Direct Comparison between Two Third Convergence Order Schemes for Solving Equations
Abstract
:1. Introduction
2. Ball Convergence
- (A1)
- is differentiable; there exists a simple zero p of equation
- (A2)
- There exists a continuous and increasing function defined on with values in such that for all
- (A3)
- There exist continuous and increasing functions and on the interval with values in such that for all
- (A4)
- (A5)
- There exists such thatSet
- 1.
- In view of () and the estimate
- 2.
- The results obtained here can be used for operators F satisfying autonomous differential equations [3] of the form
- 3.
- If and are constant functions, say for some and then the radius was shown by us to be the convergence radius of Newton’s method [5,6]That is our convergence ball is at most three times larger than Rheinboldt’s. The same value for was given by Traub [15].
- 4.
- It is worth noticing that method (2) is not changing when we use simpler methods the conditions of Theorem 1 instead of the stronger conditions used in [10]. Moreover, we can compute the computational order of convergence (COC) defined byThis way we obtain in practice the order of convergence in a way that avoids the bounds involving estimates using estimates higher than the first Fréchet derivative of operator
- 5.
- 6.
- The ball convergence result for scheme (3) clearly is obtained from Theorem 1 for
- 7.
- In our earlier works with other schemes we assumed to show the existence of and using the intermediate value theorem. Bur these initial conditions on and This is not necessary with our approach. This way we further expand the applicability of scheme (2) and (3). The same is true for scheme (4) whose ball convergence follows.
3. Numerical Examples
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Amat, S.; Busquier, S.; Negra, M. Adaptive approximation of nonlinear operators. Numer. Funct. Anal. Optim. 2004, 25, 397–405. [Google Scholar] [CrossRef]
- Amat, S.; Argyros, I.K.; Busquier, S.; Magreñán, A.A. Local convergence and the dynamics of a two-point four parameter Jarratt-like method under weak conditions. Numer. Alg. 2017. [Google Scholar] [CrossRef]
- Argyros, I.K. Computational Theory of Iterative Methods, Series: Studies in Computational Mathematics; Chui, C.K., Wuytack, L., Eds.; Elsevier Publ. Company: New York, NY, USA, 2007. [Google Scholar]
- Argyros, I.K.; George, S. Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications; Volume III: Newton’s Method Defined on Not Necessarily Bounded Domain; Nova Publishes: New York, NY, USA, 2019. [Google Scholar]
- Argyros, I.K.; George, S. Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications; Volume-IV: Local Convergence of a Class of Multi-Point Super–Halley Methods; Nova Publishes: New York, NY, USA, 2019. [Google Scholar]
- Argyros, I.K.; George, S.; Magreñán, A.A. Local convergence for multi-point- parametric Chebyshev-Halley- type method of higher convergence order. J. Comput. Appl. Math. 2015, 282, 215–224. [Google Scholar] [CrossRef]
- Argyros, I.K.; Magreñán, A.A. Iterative Method and Their Dynamics with Applications; CRC Press: New York, NY, USA, 2017. [Google Scholar]
- Argyros, I.K.; Magreñán, A.A. A study on the local convergence and the dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Alg. 2015, 71, 1–23. [Google Scholar] [CrossRef]
- Argyros, I.K.; Regmi, S. Undergraduate Research at Cameron University on Iterative Procedures in Banach and Other Spaces; Nova Science Publishers: New York, NY, USA, 2019. [Google Scholar]
- Babajee, D.K.R.; Davho, M.E.; Darvishi, M.T.; Karami, A.; Barati, A. Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations. J. Comput. Appl. Math. 2010, 233, 2002–2012. [Google Scholar] [CrossRef] [Green Version]
- Alzahrani, A.K.H.; Behl, R.; Alshomrani, A.S. Some higher-order iteration functions for solving nonlinear models. Appl. Math. Comput. 2018, 334, 80–93. [Google Scholar] [CrossRef]
- Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. Stable high-order iterative methods for solving nonlinear models. Appl. Math. Comput. 2017, 303, 70–88. [Google Scholar] [CrossRef]
- Choubey, N.; Panday, B.; Jaiswal, J.P. Several two-point with memory iterative methods for solving nonlinear equations. Afrika Matematika 2018, 29, 435–449. [Google Scholar] [CrossRef]
