Two New Predictor-Corrector Iterative Methods with Third- and Ninth-Order Convergence for Solving Nonlinear Equations

In this paper, we suggest and analyze two new predictor-corrector iterative methods with third and ninth-order convergence for solving nonlinear equations. The first method is a development of [M. A. Noor, K. I. Noor and K. Aftab, Some New Iterative Methods for Solving Nonlinear Equations, World Applied Science Journal, 20(6),(2012):870-874.] based on the trapezoidal integration rule and the centroid mean. The second method is an improvement of the first new proposed method by using the technique of updating the solution. The order of convergence and corresponding error equations of new proposed methods are proved. Several numerical examples are given to illustrate the efficiency and performance of these new methods and compared them with the Newton's method and other relevant iterative methods. Keywords: Nonlinear equations, Predictor–corrector methods, Trapezoidal integral rule, Centroid mean, Technique of updating the solution; Order of convergence

Design, Analysis, and Applications of Iterative Methods for Solving Nonlinear Systems

In this chapter, we present an overview of some multipoint iterative methods for solving nonlinear systems obtained by using different techniques such as composition of known methods, weight function procedure, and pseudo-composition, etc. The dynamical study of these iterative schemes provides us valuable information about their stability and reliability. A numerical test on a specific problem coming from chemistry is performed to compare the described methods with classical ones and to confirm the theoretical results

A convex combination approach for mean-based variants of Newton's method

[EN] Several authors have designed variants of Newton¿s method for solving nonlinear equations by using different means. This technique involves a symmetry in the corresponding fixed-point operator. In this paper, some known results about mean-based variants of Newton¿s method (MBN) are re-analyzed from the point of view of convex combinations. A new test is developed to study the order of convergence of general MBN. Furthermore, a generalization of the Lehmer mean is proposed and discussed. Numerical tests are provided to support the theoretical results obtained and to compare the different methods employed. Some dynamical planes of the analyzed methods on several equations are presented, revealing the great difference between the MBN when it comes to determining the set of starting points that ensure convergence and observing their symmetry in the complex plane.This research was partially funded by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and by Generalitat Valenciana PROMETEO/2016/089 (Spain).Cordero Barbero, A.; Franceschi, J.; Torregrosa Sánchez, JR.; Zagati, AC. (2019). A convex combination approach for mean-based variants of Newton's method. Symmetry (Basel). 11(9):1-16. https://doi.org/10.3390/sym11091106S116119Weerakoon, S., & Fernando, T. G. I. (2000). A variant of Newton’s method with accelerated third-order convergence. Applied Mathematics Letters, 13(8), 87-93. doi:10.1016/s0893-9659(00)00100-2Özban, A. . (2004). Some new variants of Newton’s method. Applied Mathematics Letters, 17(6), 677-682. doi:10.1016/s0893-9659(04)90104-8Zhou, X. (2007). A class of Newton’s methods with third-order convergence. Applied Mathematics Letters, 20(9), 1026-1030. doi:10.1016/j.aml.2006.09.010Singh, M. K., & Singh, A. K. (2017). A New-Mean Type Variant of Newton´s Method for Simple and Multiple Roots. International Journal of Mathematics Trends and Technology, 49(3), 174-177. doi:10.14445/22315373/ijmtt-v49p524Verma, K. L. (2016). On the centroidal mean Newton’s method for simple and multiple roots of nonlinear equations. International Journal of Computing Science and Mathematics, 7(2), 126. doi:10.1504/ijcsm.2016.076403Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.062Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2013). Drawing Dynamical and Parameters Planes of Iterative Families and Methods. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/78015

