation of Newton's method, Amer. Math. Monthly 84 (1977), no. 9, 726-728. MR 0461884 (57 #1866)
  • [4] Alfredo Deaño, Amparo Gil, and Javier Segura, New inequalities from classical Sturm theorems, J. Approx. Theory 131 (2004), no. 2, 208-230. MR 2106538 (2006c:33007),
  • [5] Toshio Fukushima, Numerical inversion of a general incomplete elliptic integral, J. Comput. Appl. Math. 237 (2013), no. 1, 43-61. MR 2966887,
  • [6] A. Gil, J. Segura, and N. M. Temme, An efficient algorithm for the inversion of the cumulative central beta distribution, accepted for publication in Numer. Algorithms, DOI 10.1007/s11075-016-0139-2.
  • [7] Amparo Gil, Javier Segura, and Nico M. Temme, Efficient and accurate algorithms for the computation and inversion of the incomplete gamma function ratios, SIAM J. Sci. Comput. 34 (2012), no. 6, A2965-A2981. MR 3023741,
  • [8] Amparo Gil, Javier Segura, and Nico M. Temme, Gammachi: a package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral), Comput. Phys. Commun. 191 (2015), 132-139.
  • [9] A. Melman, Geometry and convergence of Euler's and Halley's methods, SIAM Rev. 39 (1997), no. 4, 728-735. MR 1491054 (98h:65019),
  • [10] R. B. Paris, Incomplete gamma and related functions, NIST Handbook of Mathematical Functions, U.S. Dept. Commerce, Washington, DC, 2010, pp. 175-192. MR 2655348
  • [11] G. S. Salehov, On the convergence of the process of tangent hyperbolas, Doklady Akad. Nauk SSSR (N.S.) 82 (1952), 525-528 (Russian). MR 0048909 (14,91f)
  • [12] T. R. Scavo and J. B. Thoo, On the geometry of Halley's method, Amer. Math. Monthly 102 (1995), no. 5, 417-426. MR 1327786 (96f:01019),
  • [13] Javier Segura, Reliable computation of the zeros of solutions of second order linear ODEs using a fourth order method, SIAM J. Numer. Anal. 48 (2010), no. 2, 452-469. MR 2646104 (2011d:65118),
  • [14] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0169356 (29 #6607)

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Additional Information

Javier Segura
Affiliation: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, 39005-Santander, Spain

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

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