- Cordero, A.; Torregrosa, J.R. Variants of Newton’s method for functions of several variables. Appl. Math. Comput. 2006, 183, 199–208. [Google Scholar] [CrossRef]
- Cordero, A.; Hueso, J.L.; Martínez, E.; Torregrosa, J.R. A modified Newton-Jarratt’s composition. Numer. Alg. 2010, 55, 87–99. [Google Scholar] [CrossRef]
- Cordero, A.; Torregrosa, J.R. Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 2007, 190, 686–698. [Google Scholar] [CrossRef]
- Cordero, A.; Hueso, J.L.; Martínez, E.; Torregrosa, J.R. Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 2012, 25, 2369–2374. [Google Scholar] [CrossRef] [Green Version]
- Darvishi, M.T.; Barati, A. Super cubic iterative methods to solve systems of nonlinear equations. Appl. Math. Comput. 2007, 188, 1678–1685. [Google Scholar] [CrossRef]
- Esmaeili, H.; Ahmadi, M. An efficient three-step method to solve system of non linear equations. Appl. Math. Comput. 2015, 266, 1093–1101. [Google Scholar]
- Fang, X.; Ni, Q.; Zeng, M. A modified quasi-Newton method for nonlinear equations. J. Comput. Appl. Math. 2018, 328, 44–58. [Google Scholar] [CrossRef]
- Fousse, L.; Hanrot, G.; Lefvre, V.; Plissier, P.; Zimmermann, P. MPFR: A multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 2007, 33, 15. [Google Scholar] [CrossRef]
- Homeier, H.H.H. A modified Newton method with cubic convergence: The multivariate case. J. Comput. Appl. Math. 2004, 169, 161–169. [Google Scholar] [CrossRef] [Green Version]
- Iliev, A.; Kyurkchiev, N. Nontrivial Methods in Numerical Analysis: Selected Topics in Numerical Analysis; LAP LAMBERT Academic Publishing: Saarbrucken, Germany, 2010; ISBN 978-3-8433-6793-6. [Google Scholar]
- Lotfi, T.; Bakhtiari, P.; Cordero, A.; Mahdiani, K.; Torregrosa, J.R. Some new efficient multipoint iterative methods for solving nonlinear systems of equations. Int. J. Comput. Math. 2015, 92, 1921–1934. [Google Scholar] [CrossRef] [Green Version]
- Magreñán, A.A. Different anomalies in a Jarratt family of iterative root finding methods. Appl. Math. Comput. 2014, 233, 29–38. [Google Scholar]
- Magreñán, A.A. A new tool to study real dynamics: The convergence plane. Appl. Math. Comput. 2014, 248, 29–38. [Google Scholar] [CrossRef] [Green Version]
- Noor, M.A.; Waseem, M. Some iterative methods for solving a system of nonlinear equations. Comput. Math. Appl. 2009, 57, 101–10619. [Google Scholar] [CrossRef] [Green Version]
- Ortega, J.M.; Rheinboldt, W.C. Iterative Solutions of Nonlinear Equations in Several Variables; Academic Press: New York, NY, USA, 1970. [Google Scholar]
- Ostrowski, A.M. Solution of Equation and Systems of Equations; Academic Press: New York, NY, USA, 1960. [Google Scholar]
- Rheinboldt, W.C. An adaptive continuation process for solving systems of nonlinear equations. In Mathematical Models and Numerical Solvers; Tikhonov, A.N., Ed.; Banach Center: Warsaw, Poland, 1977; pp. 129–142. [Google Scholar]
- Petkovic, M.S.; Neta, B.; Petkovic, L.j.D.; Dzunic, J. Multi-Point Methods for Solving Nonlinear Equations; Elsevier/Academic Press: Amsterdam, The Netherlands; Boston, MA, USA; Heidelberg, Germany; London, UK; New York, NY, USA, 2013. [Google Scholar]
- Sharma, J.R.; Sharma, R.; Bahl, A. An improved Newton-Traub composition for solving systems of nonlinear equa- tions. Appl. Math. Comput. 2016, 290, 98–110. [Google Scholar]
- Sharma, J.R.; Arora, H. Improved Newton-like methods for solving systems of nonlinear equations. SeMA 2017, 74, 147–163. [Google Scholar] [CrossRef]
- Sharma, J.R.; Arora, H. Efficient derivative-free numerical methods for solving systems of nonlinear equations. Comput. Appl. Math. 2016, 35, 269–284. [Google Scholar] [CrossRef]
- Traub, J.F. Iterative Methods for the Solution of Equations; Chelsea Publishing Company: New York, NY, USA, 1982. [Google Scholar]
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Regmi, S.; Argyros, I.K.; George, S. Direct Comparison between Two Third Convergence Order Schemes for Solving Equations. Symmetry 2020, 12, 1080. https://doi.org/10.3390/sym12071080
Regmi S, Argyros IK, George S. Direct Comparison between Two Third Convergence Order Schemes for Solving Equations. Symmetry. 2020; 12(7):1080. https://doi.org/10.3390/sym12071080
Chicago/Turabian StyleRegmi, Samundra, Ioannis K. Argyros, and Santhosh George. 2020. "Direct Comparison between Two Third Convergence Order Schemes for Solving Equations" Symmetry 12, no. 7: 1080. https://doi.org/10.3390/sym12071080