MODIFIKASI METODE HANSEN-PATRICK DENGAN ORDE KONVERGENSI OPTIMAL UNTUK MENYELESAIKAN PERSAMAAN NONLINIER

The Hansen-Patrick method is a third-order  iterative  method used to solve nonlinear equation. The method requires three evaluation of functions and has an efficiency index 31/3 » 1,4224. This study discusses a modification of the Hansen-Patrick method using the second order Taylor series. The second derivative is reduced using hyperbolic function with one parameter h. The aim of modification is to improve the convergence order of the Hansen-Patrick’s method. Based on the convergence analysis, the method has a fourth-order of convergence and envolve three evaluation of functions. So, its efficiency index is 41/3 » 1,5874. Numerical simulation is given to illustrate performance  of the iterative method using six real functions. The performance of the iterative method include : a computational order of convergence, the number of iteration, evaluation of function, absolute error, and value of function, will be compared with Newton’s method, Halley’s method, Newton-Steffensen’s method, and Hansen-Patrick method. The numerical simulation shows that the performance of the method better than other

Modifikasi Metode Iterasi Dua Langkah Menggunakan Kombinasi Linear Tiga Parameter Real

Makalah ini membahas modifikasi  metode iterasi dua langkah dengan menggunakan kombinasi linier tiga parameter dan tiga metode iterasi berorde konvergensi tiga yang masing-masing dihasilkan dari penjumlahan metode Potra-Ptak dan metode varian Newton, modifikasi metode varian Newton  rata-rata kontra harmonik, dan Metode Newton-Steffensen. Berdasarkan hasil kajian diperoleh bahwa metode iterasi baru memiliki orde konvergensi empat untuk q 1 = -2, q 2 = 3 - q 3 dan q3 ÎÂ yang melibatkan tiga evaluasi fungsi dengan indeks efisiensi sebesar 41/3 » 1,5874. Simulasi numerik diberikan untuk menunjukkan performa metode iterasi baru dibandingkan dengan metode Newton, metode Potra-Ptak, dan metode Chebyshe

A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots

[EN] The aim of this paper is to introduce new high order iterative methods for multiple roots of the nonlinear scalar equation; this is a demanding task in the area of computational mathematics and numerical analysis. Specifically, we present a new Chebyshev¿Halley-type iteration function having at least sixth-order convergence and eighth-order convergence for a particular value in the case of multiple roots. With regard to computational cost, each member of our scheme needs four functional evaluations each step. Therefore, the maximum efficiency index of our scheme is 1.6818 for ¿ = 2,which corresponds to an optimal method in the sense of Kung and Traub¿s conjecture. We obtain the theoretical convergence order by using Taylor developments. Finally, we consider some real-life situations for establishing some numerical experiments to corroborate the theoretical results.This research was partially supported by Ministerio de Economia y Competitividad under Grant MTM2014-52016-C2-1-2-P and by the project of Generalitat Valenciana Prometeo/2016/089Behl, R.; Martínez Molada, E.; Cevallos-Alarcon, FA.; Alarcon-Correa, D. (2019). A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots. Mathematics. 