Study of optimal eighth order weighted-Newton methods in Banach spaces
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In this work, we generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study its local convergence to approximate a locally-unique solution of a system of nonlinear equations. The convergence in this study is shown under hypotheses only on the first derivative. Our analysis avoids the usual Taylor expansions requiring higher order derivatives but uses generalized Lipschitz-type conditions only on the first derivative. Moreover, our new approach provides computable radius of convergence as well as error bounds on the distances involved and estimates on the uniqueness of the solution based on some functions appearing in these generalized conditions. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives which may not exist or may be very expensive or impossible to compute. The convergence order is computed using computational order of convergence or approximate computational order of convergence which do not require usage of higher derivatives. This technique can be applied to any iterative method using Taylor expansions involving high order derivatives. The study of the local convergence based on Lipschitz constants is important because it provides the degree of difficulty for choosing initial points. In this sense the applicability of the method is expanded. Finally, numerical examples are provided to verify the theoretical results and to show the convergence behavior.
Abstract
In this work, we generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study its local convergence to approximate a locally-unique solution of a system of nonlinear equations. The convergence in this study is shown under hypotheses only on the first derivative. Our analysis avoids the usual Taylor expansions requiring higher order derivatives but uses generalized Lipschitz-type conditions only on the first derivative. Moreover, our new approach provides computable radius of convergence as well as error bounds on the distances involved and estimates on the uniqueness of the solution based on some functions appearing in these generalized conditions. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives which may not exist or may be very expensive or impossible to compute. The convergence order is computed using computational order of convergence or approximate computational order of convergence which do not require usage of higher derivatives. This technique can be applied to any iterative method using Taylor expansions involving high order derivatives. The study of the local convergence based on Lipschitz constants is important because it provides the degree of difficulty for choosing initial points. In this sense the applicability of the method is expanded. Finally, numerical examples are provided to verify the theoretical results and to show the convergence behavior.
참고문헌 (31)
- S. Amat, "Third-order iterative methods with applications to Hammerstein equations: a unified approach", J. Comput. Appl. Math., 235(9), pp.2936~2943 (2011)[DOI]
- S. Amat, "Chaotic dynamics of a third-order Newton-type method", J. Math. Anal. Appl., 366(1), pp.24~32 (2010)[DOI]
- S. Amat, "Semilocal convergence of a sixth order iterative method for quadratic equations", Appl. Numer. Math., 62(7), pp.833~841 (2012)[DOI]
- I. K. Argyros, "Computational Theory of Iterative Methods, Studies in Computational Mathematics, 15" (2007), Elsevier B. V.
- I. K. Argyros, "Weaker conditions for the convergence of Newton's method", J. Complexity, 28(3), pp.364~387 (2012)[DOI]
- I. K. Argyros, "Iterative methods and their dynamics with applications" (2017), CRC Press
- I. K. Argyros, "Improved local analysis for a certain class of iterative methods with cubic convergence", Numer. Algorithms, 59(4), pp.505~521 (2012)[DOI]
- I. K. Argyros, "Local convergence of Newton-Gauss method in Banach spaces", SeMA, 74(4), pp.429~439 (2016)[DOI]
- D. K. R. Babajee, "A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule", Appl. Math. Comput., 200(1), pp.452~458 (2008)
- D. K. R. Babajee, "Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations", J. Comput. Appl. Math., 233(8), pp.2002~2012 (2010)[DOI]
- C. Chun, "Third-order family of methods in Banach spaces", Comput. Math. Appl., 61(6), pp.1665~1675 (2011)[DOI]
- A. Cordero, "Variants of Newton's method using fifth-order quadrature formulas", Appl. Math. Comput., 190(1), pp.686~698 (2007)
- M. T. Darvishi, "A two-step high order Newton-like method for solving systems of nonlinear equations", Int. J. Pure Appl. Math., 57(4), pp.543~555 (2009)
- M. T. Darvishi, "Some three-step iterative methods free from second order derivative for finding solutions of systems of nonlinear equations", Int. J. Pure Appl. Math., 57(4), pp.557~573 (2009)
- M. T. Darvishi, "Super cubic iterative methods to solve systems of nonlinear equations", Appl. Math. Comput., 188(2), pp.1678~1685 (2007)
- J. A. Ezquerro, "Recurrence relations for Chebyshev-type methods", Appl. Math. Optim., 41(2), pp.227~236 (2000)[DOI]
- M. Grau-Sanchez, "Ostrowski type methods for solving systems of nonlinear equations", Appl. Math. Comput., 218(6), pp.2377~2385 (2011)
- J. M. Gutierrez, "On the semilocal convergence of Newton-Kantorovich method under center-Lipschitz conditions", Appl. Math. Comput., 221, pp.79~88 (2013)
- M. A. Hernandez, "On the semilocal convergence of a three steps Newton- type iterative process under mild convergence conditions", Numer. Algorithms, 70(2), pp.377~392 (2015)[DOI]
- H. H. H. Homeier, "A modified Newton method with cubic convergence: the multivariate case", J. Comput. Appl. Math., 169(1), pp.161~169 (2004)[DOI]
- K. I. Noor, "Iterative methods with fourth-order convergence for nonlinear equations", Appl. Math. Comput., 189(1), pp.221~227 (2007)
- P. K. Parida, "Recurrence relations for a Newton-like method in Banach spaces", J. Comput. Appl. Math., 206(2), pp.873~887 (2007)[DOI]
- M. S. Petkovic, "Multipoint methods for solving nonlinear equations", pp.2013, Elsevier
- H. Ren, "Convergence ball and error analysis of a family of iterative methods with cubic convergence", Appl. Math. Comput., 209(2), pp.369~378 (2009)
- W. C. Rheinboldt, "An adaptive continuation process for solving systems of nonlinear equations, in Mathematical models and numerical methods (Papers, Fifth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1975)", pp.129~142 (1978), Banach Center Publ.
- J. R. Sharma, "On efficient weighted-Newton methods for solving systems of nonlinear equations", Appl. Math. Comput., 222, pp.497~506 (2013)
- J. R. Sharma, "An efficient family of weighted-Newton methods with optimal eighth order convergence", Appl. Math. Lett., 29, pp.1~6 (2014)
- B.-C. Shin, "A comparison of the Newton-Krylov method with high order Newton-like methods to solve nonlinear systems", Appl. Math. Comput., 217(7), pp.3190~3198 (2010)
- J. F. Traub, "Iterative Methods for the Solution of Equations, Prentice-Hall Series in Automatic Computation" (1964), Prentice-Hall, Inc.
- S. Weerakoon, "A variant of Newton's method with accelerated third-order convergence", Appl. Math. Lett., 13(8), pp.87~93 (2000)
- S. Wolfram, "The Mathematica Book, fifth edition" (2003), Wolfram Media, Inc.