Fourth-order variants of Newton's method without second derivatives for solving non-linear equations

Engineering Computations (Swansea, Wales)

Volumn 29, Issue 4, 2012, Pages 356-365

  Dehghan, Mehdi ^a   Hajarian, Masoud ^b  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

^b SHAHID BEHESHTI UNIVERSITY   (Iran)

Author keywords

Fourth order convergence; Iterative methods; Mathematics; Newton's method; Order of convergence

Indexed keywords

CONVERGENCE RESULTS; DEFINITE INTEGRAL; DESIGN/METHODOLOGY/APPROACH; FOURTH-ORDER; ITERATIVE ALGORITHM; MODIFIED NEWTON'S METHOD; NEWTON'S METHODS; NUMERICAL EXAMPLE; NUMERICAL RESULTS; ORDER OF CONVERGENCE; SECOND DERIVATIVES; SECOND-DERIVATIVE-FREE; SIMPLE ROOT;

ALGORITHMS; ITERATIVE METHODS; MATHEMATICAL TECHNIQUES; NEWTON-RAPHSON METHOD; NUMERICAL METHODS;

LINEAR EQUATIONS;

EID: 84861796441     PISSN: 02644401     EISSN: None     Source Type: Journal    
DOI: 10.1108/02644401211227590     Document Type: Article

References (27)

1. Cordero, A. and Torregrosa, J.R. (2010), "On interpolation variants of Newton's method for functions of several variables", J. Comput. Appl. Math., Vol. 234, pp. 34-43.
2. Cordero, A. and Torregrosa, J.R. (2011), "A class of Steffensen type methods with optimal order of convergence", Appl. Math. Comput., Vol. 217, pp. 7653-9.
3. Cordero, A., Torregrosa, J.R. and Vassileva, M.P. (2011), "Three-step iterative methods with optimal eighth-order convergence", J. Comput. Appl. Math., Vol. 235, pp. 3189-94.
4. Cordero, A., Hueso, J.L., Martinez, E. and Torregrosa, J.R. (n.d.), "Steffensen type methods for solving nonlinear equations", J. Comput. Appl. Math., May (in press).
5. Dehghan, M. and Hajarian, M. (2010a), "New iterative method for solving non-linear equations with fourth-order convergence", Int. J. Comput. Math., Vol. 87, pp. 834-9.
6. Dehghan, M. and Hajarian, M. (2010b), "Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations", Comput. Appl. Math., Vol. 29, pp. 19-30.
7. Dehghan, M. and Hajarian, M. (2011a), "On derivative free cubic convergence iterative methods for solving nonlinear equations", Computational Mathematics and Mathematical Physics, Vol. 51, pp. 513-19.
8. Dehghan, M. and Hajarian, M. (2011b), "On some cubic convergence iterative formulae without derivatives for solving nonlinear equations", Int. J. Numer. Meth. Biomed. Eng., Vol. 27, pp. 722-31.
9. Ding, F. and Chen, T. (2005), "Gradient based iterative algorithms for solving a class of matrix equations", IEEE Trans. Autom. Contr., Vol. 50, pp. 1216-21.
10. Ding, F. and Chen, T. (2006), "On iterative solutions of general coupled matrix equations", SIAM J. Contr. Optim., Vol. 44, pp. 2269-84.
11. Ding, F., Liu, G. and Liu, X.P. (2010a), "Partially coupled stochastic gradient identification methods for nonuniformly sampled systems", IEEE Transactions on Automatic Control, Vol. 55 No. 8, pp. 1976-81.
12. Ding, F., Liu, P.X. and Ding, J. (2008), "Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle", Appl. Math. Comput., Vol. 197, pp. 41-50.
13. Ding, F., Liu, P.X. and Liu, G. (2010b), "Gradient based and least-squares based iterative identification methods for OE and OEMA systems", Digital Signal Processing, Vol. 20 No. 3, pp. 664-77.
14. Ding, F., Liu, P.X. and Liu, G. (2011), "Identification methods for Hammerstein nonlinear systems", Digital Signal Processing, Vol. 21 No. 2, pp. 215-38.
15. Ding, F., Shi, Y. and Chen, T. (2007), "Auxiliary model based least-squares identification methods for Hammerstein output-error systems", Systems & Control Letters, Vol. 56 No. 5, pp. 373-80.
16. Frontini, M. and Sormani, E. (2003), "Some variants of Newton's method with third-order convergence", Appl. Math. Comput., Vol. 140, pp. 419-26.
17. Han, L.L. and Ding, F. (2009), "Multi-innovation stochastic gradient algorithms for multi-input multi-output systems", Digital Signal Processing, Vol. 19 No. 4, pp. 545-54.
18. Kou, J. (2008), "Some variants of Cauchy's method with accelerated fourth-order convergence", J. Comput. Appl. Math., Vol. 213, pp. 71-8.
19. Kou, J. and Li, Y. (2007), "A family of modified super-Halley methods with fourth-order convergence", Appl. Math. Comput., Vol. 189, pp. 366-70.
20. Liu, Y.J., Xiao, Y.S. and Zhao, X.L. (2009), "Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model", Applied Mathematics and Computation, Vol. 215 No. 4, pp. 1477-83.
21. Liu, Y.J., Wang, D.Q. and Ding, F. (2010), "Least-squares based iterative algorithms for identifying Box-Jenkins models with finite measurement data", Digital Signal Processing, Vol. 20 No. 5, pp. 1458-67.
22. Margaris, A. and Goulianas, K. (n.d.), "Finding all roots of 2 £ 2 nonlinear algebraic systems using back-propagation neural networks", Neural Computing & Applications, November.
23. Rafiq, A., Awais, M. and Zafar, F. (2007), "Modified efficient variant of super-Halley method", Appl. Math. Comput., Vol. 189, pp. 2004-10.
24. Wang, D.Q. and Ding, F. (2011), "Least squares based and gradient based iterative identification for Wiener nonlinear systems", Signal Processing, Vol. 91 No. 5, pp. 1182-9.
25. Wang, D.Q., Yang, G.W. and Ding, R.F. (2010), "Gradient-based iterative parameter estimation for Box-Jenkins systems", Computersand Mathematics withApplications, Vol. 60 No. 5, pp. 1200-8.
26. Weerakoon, S. and Fernando, G.I. (2000), "A variant of Newton's method with accelerated third-order convergence", Appl. Math. Lett., Vol. 17, pp. 87-93.
27. Zhang, J.B., Ding, F. and Shi, Y. (2009), "Self-tuning control based on multi-innovation stochastic gradient parameter estimation", Systems & Control Letters, Vol. 58 No. 1, pp. 69-75.