On Extension of MSP Iterative Scheme
Abstract
In this paper, we modify the iterative method for solving nonlinear equations which is based on the idea of the fixed point iteration, namely MSP-iteration. The motivation is to simplify the computation via reducing the number of function evaluations and avoiding the derivative of the function. We propose two methods and illustrate the numerical results with several examples from the references for solving nonlinear equations. The results indicate that our proposed methods provide the good performance in the case iteration counting compared with SP-iteration and MSP-iteration. Keywords : Nonlinear equations, iterative method, fixed point iteration, SP-iteration, MSP-iterationReferences
Borwein D., & Borwein, J. (1991). Fixed point iterations for real functions, J. Math. Anal. Appl., 157, 112–126.
Babu, G.V., & Prasad, K.N. (2006). Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators. Fixed Point Theory Appl., 49615, 1–6.
Chapra, S. C., & Canale, R. P. (2010). Numerical Methods for Engineers, 6th edition, New York: McGraw-Hill.
Chugh, R., & Kumar, V. (2011). Strong Convergence of SP Iterative Scheme for Quasi-Contractive Operators, Int. J. Comput. Appl., 31(5), 21-27.
Frontini, M., & Sormani, E. (2003). Some variant of Newton’s method with third-order convergence. Appl. Math. Comput., 140, 419-426.
Ibrahim, K., & Murat, O. (2013). A general iterative method for approximation of fixed points and their applications, Advances in Fixed Point Theory., 3, 510-526.
Kang, S.M., Rafiq, A., & Kwun, Y. C. (2013). A new second-order iteration method for solving nonlinear equations, Abstr. Appl. Anal. 2013 Article ID 487062, 1-4.
Kung, H., & Traub, J. (1974). Optimal order of one point and multipoint iteration, ACM., 21, 643-651.
Makaje, N., & Phon-On, A. (2016). A modified SP-iterative scheme for solving nonlinear equations. Far East J. Math. Sci., 99(7), 1021-1036.
Maheshwari, A.K. (2009). A fourth-order iterative method for solving nonlinear equations. Appl. Math. Comput., 211, 383-391.
Noor, M.A. (2000). New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251, 217–229.
Phuengrattana, W., & Suantai, S. (2011). On the rate of convergence of Mann, Ishikawa, Noor and SP iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math., 235, 3006-3014.
Rafiq, A. (2006). On the convergence of the three-step iteration process in the class of quasi-contractive operators. Acta Math. Acad. Paedagog. Nyhazi., 22, 305-309.
Wang, P. (2011). A third –order family of Newton-like iteration methods for solving nonlinear equations. J. Num. Math. Stochast., 3, 13-19.
Weerakoon, S., & Fernando, T.G.I. (2000). A variant of Newton’s method with accelerated third order convergence. Appl. Math. Lett., 13, 87-93.
Babu, G.V., & Prasad, K.N. (2006). Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators. Fixed Point Theory Appl., 49615, 1–6.
Chapra, S. C., & Canale, R. P. (2010). Numerical Methods for Engineers, 6th edition, New York: McGraw-Hill.
Chugh, R., & Kumar, V. (2011). Strong Convergence of SP Iterative Scheme for Quasi-Contractive Operators, Int. J. Comput. Appl., 31(5), 21-27.
Frontini, M., & Sormani, E. (2003). Some variant of Newton’s method with third-order convergence. Appl. Math. Comput., 140, 419-426.
Ibrahim, K., & Murat, O. (2013). A general iterative method for approximation of fixed points and their applications, Advances in Fixed Point Theory., 3, 510-526.
Kang, S.M., Rafiq, A., & Kwun, Y. C. (2013). A new second-order iteration method for solving nonlinear equations, Abstr. Appl. Anal. 2013 Article ID 487062, 1-4.
Kung, H., & Traub, J. (1974). Optimal order of one point and multipoint iteration, ACM., 21, 643-651.
Makaje, N., & Phon-On, A. (2016). A modified SP-iterative scheme for solving nonlinear equations. Far East J. Math. Sci., 99(7), 1021-1036.
Maheshwari, A.K. (2009). A fourth-order iterative method for solving nonlinear equations. Appl. Math. Comput., 211, 383-391.
Noor, M.A. (2000). New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251, 217–229.
Phuengrattana, W., & Suantai, S. (2011). On the rate of convergence of Mann, Ishikawa, Noor and SP iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math., 235, 3006-3014.
Rafiq, A. (2006). On the convergence of the three-step iteration process in the class of quasi-contractive operators. Acta Math. Acad. Paedagog. Nyhazi., 22, 305-309.
Wang, P. (2011). A third –order family of Newton-like iteration methods for solving nonlinear equations. J. Num. Math. Stochast., 3, 13-19.
Weerakoon, S., & Fernando, T.G.I. (2000). A variant of Newton’s method with accelerated third order convergence. Appl. Math. Lett., 13, 87-93.
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2019-05-23
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