7(4):1-12. https://doi.org/10.3390/math7040339S11274Gutiérrez, J. M., & Hernández, M. A. (1997). A family of Chebyshev-Halley type methods in Banach spaces. Bulletin of the Australian Mathematical Society, 55(1), 113-130. doi:10.1017/s0004972700030586Kanwar, V., Singh, S., & Bakshi, S. (2008). Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations. Numerical Algorithms, 47(1), 95-107. doi:10.1007/s11075-007-9149-4Argyros, I. K., Ezquerro, J. A., Gutiérrez, J. M., Hernández, M. A., & Hilout, S. (2011). On the semilocal convergence of efficient Chebyshev–Secant-type methods. Journal of Computational and Applied Mathematics, 235(10), 3195-3206. doi:10.1016/j.cam.2011.01.005Xiaojian, Z. (2008). Modified Chebyshev–Halley methods free from second derivative. Applied Mathematics and Computation, 203(2), 824-827. doi:10.1016/j.amc.2008.05.092Amat, S., Hernández, M. A., & Romero, N. (2008). A modified Chebyshev’s iterative method with at least sixth order of convergence. Applied Mathematics and Computation, 206(1), 164-174. doi:10.1016/j.amc.2008.08.050Kou, J., & Li, Y. (2007). Modified Chebyshev–Halley methods with sixth-order convergence. Applied Mathematics and Computation, 188(1), 681-685. doi:10.1016/j.amc.2006.10.018Li, D., Liu, P., & Kou, J. (2014). An improvement of Chebyshev–Halley methods free from second derivative. Applied Mathematics and Computation, 235, 221-225. doi:10.1016/j.amc.2014.02.083Sharma, J. R. (2015). Improved Chebyshev–Halley methods with sixth and eighth order convergence. Applied Mathematics and Computation, 256, 119-124. doi:10.1016/j.amc.2015.01.002Neta, B. (2010). Extension of Murakami’s high-order non-linear solver to multiple roots. International Journal of Computer Mathematics, 87(5), 1023-1031. doi:10.1080/00207160802272263Zhou, X., Chen, X., & Song, Y. (2011). Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. Journal of Computational and Applied Mathematics, 235(14), 4199-4206. doi:10.1016/j.cam.2011.03.014Hueso, J. L., Martínez, E., & Teruel, C. (2014). Determination of multiple roots of nonlinear equations and applications. Journal of Mathematical Chemistry, 53(3), 880-892. doi:10.1007/s10910-014-0460-8Behl, R., Cordero, A., Motsa, S. S., & Torregrosa, J. R. (2015). On developing fourth-order optimal families of methods for multiple roots and their dynamics. Applied Mathematics and Computation, 265, 520-532. doi:10.1016/j.amc.2015.05.004Behl, R., Cordero, A., Motsa, S. S., Torregrosa, J. R., & Kanwar, V. (2015). An optimal fourth-order family of methods for multiple roots and its dynamics. Numerical Algorithms, 71(4), 775-796. doi:10.1007/s11075-015-0023-5Geum, Y. H., Kim, Y. I., & Neta, B. (2015). A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics. Applied Mathematics and Computation, 270, 387-400. doi:10.1016/j.amc.2015.08.039Geum, Y. H., Kim, Y. I., & Neta, B. (2016). A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Applied Mathematics and Computation, 283, 120-140. doi:10.1016/j.amc.2016.02.029Behl, R., Alshomrani, A. S., & Motsa, S. S. (2018). An optimal scheme for multiple roots of nonlinear equations with eighth-order convergence. Journal of Mathematical Chemistry, 56(7), 2069-2084. doi:10.1007/s10910-018-0857-xMcNamee, J. M. (1998). A comparison of methods for accelerating convergence of Newton’s method for multiple polynomial roots. ACM SIGNUM Newsletter, 33(2), 17-22. doi:10.1145/290590.290592Cordero, 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

