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On the computational efficiency index and some iterative methods for solving systems of nonlinear equations. (English) Zbl 1231.65090
Summary: Two new iterative methods are built up and analyzed. A generalization of the efficiency index used in the scalar case to several variables in iterative methods for solving systems of nonlinear equations is revisited. Analytic proofs of the local order of convergence based on developments of multilineal functions and numerical concepts that will be used to illustrate the analytic results are given. An approximation of the computational order of convergence is computed independently of the knowledge of the root and the necessary time to get one correct decimal is studied in our examples.
MSC:
65H10Systems of nonlinear equations (numerical methods)
65Y20Complexity and performance of numerical algorithms
Software:
MPFR
References:
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[3]Grau-Sánchez, M.: Improvement of the efficiency of some three-step iterative like-Newton methods, Numer. math. 107, 131-146 (2007) · Zbl 1123.65037 · doi:10.1007/s00211-007-0088-8
[4]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 · doi:10.1016/S0893-9659(00)00100-2
[5]Ostrowski, A. M.: Solutions of equations and system of equations, (1960) · Zbl 0115.11201
[6]Bailey, D. H.; Borwein, J. M.: High-precision computation and mathematical physics, (2008)
[7]Özban, A. Y.: Some new variants of Newton’s method, Appl. math. Lett. 17, 677-682 (2004)
[8]Fousse, L.; Hanrot, G.; Lefèvre, V.; Pélissier, P.; Zimmermann, P.: MPFR: a multiple-precision binary floating-point library with correct rounding, ACM trans. Math. software 33, No. 2 (2007)
[9]
[10]Grau-Sánchez, M.; Noguera, M.; Gutiérrez, J. M.: On some computational orders of convergence, Appl. math. Lett. 23, 472-478 (2010) · Zbl 1189.65092 · doi:10.1016/j.aml.2009.12.006
[11]Polyanin, A. D.; Manzhirov, A. V.: Handbook of integral equations, (1998)