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.
Reviewer: Zhen Mei (Toronto)


65H05 Numerical computation of solutions to single equations
65H10 Numerical computation of solutions to systems of equations
Full Text: DOI


[1] M. Drexler, Newton method as a global solver for non-linear problems, Ph.D. thesis, University of Oxford, 1997.
[2] Bathi Kasturiarachi, A., Leap-frogging Newton method, Int. J. math. educ. sci. technol., 33, 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), McGraw-Hill New York · 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) · Zbl 0748.65044
[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) · Zbl 0893.47043
[8] Verona, Juan L., Graphic and numerical comparison between iterative methods, The math. intell., 24, 1, 37-46, (2002) · Zbl 1003.65046
[9] Traub, J.F., Iterative methods for the solution of equations, (1976), Prentice Hall Englewood Cliffs, New Jersey · Zbl 0121.11204
[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) · Zbl 1333.65052
[13] Nedzhibov, Gyurhan, On a few iterative methods for solving nonlinear equations. application of mathematics in engineering and economics’28, () · Zbl 1253.65078
[14] Frontini, M.; Sormani, E., Some variant of newton’s method with third-order convergence, Appl. math. comput., 140, 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, 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, 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, 1, 161-169, (2004) · Zbl 1059.65044
[18] Homeier, H.H.H., On Newton-type methods with cubic convergence, J. comput. appl. math., 176, 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. · Zbl 1154.65032
[20] Kreyszig, Erwin, Introductory functional analysis with applications, (1989), Wiley Classics Library, pp. 300-302 · Zbl 0706.46001
[21] Taylor, A.E., Introduction to functional analysis, (1957), Wiley New York
[22] Ortega, J.M.; Reinbolt, W.C., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[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, 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.
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