Multipoint efficient iterative methods and the dynamics of Ostrowski's method

This is an Author's Accepted Manuscript of an article published in José L. Hueso, Eulalia Martínez & Carles Teruel (2019) Multipoint efficient iterative methods and the dynamics of Ostrowski's method, International Journal of Computer Mathematics, 96:9, 1687-1701, DOI: 10.1080/00207160.2015.1080354 in the International Journal of Computer Mathematics, SEP 2 2019 [copyright Taylor & Francis], available online at: http://www.tandfonline.com/10.1080/00207160.2015.1080354[EN] In this work, we introduce a modification into the technique, presented in A. Cordero, J.L. Hueso, E. Martinez, and J.R. Torregrosa [Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett. 25 (2012), pp. 2369-2374], that increases by two units the convergence order of an iterative method. The main idea is to compose a given iterative method of order p with a modification of Newton's method that introduces just one evaluation of the function, obtaining a new method of order p+2, avoiding the need to compute more than one derivative, so we improve the efficiency index in the scalar case. This procedure can be repeated n times, with the same approximation to the derivative, obtaining new iterative methods of order p+2n. We perform different numerical tests that confirm the theoretical results. By applying this procedure to Newton's method one obtains the well known fourth order Ostrowski's method. We finally analyse its dynamical behaviour on second and third degree real polynomials.This research was supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C22 and by the project of Generalitat Valenciana Prometeo/2016/089.Hueso, JL.; Martínez Molada, E.; Teruel-Ferragud, C. (2019). Multipoint efficient iterative methods and the dynamics of Ostrowski's method. International Journal of Computer Mathematics. 96(9):1687-1701. https://doi.org/10.1080/00207160.2015.1080354S16871701969Amat, S., Busquier, S., & Plaza, S. (2010). Chaotic dynamics of a third-order Newton-type method. Journal of Mathematical Analysis and Applications, 366(1), 24-32. doi:10.1016/j.jmaa.2010.01.047Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.062Cordero, A., Martínez, E., & Torregrosa, J. R. (2009). Iterative methods of order four and five for systems of nonlinear equations. Journal of Computational and Applied Mathematics, 231(2), 541-551. doi:10.1016/j.cam.2009.04.015Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2012). Increasing the convergence order of an iterative method for nonlinear systems. Applied Mathematics Letters, 25(12), 2369-2374. doi:10.1016/j.aml.2012.07.005Jarratt, P. (1966). Some fourth order multipoint iterative methods for solving equations. Mathematics of Computation, 20(95), 434-434. doi:10.1090/s0025-5718-66-99924-

Memory in the iterative processes for nonlinear problems

In this paper, we study different ways for introducing memory to a parametric family of optimal two-step iterative methods. We study the convergence and the stability, by means of real dynamics, of the methods obtained by introducing memory in order to compare them. We also perform several numerical experiments to see how the methods behave

Generalizing Traub's method to a parametric iterative class for solving multidimensional nonlinear problems

[EN] In this work, we modify the iterative structure of Traub's method to include a real parameter alpha$$\alpha$$. A parametric family of iterative methods is obtained as a generalization of Traub, which is also a member of it. The cubic order of convergence is proved for any value of alpha$$\alpha$$. Then, a dynamical analysis is performed after applying the family for solving a system cubic polynomials by means of multidimensional real dynamics. This analysis allows to select the best members of the family in terms of stability as a preliminary study to be generalized to any nonlinear function. Finally, some iterative schemes of the family are used to check numerically the previous developments when they are used to approximate the solutions of academic nonlinear problems and a chemical diffusion reaction problem.ERDF A way of making Europe, Grant/Award Number: PGC2018-095896-B-C22; MICoCo of Universidad Internacional de La Rioja (UNIR), Grant/Award Number: PGC2018-095896-B-C22Chicharro, FI.; Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR. (2023). Generalizing Traub's method to a parametric iterative class for solving multidimensional nonlinear problems. Mathematical Methods in the Applied Sciences. 1-14. https://doi.org/10.1002/mma.937111

Numerical Recipes in Python

Numerical Recipes in Python is to serve as Laboratory Manual of Simplified Numerical Analysis (Python Version): A companion book of the principal book: Simplified Numerical Analysis (Fourth Edition) by Dr. Amjad Ali

Multistep High-Order Methods for Nonlinear Equations Using Pade-Like Approximants

