×

zbMATH — the first resource for mathematics

Convergence behavior for Newton-Steffensen’s method under \(\gamma\)-condition of second derivative. (English) Zbl 07095233
Summary: The present paper is concerned with the semilocal as well as the local convergence problems of Newton-Steffensen’s method to solve nonlinear operator equations in Banach spaces. Under the assumption that the second derivative of the operator satisfies \(\gamma\)-condition, the convergence criterion and convergence ball for Newton-Steffensen’s method are established.
MSC:
65 Numerical analysis
47 Operator theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Blum, L.; Cucker, F.; Shub, M.; Smale, S., Complexity and Real Computation, xvi+453, (1998), New York, NY, USA: Springer, New York, NY, USA
[2] He, J.-S.; Wang, J.-H.; Li, C., Newton’s method for underdetermined systems of equations under the γ-condition, Numerical Functional Analysis and Optimization, 28, 5-6, 663-679, (2007) · Zbl 1120.65070
[3] Li, C.; Hu, N.; Wang, J., Convergence behavior of Gauss-Newton’s method and extensions of the Smale point estimate theory, Journal of Complexity, 26, 3, 268-295, (2010) · Zbl 1192.65057
[4] Smale, S., The fundamental theorem of algebra and complexity theory, Bulletin of the American Mathematical Society, 4, 1, 1-36, (1981) · Zbl 0456.12012
[5] Smale, S., Newton’s method estimates from data at one point, The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics, 185-196, (1986), New York, NY, USA: Springer, New York, NY, USA
[6] Smale, S., Complexity theory and numerical analysis, Acta Numerica, 6, 523-551, (1997) · Zbl 0883.65125
[7] Wang, X., Convergence on the iteration of Halley family in weak conditions, Chinese Science Bulletin, 42, 7, 552-555, (1997) · Zbl 0884.30004
[8] Wang, X. H.; Han, D. F., Criterion α and Newton’s method under weak conditions, Mathematica Numerica Sinica, 19, 1, 96-105, (1997)
[9] Candela, V.; Marquina, A., Recurrence relations for rational cubic methods—I. The Halley method, Computing, 44, 2, 169-184, (1990) · Zbl 0701.65043
[10] Candela, V.; Marquina, A., Recurrence relations for rational cubic methods—II. The Chebyshev method, Computing, 45, 4, 355-367, (1990) · Zbl 0714.65061
[11] Hernández, M. A.; Salanova, M. A., A family of Chebyshev-Halley type methods, International Journal of Computer Mathematics, 47, 59-63, (1993) · Zbl 0812.65038
[12] Wang, H.; Li, C.; Wang, X., On relationship between convergence ball of Euler iteration in Banach spaces and its dynamical behavior on Riemann spheres, Science in China. Mathematics, 46, 3, 376-382, (2003) · Zbl 1217.37045
[13] Wang, X.; Li, C., On the united theory of the family of Euler-Halley type methods with cubical convergence in Banach spaces, Journal of Computational Mathematics, 21, 2, 195-200, (2003) · Zbl 1057.65033
[14] Xu, X.; Ling, Y., Semilocal convergence for Halley’s method under weak Lipschitz condition, Applied Mathematics and Computation, 215, 8, 3057-3067, (2009) · Zbl 1187.65059
[15] Xu, X.; Ling, Y., Semilocal convergence for a family of Chebyshev-Halley like iterations under a mild differentiability condition, Journal of Applied Mathematics and Computing, 40, 1-2, 627-647, (2012) · Zbl 1294.47093
[16] Ye, X.; Li, C., Convergence of the family of the deformed Euler-Halley iterations under the Hölder condition of the second derivative, Journal of Computational and Applied Mathematics, 194, 2, 294-308, (2006) · Zbl 1100.47057
[17] Sharma, J. R., A composite third order Newton-Steffensen method for solving nonlinear equations, Applied Mathematics and Computation, 169, 1, 242-246, (2005) · Zbl 1084.65054
[18] Wang, X. H.; Han, D. F., On dominating sequence method in the point estimate and Smale theorem, Science in China. Mathematics, 33, 2, 135-144, (1990) · Zbl 0699.65046
[19] Wang, X., Convergence of Newton’s method and inverse function theorem in Banach space, Mathematics of Computation, 68, 225, 169-186, (1999) · Zbl 0923.65028
[20] Wang, X. H.; Han, D. F.; Sun, F. Y., Point estimates for some deformation Newton iterations, Mathematica Numerica Sinica, 12, 2, 145-156, (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.