Families of Iterative Methods with Higher Convergence Orders for Solving Nonlinear Equations
Keywords:
Nonlinear Equations, Iterative Method, Newton's Method, Order of Convergence.
Abstract
Recently, there has been progress in developing Newton-type methods with higher convergence to solve nonlinear equations. This paper develops new classes of iterative methods with higher convergence orders (more than four) by a construction of two iterative methods of integer-order of convergence, π, π and π > π. The construction of such methods provides new classes of methods of order 2π + π of convergence. Numerical examples are provided, as well as the use of other existing methods to demonstrate the performance of the presented methods.
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References
[1] Al-subaihi, IA (2014). Optimal Fourth-Order Iterative Methods for Solving Nonlinear Equations, IJMTT, 13(1), 13-18.
[2] Al-subaihi, IA, & Alqarni, Awatif J (2014). Higher-Order Iterative Methods for Solving Nonlinear Equations, Life Science Journal, 11(2), 85-91.
[3] Al-subaihi, IA. & Siyyam, Hani I (2014). Efficient Techniques for Constructing Optimal Iterative Methods with Eighth and Sixteenth Order of Convergence for Solving Nonlinear Equations, SYLWAN Journal, 158( 5).
[4] Bi, WH, Ren, HM, Wu, QB (2009). Three-step iterative methods with eighth-order convergence for solving nonlinear equations, Journal of Computational and Applied Mathematics, 225(1), 105β112.
[5] Chun, C (2008). Some fourth-order iterative methods for solving nonlinear equations, Appl. Math. Comute. 195, 454β459.
[6] Cordero, A, Hueso, J. L. Martines, E & Torregrosa JR (2010). A family of iterative methods with sixth and seventh order convergence for nonlinear equations, Math. Comput. Model, 52, 1490β1496.
[7] Fang, L & He, GP (2009). Some modifications of Newtonβs method with higher-order convergence for solving nonlinear equations, Journal of computational and Applied Mathematics, 228(1), 296β303.
[8] Heydari, M, Hosseini, S & Loghmani, G (2011). On two families of iterative methods for solving nonlinear equations with optimal order, Appl. Anal. Discrete Math. 5, 93β109.
[9] R. King, A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10 (1973) 876β879.
[10] Kou, J, Li, Y & Wang, X (2007) . A composite fourth-order iterative method for solving non-linear equattions, Applied Mathematics and Computation, 184, 471β475.
[11] LI, Z, Peng, C, Zhou , T & Gao, J (2011). A New Newton-type Method for Solving Nonlinear Equations with Any Integer Order of Convergence, Journal of Computational Information Systems, 7(7), 2371β2378.
[12] Li, XW, Mu, CL, Ma, JW & Wang, C (2010) . Sixteenth-order method for nonlinear equations, Applied Mathematics and Computation, 215(10), 3754β3758.
[13] Liu , LP & Wang, X (2010) . Eighth-order methods with high efficiency index for solving nonlinear equations, Applied Mathematics and Computation, 215(9) ,3449β3454.
Volume 7, Issue 4 available at www.scitecresearch.com/journals/index.php/jprm 1139|
[14] Petkovic, S. Miodrag (2009)On Optimal Multipoint Methods for Solving Nonlinear Equations, Novi Sad J. Math. 39(1), 123β130.
[15] Sharma, JR & Sharma, R (2010). A new family of modifed Ostrowskiβs methods with accelerated eighth order convergence, Numer Algor. 54, 445β458.
[16] Siyyam, H I, Shatnawi, Mohd Taib & Al-Subaihi, IA (2013). A new one parameter family of iterative methods with eighth-order of convergence for solving nonlinear equations. International Journal of Pure and Applied Mathematics, 84(5), 451β461.
[17] Traub, JF, 1964. Iterative Methods for the Solution of Equations, Prentice Hall, New York.
[18] Weerakoon, S & Fernando, G (2000). A variant of Newtonβs method with accelerated third-order convergence, Appl. Math. Lett. 17(8), 87β93.
[2] Al-subaihi, IA, & Alqarni, Awatif J (2014). Higher-Order Iterative Methods for Solving Nonlinear Equations, Life Science Journal, 11(2), 85-91.
[3] Al-subaihi, IA. & Siyyam, Hani I (2014). Efficient Techniques for Constructing Optimal Iterative Methods with Eighth and Sixteenth Order of Convergence for Solving Nonlinear Equations, SYLWAN Journal, 158( 5).
[4] Bi, WH, Ren, HM, Wu, QB (2009). Three-step iterative methods with eighth-order convergence for solving nonlinear equations, Journal of Computational and Applied Mathematics, 225(1), 105β112.
[5] Chun, C (2008). Some fourth-order iterative methods for solving nonlinear equations, Appl. Math. Comute. 195, 454β459.
[6] Cordero, A, Hueso, J. L. Martines, E & Torregrosa JR (2010). A family of iterative methods with sixth and seventh order convergence for nonlinear equations, Math. Comput. Model, 52, 1490β1496.
[7] Fang, L & He, GP (2009). Some modifications of Newtonβs method with higher-order convergence for solving nonlinear equations, Journal of computational and Applied Mathematics, 228(1), 296β303.
[8] Heydari, M, Hosseini, S & Loghmani, G (2011). On two families of iterative methods for solving nonlinear equations with optimal order, Appl. Anal. Discrete Math. 5, 93β109.
[9] R. King, A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10 (1973) 876β879.
[10] Kou, J, Li, Y & Wang, X (2007) . A composite fourth-order iterative method for solving non-linear equattions, Applied Mathematics and Computation, 184, 471β475.
[11] LI, Z, Peng, C, Zhou , T & Gao, J (2011). A New Newton-type Method for Solving Nonlinear Equations with Any Integer Order of Convergence, Journal of Computational Information Systems, 7(7), 2371β2378.
[12] Li, XW, Mu, CL, Ma, JW & Wang, C (2010) . Sixteenth-order method for nonlinear equations, Applied Mathematics and Computation, 215(10), 3754β3758.
[13] Liu , LP & Wang, X (2010) . Eighth-order methods with high efficiency index for solving nonlinear equations, Applied Mathematics and Computation, 215(9) ,3449β3454.
Volume 7, Issue 4 available at www.scitecresearch.com/journals/index.php/jprm 1139|
[14] Petkovic, S. Miodrag (2009)On Optimal Multipoint Methods for Solving Nonlinear Equations, Novi Sad J. Math. 39(1), 123β130.
[15] Sharma, JR & Sharma, R (2010). A new family of modifed Ostrowskiβs methods with accelerated eighth order convergence, Numer Algor. 54, 445β458.
[16] Siyyam, H I, Shatnawi, Mohd Taib & Al-Subaihi, IA (2013). A new one parameter family of iterative methods with eighth-order of convergence for solving nonlinear equations. International Journal of Pure and Applied Mathematics, 84(5), 451β461.
[17] Traub, JF, 1964. Iterative Methods for the Solution of Equations, Prentice Hall, New York.
[18] Weerakoon, S & Fernando, G (2000). A variant of Newtonβs method with accelerated third-order convergence, Appl. Math. Lett. 17(8), 87β93.
Published
2016-05-29
How to Cite
Al-Subaihi, I. (2016). Families of Iterative Methods with Higher Convergence Orders for Solving Nonlinear Equations. Journal of Progressive Research in Mathematics, 7(4), 1129-1141. Retrieved from https://scitecresearch.com/journals/index.php/jprm/article/view/771
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