Grau-Sánchez, Miquel; Grau, Àngela; Noguera, Miquel On the computational efficiency index and some iterative methods for solving systems of nonlinear equations. (English) Zbl 1231.65090 J. Comput. Appl. Math. 236, No. 6, 1259-1266 (2011). 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. Cited in 1 ReviewCited in 60 Documents MSC: 65H10 Numerical computation of solutions to systems of equations 65Y20 Complexity and performance of numerical algorithms Keywords:order of convergence; system of nonlinear equations; iterative methods; computational efficiency index; computational order of convergence Software:MPFR PDFBibTeX XMLCite \textit{M. Grau-Sánchez} et al., J. Comput. Appl. Math. 236, No. 6, 1259--1266 (2011; Zbl 1231.65090) Full Text: DOI References: [1] Traub, J. F., Iterative Methods for the Solution of Equations (1964), Prentice-Hall: Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0121.11204 [2] Grau, M.; Díaz-Barrero, J. L., An improvement of the Euler-Chebyshev iterative method, J. Math. Anal. Appl., 315, 1-7 (2006) · Zbl 1113.65048 [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 [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 [5] Ostrowski, A. M., Solutions of Equations and System of Equations (1960), Academic Press: Academic Press New York · Zbl 0115.11201 [6] Bailey, D. H.; Borwein, J. M., High-precision computation and mathematical physics, (XII Advanced Computing and Analysis Techniques in Physics Research (2008)) · Zbl 1248.65147 [7] Özban, A. Y., Some new variants of Newton’s method, Appl. Math. Lett., 17, 677-682 (2004) · Zbl 1065.65067 [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, 2 (2007), Art. 13 (15 pp) · Zbl 1365.65302 [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 [11] Polyanin, A. D.; Manzhirov, A. V., Handbook of Integral Equations (1998), CRC Press: CRC Press Boca Ratón · Zbl 1021.45001 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.