On an Aitken-Newton type method

7 years ago

(original), divided differences, inverse interpolation, ISI/JCR, local convergence, Newton method, nonlinear equations in R, Numerical Analysis, paper, solving F(x)=0 iteratively

Abstract

We study the solving of nonlinear equations by an iterative method of Aitken type, which has the interpolation nodes controlled by the Newton method. We obtain a local convergence result which shows that the q-convergence order of this method is 6 and its efficiency index is \(\sqrt[5]6\), which is higher than the efficiency index of the Aitken or Newton methods. Monotone sequences are obtained for initial approximations farther from the solution, if they satisfy the Fourier condition and the nonlinear mapping satisfies monotony and convexity assumptions on the domain.

Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Keywords

Nonlinear equations in R; Aitken method; Newton method; inverse interpolatory polynomials; divided differences; monotone sequences.

Paper coordinates

I. Păvăloiu, E. Cătinaş, On an Aitken-Newton type method, Numer. Algor., 62 (2013) no. 2, pp. 253-260
10.1007/s11075-012-9577-7

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About this paper

Journal

Numerical Algorithms

Publisher Name

Springer

DOI

10.1007/s11075-012-9577-7

Print ISSN

1017-1398

Online ISSN

1572-9265

Google Scholar Profile

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Amat, S., Busquier, S.: A two step Steffenssen’s method under modified convergence conditions. J. Math. Anal. Appl. 324, 1084–1092 (2006) Article MathSciNet MATH Google Scholar
Beltyukov, B.A.: An analogue of the Aitken–Steffensen method with controlled step. URSS Comput. Math. Math. Phys. 27(3), 103–112 (1987) Article MathSciNet Google Scholar
Chun, C.: A geometric construction of iterative formulas of order three. Appl. Math. Lett. 23, 512–516 (2010) Article MathSciNet MATH Google Scholar
Cordero, A., Torregrosa, J.R.: Variants of Newton’s method for functions of several variables. Appl. Math. Comput. 183, 199–2008 (2006) Article MathSciNet MATH Google Scholar
Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007) Article MathSciNet MATH Google Scholar
Cordero A., Torregrosa R.J.: A class of Steffenssen type method with optimal order of convergence. Appl. Math. Comput. 217, 7653–7659 (2011) Article MathSciNet MATH Google Scholar
Hongmin, R., Qingbio, W., Welhong, B.: A class of two-step Steffensen type methods with fourth order convergence. Appl. Math. Comput. 209(2), 206–210 (2009) Article MathSciNet MATH Google Scholar
Jain, P.: Steffensen type method for solving non-linear equations. Appl. Math. Comput. 194, 527–533 (2007) Article MathSciNet MATH Google Scholar
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970) MATH Google Scholar
Ostrowski, M.A.: Solution of Equations in Euclidian and Banach Spaces. Academic Press, New York and London (1973) Google Scholar
Păvăloiu, I.: Approximation of the root of equations by Aitken–Steffensen-type monotonic sequences. Calcolo 32, 69–82 (1995) Article MathSciNet MATH Google Scholar
Păvăloiu, I., Cătinaş, E.: On a Steffensen–Hermite method of order three. Appl. Math. Comput. 215(7), 2663–2672 (2009) Article MathSciNet MATH Google Scholar
Păvăloiu, I., Cătinaş, E.: On a Steffensen type method. In: Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, Proceedings, pp. 369–375 (2007)
Păvăloiu, I.: Optimal algorithms, concerning the solving of equations by interpolation. In: Popoviciu, E. (ed.) Research on Theory of Allure, Approximation, Convexity and Optimization, Editura Srima, Cluj-Napoca, pp. 222–248 (1999)
Păvăloiu, I.: Aitken–Steffensen-type method for nondifferentiable functions (I). Rev. Anal. Numér. Theor. Approx. 31(1), 109–114 (2002) MATH Google Scholar
Păvăloiu, I.: Aitken–Steffensen-type method for nonsmooth functions (II). Rev. Anal. Numér. Théor. Approx. 31(2), 195–198 (2002) MATH Google Scholar
Păvăloiu, I.: Aitken–Steffensen-type method for nonsmooth functions (III). Rev. Anal. Numér. Théor. Approx. 32(1), 73–78 (2003) MATH Google Scholar
Quan, Z., Peng, Z., Li, Z., Wenchao, M.: Variants of Steffensen secant method and applications. Appl. Math. Comput. 216(12), 3486–3496 (2010) Article MathSciNet MATH Google Scholar
Sharma, R.J.: A composite thid order Newton–Steffensen method for solving nonlinear equations. Appl. Math. Comput. 169, 242–246 (2005) Article MathSciNet MATH Google Scholar
Traub, Y.F.: Iterative Method for Solution of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964) Google Scholar
Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000) Article MathSciNet MATH Google Scholar