Composite materials in compression

A geometric and material non-linear finite element code is built using Matlab. The theoretical derivations for building the code is outlined and explained by pseudo code. Three different solvers are introduced; the Newton Raphson method, the modified Newton Raphson method, and the arc length method. The code is tested for material non-linearity an geometric non-linearity separately using standard reference solutions. The future work is outlined as a continuity of this report

Basins of attraction for various Steffensen-Type methods

The dynamical behavior of different Steffensen-type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen-type methods, that is, derivative-free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen-type methods are improved by increasing the convergence order. The computer programming package MATHEMATICA provides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.The authors are indebted to the referees for some interesting comments and suggestions. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Cordero Barbero, A.; Soleymani, F.; Torregrosa Sánchez, JR.; Shateyi, S. (2014). Basins of attraction for various Steffensen-Type methods. Journal of Applied Mathematics. 2014. https://doi.org/10.1155/2014/539707S2014Soleymani, F. (2011). Optimal fourth-order iterative method free from derivative. Miskolc Mathematical Notes, 12(2), 255. doi:10.18514/mmn.2011.303Zheng, Q., Zhao, P., Zhang, L., & Ma, W. (2010). Variants of Steffensen-secant method and applications. Applied Mathematics and Computation, 216(12), 3486-3496. doi:10.1016/j.amc.2010.04.058Neta, B., Scott, M., & Chun, C. (2012). Basins of attraction for several methods to find simple roots of nonlinear equations. Applied Mathematics and Computation, 218(21), 10548-10556. doi:10.1016/j.amc.2012.04.017Neta, B., & Scott, M. (2013). On a family of Halley-like methods to find simple roots of nonlinear equations. Applied Mathematics and Computation, 219(15), 7940-7944. doi:10.1016/j.amc.2013.02.035Neta, B., & Chun, C. (2013). On a family of Laguerre methods to find multiple roots of nonlinear equations. Applied Mathematics and Computation, 219(23), 10987-11004. doi:10.1016/j.amc.2013.05.002Neta, B., Chun, C., & Scott, M. (2014). Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations. Applied Mathematics and Computation, 227, 567-592. doi:10.1016/j.amc.2013.11.017Amat, S., Busquier, S., & Plaza, S. (2005). Dynamics of the King and Jarratt iterations. Aequationes mathematicae, 69(3), 212-223. doi:10.1007/s00010-004-2733-yChicharro, F., Cordero, A., Gutiérrez, J. M., & Torregrosa, J. R. (2013). Complex dynamics of derivative-free methods for nonlinear equations. Applied Mathematics and Computation, 219(12), 7023-7035. doi:10.1016/j.amc.2012.12.075Cordero, A., García-Maimó, J., Torregrosa, J. R., Vassileva, M. P., & Vindel, P. (2013). Chaos in King’s iterative family. Applied Mathematics Letters, 26(8), 842-848. doi:10.1016/j.aml.2013.03.012Chun, C., Lee, M. Y., Neta, B., & Džunić, J. (2012). On optimal fourth-order iterative methods free from second derivative and their dynamics. Applied Mathematics and Computation, 218(11), 6427-6438. doi:10.1016/j.amc.2011.12.013Cordero, A., Torregrosa, J. R., & Vindel, P. (2013). Dynamics of a family of Chebyshev–Halley type methods. Applied Mathematics and Computation, 219(16), 8568-8583. doi:10.1016/j.amc.2013.02.042Soleimani, F., Soleymani, F., & Shateyi, S. (2013). Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations. Discrete Dynamics in Nature and Society, 2013, 1-10. doi:10.1155/2013/301718Susanto, H., & Karjanto, N. (2009). Newton’s method’s basins of attraction revisited. Applied Mathematics and Computation, 215(3), 1084-1090. doi:10.1016/j.amc.2009.06.041Vrscay, E. R., & Gilbert, W. J. (1987). Extraneous fixed points, basin boundaries and chaotic dynamics for Schr�der and K�nig rational iteration functions. Numerische Mathematik, 52(1), 1-16. doi:10.1007/bf01401018Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-142. doi:10.1090/s0273-0979-1984-15240-6Varona, J. L. (2002). Graphic and numerical comparison between iterative methods. The Mathematical Intelligencer, 24(1), 37-46. doi:10.1007/bf03025310Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860McMullen, C. (1987). Families of Rational Maps and Iterative Root-Finding Algorithms. The Annals of Mathematics, 125(3), 467. doi:10.2307/1971408Smale, S. (1985). On the efficiency of algorithms of analysis. Bulletin of the American Mathematical Society, 13(2), 87-122. doi:10.1090/s0273-0979-1985-15391-1Liu, Z., Zheng, Q., & Zhao, P. (2010). A variant of Steffensen’s method of fourth-order convergence and its applications. Applied Mathematics and Computation, 216(7), 1978-1983. doi:10.1016/j.amc.2010.03.028Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2012). A Family of Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations. Abstract and Applied Analysis, 2012, 1-15. doi:10.1155/2012/836901Zheng, Q., Li, J., & Huang, F. (2011). An optimal Steffensen-type family for solving nonlinear equations. Applied Mathematics and Computation, 217(23), 9592-9597. doi:10.1016/j.amc.2011.04.035Soleymani, F., Karimi Vanani, S., & Jamali Paghaleh, M. (2012). A Class of Three-Step Derivative-Free Root Solvers with Optimal Convergence Order. Journal of Applied Mathematics, 2012, 1-15. doi:10.1155/2012/56874

Newton and numerical mathematics

Tématem bakalářské práce jsou Newtonovy metody pro numerické řešení různých problémů. Zejména je vysvětlena problematika řešení nelineárních rovnic, soustav nelineárních rovnic a numerický výpočet integrálů. Je předvedena Newtonova metoda pro řešení nelineárních rovnic a mnohé její modifikace a také její zobecnění pro soustavy nelineárních rovnic. Užitečnost metod je demonstrována na různých příkladech. Na závěr jsou uvedeny Newton-Cotesovy kvadraturní formule pro numerické integrování.Topic of this bachelor thesis are Newton's methods for numerical solutions of various problems. Especially the problems of solving nonlinear equations and systems of nonlinear equations, as well as numerical integration are explained. The Newton's method for solving nonlinear equations is presented, as well as its many modifications and its generalisation for systems of nonlinear equations. Usefulness of methods is demonstrated on various examples. In the end, Newton-Cotes quadrature formulae for numerical integration are presented.

Steffensen's method and Steffensen type methods

metody Steffensenova typu, nelineární rovnice, Newtonova metoda, Steffensenova metodaSteffensen type methods, nonlinear equations, Newton's method, Steffensen's method