A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equations

A new technique to obtain derivative-free methods with optimal order of convergence in the sense of the Kung-Traub conjecture for solving nonlinear smooth equations is described. The procedure uses Steffensen-like methods and Pade approximants. Some numerical examples are provided to show the good performance of the new methods. (c) 2012 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID-06-2010-2285.Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2013). A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equations. Journal of Computational and Applied Mathematics. 252:95-102. https://doi.org/10.1016/j.cam.2012.03.030S9510225

An efficient two-parametric family with memory for nonlinear equations

A new two-parametric family of derivative-free iterative methods for solving nonlinear equations is presented. First, a new biparametric family without memory of optimal order four is proposed. The improvement of the convergence rate of this family is obtained by using two self-accelerating parameters. These varying parameters are calculated in each iterative step employing only information from the current and the previous iteration. The corresponding R-order is 7 and the efficiency index 7(1/3) = 1.913. Numerical examples and comparison with some existing derivative-free optimal eighth-order schemes are included to confirm the theoretical results. In addition, the dynamical behavior of the designed method is analyzed and shows the stability of the scheme.The second author wishes to thank the Islamic Azad University, Hamedan Branch, where the paper was written as a part of the research plan, for financial support.Cordero Barbero, A.; Lotfi, T.; Bakhtiari, P.; Torregrosa Sánchez, JR. (2015). An efficient two-parametric family with memory for nonlinear equations. Numerical Algorithms. 68(2):323-335. doi:10.1007/s11075-014-9846-8S323335682Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equation. J. Comput. Appl. Math. 252, 95–102 (2013)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Pseudocomposition: a technique to design predictor-corrector methods for systems of nonlinear equations. Appl. Math. Comput. 218, 11496–11508 (2012)Džunić, J.: On efficient two-parameter methods for solving nonlinear equations. Numer. Algorithms. 63(3), 549–569 (2013)Džunić, J., Petković, M.S.: On generalized multipoint root-solvers with memory. J. Comput. Appl. Math. 236, 2909–2920 (2012)Petković, M.S., Neta, B., Petković, L.D., Džunić, J. (ed.).: Multipoint methods for solving nonlinear equations. Elsevier (2013)Sharma, J.R., Sharma, R.: A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer. Algorithms 54, 445–458 (2010)Soleymani, F., Shateyi, S.: Two optimal eighth-order derivative-free classes of iterative methods. Abstr. Appl. Anal. 2012(318165), 14 (2012). doi: 10.1155/2012/318165Soleymani, F., Sharma, R., Li, X., Tohidi, E.: An optimized derivative-free form of the Potra-Pták methods. Math. Comput. Model. 56, 97–104 (2012)Thukral, R.: Eighth-order iterative methods without derivatives for solving nonlinear equations. ISRN Appl. Math. 2011(693787), 12 (2011). doi: 10.5402/2011/693787Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)Wang, X., Džunić, J., Zhang, T.: On an efficient family of derivative free three-point methods for solving nonlinear equations. Appl. Math. Comput. 219, 1749–1760 (2012)Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)Ortega, J.M., Rheinboldt, W.G. (ed.).: Iterative Solutions of Nonlinear Equations in Several Variables, Ed. Academic Press, New York (1970)Jay, I.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)Blanchard, P.: Complex Analytic Dynamics on the Riemann Sphere. Bull. AMS 11(1), 85–141 (1984)Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. arXiv: 1307.6705 [math.NA

Dynamics and fractal dimension of Steffensen-type methods

In this paper, the dynamical behavior of different optimal iterative schemes for solving nonlinear equations with increasing order, is studied. The tendency of the complexity of the Julia set is analyzed and referred to the fractal dimension. In fact, this fractal dimension can be shown to be a powerful tool to compare iterative schemes that estimate the solution of a nonlinear equation. Based on the box-counting algorithm, several iterative derivative-free methods of different convergence orders are compared.Chicharro López, FI.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2015). Dynamics and fractal dimension of Steffensen-type methods. Algorithms. 8(2):271-279. doi:10.3390/a8020271S27127982Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860Varona, J. L. (2002). Graphic and numerical comparison between iterative methods. The Mathematical Intelligencer, 24(1), 37-46. doi:10.1007/bf03025310Amat, S., Busquier, S., Bermúdez, C., & Plaza, S. (2012). On two families of high order Newton type methods. Applied Mathematics Letters, 25(12), 2209-2217. doi:10.1016/j.aml.2012.06.004Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-142. doi:10.1090/s0273-0979-1984-15240-6Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2013). A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equations. Journal of Computational and Applied Mathematics, 252, 95-102. doi:10.1016/j.cam.2012.03.030Epureanu, B. I., & Greenside, H. S. (1998). Fractal Basins of Attraction Associated with a Damped Newton’s Method. SIAM Review, 40(1), 102-109. doi:10.1137/s003614459631003

Nonlinear Parabolic Equations arising in Mathematical Finance

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387

Fractional Order Version of the HJB Equation

We consider an extension of the well-known Hamilton-Jacobi-Bellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Nonlinear Dynamics', ISSN 1555-1415, eISSN 1555-1423, CODEN: JCNDDM. Submitted 28-June-2018; Revised 15-Sept-2018; Accepted 28-Oct-201