[EN] We present new high-order optimal iterativemethods for solving a nonlinear equation, f(x) = 0, by using Pade-like approximants. We compose optimal methods of order 4 with Newton's step and substitute the derivative by using an appropriate rational approximant, getting optimal methods of order 8. In the same way, increasing the degree of the approximant, we obtain optimal methods of order 16. We also perform different numerical tests that confirm the theoretical results.This work has been supported by Ministerio de Ciencia e Innovacion de Espana MTM2014-52016-C2-02-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2017). Multistep High-Order Methods for Nonlinear Equations Using Pade-Like Approximants. Discrete Dynamics in Nature and Society. 1-6. https://doi.org/10.1155/2017/3204652S16Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860Petković, M. S., Neta, B., Petković, L. D., & Džunić, J. (2013). Basic concepts. Multipoint Methods, 1-26. doi:10.1016/b978-0-12-397013-8.00001-7Bi, W., Ren, H., & Wu, Q. (2009). Three-step iterative methods with eighth-order convergence for solving nonlinear equations. Journal of Computational and Applied Mathematics, 225(1), 105-112. doi:10.1016/j.cam.2008.07.004Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2011). Three-step iterative methods with optimal eighth-order convergence. Journal of Computational and Applied Mathematics, 235(10), 3189-3194. doi:10.1016/j.cam.2011.01.004Liu, L., & Wang, X. (2010). Eighth-order methods with high efficiency index for solving nonlinear equations. Applied Mathematics and Computation, 215(9), 3449-3454. doi:10.1016/j.amc.2009.10.040Sharma, J. R., & Sharma, R. (2009). A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numerical Algorithms, 54(4), 445-458. doi:10.1007/s11075-009-9345-5Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2010). New modifications of Potra–Pták’s method with optimal fourth and eighth orders of convergence. Journal of Computational and Applied Mathematics, 234(10), 2969-2976. doi:10.1016/j.cam.2010.04.009Wang, X., & Liu, L. (2010). New eighth-order iterative methods for solving nonlinear equations. Journal of Computational and Applied Mathematics, 234(5), 1611-1620. doi:10.1016/j.cam.2010.03.002Neta, B., & Petković, M. S. (2010). Construction of optimal order nonlinear solvers using inverse interpolation. Applied Mathematics and Computation, 217(6), 2448-2455. doi:10.1016/j.amc.2010.07.045Fidkowski, K. J., Oliver, T. A., Lu, J., & Darmofal, D. L. (2005). p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. Journal of Computational Physics, 207(1), 92-113. doi:10.1016/j.jcp.2005.01.005Amat, S., & Busquier, S. (Eds.). (2016). Advances in Iterative Methods for Nonlinear Equations. SEMA SIMAI Springer Series. doi:10.1007/978-3-319-39228-8Bruns, D. D., & Bailey, J. E. (1977). Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chemical Engineering Science, 32(3), 257-264. doi:10.1016/0009-2509(77)80203-0He, Y., & Ding, C. H. Q. (2001). The Journal of Supercomputing, 18(3), 259-277. doi:10.1023/a:1008153532043Revol, N., & Rouillier, F. (2005). Motivations for an Arbitrary Precision Interval Arithmetic and the MPFI Library. Reliable Computing, 11(4), 275-290. doi:10.1007/s11155-005-6891-yKing, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Maheshwari, A. K. (2009). A fourth order iterative method for solving nonlinear equations. Applied Mathematics and Computation, 211(2), 383-391. doi:10.1016/j.amc.2009.01.047Weerakoon, S., & Fernando, T. G. I. (2000). A variant of Newton’s method with accelerated third-order convergence. Applied Mathematics Letters, 13(8), 87-93. doi:10.1016/s0893-9659(00)00100-2Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.06

