×

Directional Chebyshev-type methods for solving equations. (English) Zbl 1307.65072

Summary: A semi-local convergence analysis for directional Chebyshev-type methods in \( m\)-variables is presented in this study. Our convergence analysis uses recurrent relations and Newton-Kantorovich-type hypotheses. Numerical examples are also provided to show the effectiveness of the proposed method.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] An, Heng-Bin; Bai, Zhong-Zhi, Directional secant method for nonlinear equations, J. Comput. Appl. Math., 175, 2, 291-304 (2005) · Zbl 1076.65046 · doi:10.1016/j.cam.2004.05.013
[2] Argyros, Ioannis K., On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 169, 2, 315-332 (2004) · Zbl 1055.65066 · doi:10.1016/j.cam.2004.01.029
[3] Argyros, Ioannis K., A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl., 298, 2, 374-397 (2004) · Zbl 1057.65029 · doi:10.1016/j.jmaa.2004.04.008
[4] Argyros, Ioannis K., Convergence and Applications of Newton-Type Iterations, xvi+506 pp. (2008), Springer: New York:Springer · Zbl 1153.65057
[5] Argyros, Ioannis K., A semilocal convergence analysis for directional Newton methods, Math. Comp., 80, 273, 327-343 (2011) · Zbl 1211.65057 · doi:10.1090/S0025-5718-2010-02398-1
[6] Argyros, Ioannis K.; Cho, Yeol Je; Hilout, Sa{\"{\i }}d, Numerical Methods for Equations and Its Applications, viii+465 pp. (2012), CRC Press: Boca Raton, FL:CRC Press · Zbl 1254.65068
[7] Argyros, I. K.; Ezquerro, J. A.; Guti{\'e}rrez, J. M.; Hern{\'a}ndez, M. A.; Hilout, S., On the semilocal convergence of efficient Chebyshev-secant-type methods, J. Comput. Appl. Math., 235, 10, 3195-3206 (2011) · Zbl 1215.65102 · doi:10.1016/j.cam.2011.01.005
[8] [673-8] A. Ben-Israel, Y. Levin, Maple programs for directional Newton methods, are avaialable at ftp://rutcor.rutgers.edu/pub/bisrael/Newton-Dir.mws. · Zbl 0985.65049
[9] Ezquerro, J. A.; Hern{\'a}ndez, M. A., An optimization of Chebyshev’s method, J. Complexity, 25, 4, 343-361 (2009) · Zbl 1183.65058 · doi:10.1016/j.jco.2009.04.001
[10] Levin, Yuri; Ben-Israel, Adi, Directional Newton methods in \(n\) variables, Math. Comp., 71, 237, 251-262 (2002) · Zbl 0985.65049 · doi:10.1090/S0025-5718-01-01332-1
[11] Luk{\'a}cs, G{\'a}bor, The generalized inverse matrix and the surface-surface intersection problem. Theory and practice of geometric modeling, Blaubeuren, 1988, 167-185 (1989), Springer: Berlin:Springer
[12] Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, xx+572 pp. (1970), Academic Press: New York:Academic Press · Zbl 0949.65053
[13] Polyak, Boris T., Introduction to Optimization, Translations Series in Mathematics and Engineering, xxvii+438 pp. (1987), Optimization Software Inc. Publications Division: New York:Optimization Software Inc. Publications Division
[14] Potra, F.-A., On the convergence of a class of Newton-like methods. Iterative solution of nonlinear systems of equations (Oberwolfach, 1982), Lecture Notes in Math. 953, 125-137 (1982), Springer: Berlin:Springer
[15] Potra, Florian Alexandru, Sharp error bounds for a class of Newton-like methods, Libertas Math., 5, 71-84 (1985) · Zbl 0581.47050
[16] Walker, Homer F.; Watson, Layne T., Least-change secant update methods for underdetermined systems, SIAM J. Numer. Anal., 27, 5, 1227-1262 (1990) · Zbl 0733.65032 · doi:10.1137/0727071
[17] 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) · Zbl 0973.65037 · doi:10.1016/S0893-9659(00)00100-2
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.