On some cubic convergence iterative formulae without derivatives for solving nonlinear equations

International Journal for Numerical Methods in Biomedical Engineering

Volumn 27, Issue 5, 2011, Pages 722-731

  Dehghan, Mehdi ^a   Hajarian, Masoud ^a  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Cubic convergence; Halleys method; Iterative method; Newton's method; Nonlinear equations

Indexed keywords

APPROXIMATE SOLUTION; CONVERGENCE ANALYSIS; CUBIC CONVERGENCE; DERIVATIVE-FREE; HALLEYS METHOD; ITERATIVE FORMULAS; NEWTON'S METHOD; NONLINEAR FUNCTIONS; NUMERICAL EXAMPLE; SCIENTIFIC COMPUTATION;

CONVERGENCE OF NUMERICAL METHODS; NEWTON-RAPHSON METHOD;

NONLINEAR EQUATIONS;

EID: 79955438766     PISSN: 20407939     EISSN: 20407947     Source Type: Journal    
DOI: 10.1002/cnm.1328     Document Type: Article

References (30)

1. Du H, Hu M, Xie J, Ling S. Control of an electrostrictive actuator using Newton's method. Precision Engineering 2005; 29:375-380.
2. Li ZH, Zhang HX. Study on gas kinetic unified algorithm for flows from rarefied transition to continuum. Journal of Computational Physics 2004; 193:708-738.
3. Wissink AM, Lyrintzis AS, Chronopoulos AT. Efficient iterative methods applied to the solution of transonic flows. Journal of Computational Physics 1996; 123:379-393.
4. Ortega JM, Rheinboldt WC. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press: New York, 1975.
5. Atkinson KE. An Introduction to Numerical Analysis (2nd edn). Wiley: New York, 1989.
6. Grau M, Noguera M. A variant of Cauchy's method with accelerated fifth-order convergence. Applied Mathematics Letters 2004; 17:509-517.
7. Yamamoto T. Historical development in convergence analysis for Newton's and Newton-like methods. Journal of Computational and Applied Mathematics 2000; 124:1-23.
8. Dehghan M, Hajarian M. Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations. Computational and Applied Mathematics, in press.
9. Kanwar V, Sharma JR, Mamta RK. A new family of Secant-like method with super-linear convergence. Applied Mathematics and Computation 2005; 171:104-107.
10. Alefeld G. On the convergence of Halley's method. American Mathematical Monthly 1981; 88:530-538.
11. Ostrowski AM. Solution of Equations in Euclidean and Banach Space. Academic Press: San Diego, 1973.
12. Traub JF. Iterative Methods for the Solution of Equations. Chelsea: New York, 1977.
13. Gutiérrez JM, Hernńndez MA. A family of Chebyshev-Halley type methods in Banach spaces. Bulletin of the Australian Mathematical Society 1997; 55:113-130.
14. Chun C. On the construction of iterative methods with at least cubic convergence. Applied Mathematics and Computation 2007; 189:1384-1392.
15. Chun C, Kim Y. Several new third-order iterative methods for solving nonlinear equations. Acta Applicandae Mathematicae 2008; DOI: 10.1007/s10440-008-9359-3.
16. Dehghan M, Hajarian M. New iterative method for solving non-linear equations with fourth-order convergence. International Journal of Computer Mathematics 2008; DOI: 10.1080/00207160802217201.
17. Grau M, Barrero JLD. An improvement of the Euler-Chebyshev iterative method. Journal of Mathematical Analysis and Applications 2006; 315:1-7.
18. Haijun W. On new third-order convergent iterative formulas. Numerical Algorithms 2008; 48:317-325.
19. Homeier HHH. On Newton-type methods with cubic convergence. Journal of Computational and Applied Mathematics 2005; 176:425-432.
20. Kanwar V, Tomar SK. Modified families of multi-point iterative methods for solving nonlinear equations. Numerical Algorithms 2007; 44:381-389.
21. Kanwar V, Tomar SK. Modified families of Newton, Halley and Chebyshev methods. Applied Mathematics and Computation 2007; 192:20-26.
22. Kou J, Li Y, Wang X. Modified Halley's method free from second derivative. Applied Mathematics and Computation 2006; 183:704-708.
23. Kasturiarachi AB. Leap frogging Newton's method. International Journal of Mathematical Education in Science and Technology 2002; 33:521-527.
24. Weerakoon S, Fernando TGI. A variant of Newton's method with accelerated third-order convergence. Applied Mathematics Letters 2000; 13:87-93.
25. Chen D, Argyros IK, Qian QS. A local convergence theorem for the Super-Halley method in a Banach space. Applied Mathematics Letters 1994; 7:49-52.
26. Gutiérrez JM, Hernńndez MA. An acceleration of Newton's method: super-Halley method. Applied Mathematics and Computation 2001; 117:223-239.
27. Ezquerro JA, Gutiérrez JM, Hernńndez MA, Salanova MA. Solving nonlinear integral equations arising in radiative transfer. Numerical Functional Analysis and Optimization 1999; 20:661-673.
28. Kou J, Li Y, Wang X. A variant of super-Halley method with accelerated fourth-order convergence. Applied Mathematics and Computation 2007; 186:535-539.
29. Kincaid D, Cheney W. Numerical Analysis (2nd edn). Brooks/Cole: Pacific Grove, CA, 1996.
30. Kou J. On Chebyshev-Halley methods with sixth-order convergence for solving non-linear equations. Applied Mathematics and Computation 2007; 190:126-131.