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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Publisher Name

Springer

Print ISSN

1017-1398

Online ISSN

1572-9265

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References

[1] Amat, S., Busquier, S., A two step Steffenssen’s method under modified convergence conditions. J. Math. Anal. Appl. 324, 1084–1092 (2006)
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