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An analysis of the properties of the variants of Newton’s method with third order convergence. (English) Zbl 1123.65036
This is a nice summary of the third order iterative methods for solving nonsingular nonlinear equations $f(x)=0$. The authors formulate eight third order methods into a single frame $x_{k+1}=x_k- f(x_k)/D_m(x_k)$ with eight different denominators $D_m(x)$, $m=1, 2,\dots, 8$, approximating the Jacobian of $f$ from different perspectives. In addition to convergence rates, the information usage and efficiency, i.e., number of function and Jacobian values required per iteration and comparison among the convergence orders, are evaluated. It is shown that these third order methods are variations of the Halley method and are all contractive in the same neighbourhood. The extension of these methods to systems of equations is also discussed. Convergence rates and neighbourhoods are illustrated from numerical and geometrical perspectives by various examples.

##### MSC:
 65H05 Single nonlinear equations (numerical methods) 65H10 Systems of nonlinear equations (numerical methods)
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##### References:
 [1] M. Drexler, Newton method as a global solver for non-linear problems, Ph.D. thesis, University of Oxford, 1997. [2] Kasturiarachi, A. Bathi: Leap-frogging Newton method. Int. J. Math. educ. Sci. technol. 33, No. 4, 521-527 (2002) [3] Gander, W.: On halley’s iteration method. Amer. math. Monthly 92, 131-134 (1985) · Zbl 0574.65041 [4] Householder, A. S.: The numerical treatment of a single nonlinear equation. (1970) · Zbl 0242.65047 [5] Hernandez, M. A.: An acceleration procedure of the Whittaker method by means of convexity. Zb. rad. Prirod.-mat. Fer. sak. Mat. 20, 27-38 (1990) [6] Ezquerro, J. A.; Hernandez, M. A.: On a convex acceleration of Newton’s method. J. optim. Theory appl. 100, 311-326 (1999) · Zbl 0915.90235 [7] Gutierrez, J. M.; Hernandez, M. A.: A family of Chebyshev -- halley type methods in Banach spaces. Bull. austral. Math. soc. 55, 113-130 (1997) [8] Verona, Juan L.: Graphic and numerical comparison between iterative methods. The math. Intell. 24, No. 1, 37-46 (2002) · Zbl 1003.65046 [9] Traub, J. F.: Iterative methods for the solution of equations. (1976) · Zbl 0399.47055 [10] Weerakoon, S.; Fernando, T. G. I.: A variant of Newton’s method with accelerated third order convergence. Appl. math. Lett. 13, 87-93 (2000) · Zbl 0973.65037 [11] Ozban, A.: Some new variants of Newton’s method. Appl. math. Lett. 17, 677-682 (2004) · Zbl 1065.65067 [12] Hasanov, V. I.; Ivanov, I. G.; Nedzhibov, G.: A new modification of Newton method. Appl. math. Eng. 27, 278-286 (2002) [13] Nedzhibov, Gyurhan: On a few iterative methods for solving nonlinear equations. Application of mathematics in engineering and economics’28. Proceedings of the XXVIII summer school Sozopol 2002 (2002) · Zbl 1253.65078 [14] Frontini, M.; Sormani, E.: Some variant of Newton’s method with third-order convergence. Appl. math. Comput. 140, No. 2-3, 419-426 (2003) · Zbl 1037.65051 [15] Frontini, M.; Sormani, E.: Modified Newton’s method with third-order convergence and multiple roots. J. comput. Appl. math. 156, No. 2, 345-354 (2003) · Zbl 1030.65044 [16] Homeier, H. H. H.: A modified Newton method for rootfinding with cubic convergence. J. comput. Appl. math. 157, No. 1, 227-230 (2003) · Zbl 1070.65541 [17] Homeier, H. H. H.: A modified Newton method with cubic convergence: the multivariate case. J. comput. Appl. math. 169, No. 1, 161-169 (2004) · Zbl 1059.65044 [18] Homeier, H. H. H.: On Newton-type methods with cubic convergence. J. comput. Appl. math. 176, No. 2, 425-432 (2005) · Zbl 1063.65037 [19] Tibor Lukic, Nebojsa M. Ralevic, Newton method with accelerated convergence modified by an aggregation operator, in: 3rd Serbian -- Hungarian Joint Symposium on Intelligent Systems, SISY, 2005. [20] Kreyszig, Erwin: Introductory functional analysis with applications. (1989) · Zbl 0706.46001 [21] Taylor, A. E.: Introduction to functional analysis. (1957) · Zbl 0077.18503 [22] Ortega, J. M.; Reinbolt, W. C.: Iterative solution of nonlinear equations in several variables. (1970) [23] Z. Gajic, S. Koskie, Newton iteration acceleration of the Nash game algorithm for power control in 3g wireless cdma networks, in: Proceedings of ITCOM 2003, 2003, pp. 115 -- 121. [24] Koskie, S.; Gajic, Z.: A Nash game algorithm for SIR-based power control in 3G wireless CDMA networks. IEEE/ACM trans. Network. 13, No. 5, 1017-1026 (2005) [25] S. Koskie, Contributions to dynamic Nash games and applications to power control for wireless networks, Ph.D. thesis, Univ. of Rutgers, 2003.