Solving nonlinear problems by Ostrowski Chun type parametric families

In this paper, by using a generalization of Ostrowski' and Chun's methods two bi-parametric families of predictor-corrector iterative schemes, with order of convergence four for solving system of nonlinear equations, are presented. The predictor of the first family is Newton's method, and the predictor of the second one is Steffensen's scheme. One of them is extended to the multidimensional case. Some numerical tests are performed to compare proposed methods with existing ones and to confirm the theoretical results. We check the obtained results by solving the molecular interaction problem.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT, Republica Dominicana.Cordero Barbero, A.; Maimo, J.; Torregrosa Sánchez, JR.; Vassileva, M. (2015). Solving nonlinear problems by Ostrowski Chun type parametric families. Journal of Mathematical Chemistry. 53(1):430-449. https://doi.org/10.1007/s10910-014-0432-zS430449531M.S. Petkovic̀, B. Neta, L.D. Petkovic̀, J. Dz̆unic̀, Multipoint Methods for Solving Nonlinear Equations (Academic, New York, 2013)M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry. J. Math. Chem. 51(9), 2361–2385 (2013)P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: Formalism and first application to atomic problems. J. Math. Chem. 22, 107–116 (1997)C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. J. Math. Chem. 49, 1384–1415 (2011)K. Maleknejad, M. Alizadeh, An efficient numerical scheme for solving hammerstein integral equation arisen in chemical phenomenon. Procedia Comput. Sci. 3, 361–364 (2011)R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem. 52, 255–267 (2014)J.F. Steffensen, Remarks on iteration. Skand. Aktuar Tidskr. 16, 64–72 (1933)J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970)H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)J.R. Sharma, R.K. Guha, R. Sharma, An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algorithms 62, 307–323 (2013)J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)M. Abad, A. Cordero, J.R. Torregrosa, Fourth- and fifth-order methods for solving nonlinear systems of equations: an application to the Global positioning system. Abstr. Appl. Anal.(2013) Article ID:586708. doi: 10.1155/2013/586708F. Soleymani, T. Lotfi, P. Bakhtiari, A multi-step class of iterative methods for nonlinear systems. Optim. Lett. 8, 1001–1015 (2014)M.T. Darvishi, N. Darvishi, SOR-Steffensen-Newton method to solve systems of nonlinear equations. Appl. Math. 2(2), 21–27 (2012). doi: 10.5923/j.am.20120202.05F. Awawdeh, On new iterative method for solving systems of nonlinear equations. Numer. Algorithms 5(3), 395–409 (2010)D.K.R. Babajee, A. Cordero, F. Soleymani, J.R. Torregrosa, On a novel fourth-order algorithm for solving systems of nonlinear equations. J. Appl. Math. (2012) Article ID:165452. doi: 10.1155/2012/165452A. Cordero, J.R. Torregrosa, M.P. Vassileva, Pseudocomposition: a technique to design predictor–corrector methods for systems of nonlinear equations. Appl. Math. Comput. 218(23), 1496–1504 (2012)A. Cordero, J.R. Torregrosa, M.P. Vassileva, Increasing the order of convergence of iterative schemes for solving nonlinear systems. J. Comput. Appl. Math. 252, 86–94 (2013)A.M. Ostrowski, Solution of Equations and System of Equations (Academic, New York, 1966)C. Chun, Construction of Newton-like iterative methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)R. King, A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)A. Cordero, J.R. Torregrosa, Low-complexity root-finding iteration functions with no derivatives of any order of convergence. J. Comput. Appl. Math. (2014). doi: 10.1016/j.cam.2014.01.024A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton Jarratts composition. Numer. Algorithms 55, 87–99 (2010)P. Jarratt, Some fourth order multipoint methods for solving equations. Math. Comput. 20, 434–437 (1966)A. Cordero, J.R. Torregrosa, Variants of Newtons method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)Z. Liu, Q. Zheng, P. Zhao, A variant of Steffensens method of fourth-order convergence and its applications. Appl. Math. Comput. 216, 1978–1983 (2010)A. Cordero, J.R. Torregrosa, A class of Steffensen type methods with optimal order of convergence. Appl. Math. Comput. 217, 7653–7659 (2011)L.B. Rall, New York, Computational Solution of Nonlinear Operator Equations (Robert E. Krieger Publishing Company Inc, New York, 1969

The Sixth Copper Mountain Conference on Multigrid Methods, part 1

The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth