Journal of applied mathematics & informatics

한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
권호 표지

VARIANTS OF NEWTON'S METHOD USING FIFTH-ORDER QUADRATURE FORMULAS: REVISITED

  • Noor, Muhammad Aslam (Mathematics Department, COMSATS Institute of Information Technology) ;
  • Waseem, Muhammad (Mathematics Department, COMSATS Institute of Information Technology)
  • 발행 : 2009.09.30
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초록

In this paper, we point out some errors in a recent paper by Cordero and Torregrosa [7]. We prove the convergence of the variants of Newton's method for solving the system of nonlinear equations using two different approaches. Several examples are given, which illustrate the cubic convergence of these methods and verify the theoretical results.

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참고문헌

  1. S. Abbasbandy, Extended Newton's method for a system of nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 170 (2005) 648-656. https://doi.org/10.1016/j.amc.2004.12.048
  2. D. K. R. Babajee and M. Z. Dauhoo, Analysis of the properties of the variants of Newton's method with third order convergence, Appl. Math. Comput. 183 (2006) 659-684. https://doi.org/10.1016/j.amc.2006.05.116
  3. D. K. R. Babajee, M. Z. Dauhoo, M. T. Darvishi and A. Barati, A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule, Appl. Math. Comput. 200 (2008) 452-458. https://doi.org/10.1016/j.amc.2007.11.009
  4. E. Babolian, J. Biazar and A.R. Vahidi, Solution of a system of nonlinear equations by Adomian decomposition method, Appl. Math. Comput. 150 (2004) 847-854. https://doi.org/10.1016/S0096-3003(03)00313-8
  5. R. L. Burden and J. D. Faires, Numerical Analysis, 7th ed., PWS Publishing Company, Boston, 2001.
  6. A. Cordero and J. R. Torregrosa, Variants of Newton's method for functions of several variables, Appl. Math. Comput. 183 (2006) 199-208. https://doi.org/10.1016/j.amc.2006.05.062
  7. A. Cordero and J. R. Torregrosa, Variants of Newton's method using fifth-order quadrature formulas, Appl. Math. Comput. 190 (2007) 686-698. https://doi.org/10.1016/j.amc.2007.01.062
  8. M. T. Darvishi and A. Barati, A third-order Newton-type method to solve systems of nonlinear equations, Appl. Math. Comput. 187 (2007) 630-635. https://doi.org/10.1016/j.amc.2006.08.080
  9. M. T. Darvishi and A. Barati, A forth-order method from quadrature formulas to solve systems of nonlinear equations, Appl. Math. Comput. 188 (2007) 257-261. https://doi.org/10.1016/j.amc.2006.09.115
  10. M. T. Darvishi and A. Barati, Super cubic iterative methods to solve systems of nonlinear equations, Appl. Math. Comput. 188 (2007) 1678-1685. https://doi.org/10.1016/j.amc.2006.11.022
  11. F. Freudensten and B. Roth, Numerical solution of systems of nonlinear equations, J. ACM 10 (1963) 550-556. https://doi.org/10.1145/321186.321200
  12. M. Frontini and E. Sormani, Third-order methods from quadrature formulas for solving systems of nonlinear equations, Appl. Math. Comput. 149 (2004) 771-782. https://doi.org/10.1016/S0096-3003(03)00178-4
  13. A. Golbabai and M. Javidi, A new family of iterative methods for solving system of nonlinear algebraic equations, Appl. Math. Comput. 190 (2007) 1717-1722. https://doi.org/10.1016/j.amc.2007.02.055
  14. H. H. H. Homeier, A modified Newton method with cubic convergence: the multivariate case, J. Comput. Appl. Math. 169 (2004) 161-169. https://doi.org/10.1016/j.cam.2003.12.041
  15. W. Haijun, New third-order method for solving systems of nonlinear equations, J. Numer. Algor. DOI 10.1007/s11075-008-9227-2.
  16. J. Kou, A third-order modification of Newton method for systems of non-linear equations, Appl. Math. Comput. 191 (2007) 117-121. https://doi.org/10.1016/j.amc.2007.02.030
  17. M. Aslam Noor, Numerical Analysis and Optimization, Lecture Notes, Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan, 2007/2008.
  18. M. Aslam Noor and M. Waseem, Some iterative methods for solving a system of nonlinear equations, Preprint, 2008.
  19. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, 1970.
  20. M. G. Sanchez, J. M. Peris and J. M. Gutierrez, Accelerated iterative methods for finding solutions of a system of nonlinear equations, Appl. Math. Comput. 190 (2007) 1815-1823. https://doi.org/10.1016/j.amc.2007.02.068
  21. R. S. Varga, Matrix Iterative Analysis, Springer, Berlin, 2000.
  22. S. Weerakoon and T. G. I. Fernando, A variant of Newton's method with accelerated thirdorder convergence, Appl. Math. Lett. 13 (2000) 87-93.