## Convergence and spectral analysis of the Frontini-Sormani family of multipoint third order methods from quadrature rule.(English)Zbl 1191.65047

Authors’ abstract: Point of attraction theory is an important tool to analyze the local convergence of iterative methods for solving systems of nonlinear equations. In this work, we prove a generalized form of Ortega-Rheinbolt result based on point of attraction theory. The new result guarantees that the solution of the nonlinear system is a point of attraction of iterative scheme, especially multipoint iterations. We then apply it to study the attraction theorem of the Frontini-Sormani family of multipoint third order methods from Quadrature Rule. Error estimates are given and compared with existing ones. We also obtain the radius of convergence of the special members of the family. Two numerical examples are provided to illustrate the theory. Further, a spectral analysis of the Discrete Fourier Transform of the numerical errors is conducted in order to find the best method of the family. The convergence and the spectral analysis of a multistep version of one of the special member of the family are studied.

### MSC:

 65H05 Numerical computation of solutions to single equations 65H10 Numerical computation of solutions to systems of equations 65T50 Numerical methods for discrete and fast Fourier transforms 42A99 Harmonic analysis in one variable 34A34 Nonlinear ordinary differential equations and systems
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### References:

  Aslam Noor, M., Waseem, M.: Some iterative methods for solving a system of nonlinear equations. Comput. Math. Appl. 57, 101–106 (2009) · Zbl 1165.65349  Babajee, D.K.R., Dauhoo, M.Z.: An analysis of the properties of the variants of Newton’s method with third order convergence. Appl. Math. Comput. 183, 659–684 (2006) · Zbl 1123.65036  Babajee, D.K.R., Dauhoo, M.Z.: Analysis of a family of two-point iterative methods with third order convergence. In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds.) International Conference on Numerical Analysis and Applied Mathematics 2006, pp. 658–661. Wiley-VCH Verlag GmbH & Co. KGaA, Greece (2006) · Zbl 1123.65036  Babajee, D.K.R., Dauhoo, M.Z.: Spectral analysis of the errors of some families of multi-step Newton-like methods. Numer. Algorithms (2008). doi: 10.1007/s11075-008-9256-x · Zbl 1180.65057  Ezquerro, J.A., Gutierrez, J.M., Hernandez, M.A, Salnova, M.A.: A biparametric family of inverse-free multipoint iterations. Comput. Appl. Math. 19(1), 109–124 (2000)  Ezquerro, J.A., Hernandez, M.A: A uniparametric Halley-type iteration with free second derivative. Int. J. Pure Appl. Math. 6(1), 103–114 (2003) · Zbl 1026.47056  Frontini, M., Sormani, E.: Some variant of Newton’s method with third order convergence. Appl. Math. Comput. 140(2–3), 419–426 (2003) · Zbl 1037.65051  Frontini, M., Sormani, E.: Modified Newton’s method with third-order convergence and multiple roots. Comput. Appl. Math. 156(2), 345–354 (2003) · Zbl 1030.65044  Frontini, M., Sormani, E.: Third order methods from quadrature formulae for solving systems of nonlinear equations. Appl. Math. Comp. 149, 771–782 (2004) · Zbl 1050.65055  Grau-Sanchez, M., Peris, J.M., Gutierrez, J.M.: Accelerated iterative methods for finding solutions of a system of nonlinear equations. Appl. Math. Comput. 190, 1815–1823 (2007) · Zbl 1122.65351  Hasanov, V.I., Ivanov, I.G., Nedzhibov, G.: A new modification of Newton method. Appl. Math. Eng. 27, 278–286 (2002) · Zbl 1333.65052  Homeier, H.H.H.: Modified Newton method with cubic convergence: the multivariate case. Comput. Appl. Math. 169, 161–169 (2004) · Zbl 1059.65044  Kou, J., Li, Y., Wang, X.: Some modifications of Newton’s method with fifth-order convergence. Comput. Appl. Math. 209, 146–152 (2007) · Zbl 1130.41005  Nedzhibov, G.: On a few iterative methods for solving nonlinear equations. In: Application of Mathematics in Engineering and Economics’28, Bulvest–2000, Sofia, pp. 56–64 (2003) · Zbl 1253.65078  Ortega, J.M., Rheinbolt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic, New York (1970)  Ozban, A.: Some new variants of Newton’s method. Appl. Math. Lett. 17, 677–682 (2004) · Zbl 1065.65067  Petkovic, L.D., Petkovic, M.S.: A note on some recent methods for solving nonlinear equations. Appl. Math. Comput. 185, 368–374 (2007) · Zbl 1121.65321  Ren, H.: A note on a paper by D.K.R. Babajee and M.Z. Dauhoo. Appl. Math. Comput. 200(1–2), 830–833 (2008) · Zbl 1155.65341  Traub, J.F.: Iterative Methods for the Solution of Equations, 2nd edn. Chelsea, New York (1964) · Zbl 0121.11204  Wait, R.: The Numerical Solution of Algebraic Equations. Wiley, New York (1979) · Zbl 0403.65007  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
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