Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs. (English) Zbl 1316.65053

The authors propose a multi-step iterative method for approximating nonsingular solutions of nonlinear systems in \({\mathbb R}^n\). Under sufficient differentiability conditions of the nonlinear function they establish high (sixth, eighth) orders of the method along the idea of a fixed point theorem. Although no second or higher order of Fréchet derivatives of the nonlinear mapping are involved in the proposed method, in each iteration step, multiple linear systems need to be solved to create intermediate approximations. The authors also discuss and compare computational efficiency indices of different methods. Numerical examples are used to illustrate the performance of their method. It would be also useful to compare the proposed method directly with the use of the Newton method in the corresponding multiple steps.
Reviewer: Zhen Mei (Toronto)


65H10 Numerical computation of solutions to systems of equations
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms
68Q25 Analysis of algorithms and problem complexity


Full Text: DOI


[1] Babajee, D.K.R.: Analysis of higher order variants of Newton’s method and their applications to differential and integral equations and in ocean acidification, Ph.D. thesis, University of Mauritius (2010) · Zbl 1235.80048
[2] Babajee, DKR; Dauhoo, MZ, Convergence and spectral analysis of the frontini-sormani family of multipoint third order methods from quadrature rule, Numer. Algor., 53, 467-484, (2010) · Zbl 1191.65047
[3] Babajee, DKR; Cordero, A; Soleymani, F; Torregrosa, JR, On a novel fourth-order algorithm for solving systems of nonlinear equations, J. Appl. Math., 2012, 12, (2012) · Zbl 1268.65072
[4] Cordero, A; Hueso, JL; Martinez, E; Torregrosa, JR, A modified Newton-jarratt’s composition, Numer. Algor., 55, 87-99, (2010) · Zbl 1251.65074
[5] Dayton, BH; Li, T-Y; Zeng, Z, Multiple zeros of nonlinear systems, Math. Comput., 80, 2143-2168, (2011) · Zbl 1242.65102
[6] Decker, DW; Kelley, CT, Sublinear convergence of the chord method at singular points, Numer. Math., 42, 147-154, (1983) · Zbl 0571.65045
[7] Hirsch, MJ; Pardalos, PM; Resende, MGC, Solving systems of nonlinear equations with continuous GRASP, Nonlinear Anal. Real World Appl., 10, 2000-2006, (2009) · Zbl 1163.90750
[8] Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, New York (1970) · Zbl 0241.65046
[9] Pourjafari, E; Mojallali, H, Solving nonlinear equations systems with a new approach based on invasive weed optimization algorithm and clustering, Swarm Evolut. Comput., 4, 33-43, (2012)
[10] Sauer, T.: Numerical analysis, Pearson, 2nd edition, USA (2012)
[11] Semenov, VS, The method of determining all real non-multiple roots of systems of nonlinear equations, Comput. Math. Math. Phys., 47, 1428-1434, (2007) · Zbl 1124.15009
[12] Soheili, A.R., Soleymani, F., Petković, M.D.: On the computation of weighted Moore-Penrose inverse using a high-order matrix method. Comput. Math. Appl. (2013). doi:10.1016/j.camwa.2013.09.007 · Zbl 1350.65033
[13] Soleymani, F; Stanimirović, PS; Ullah, MZ, On an accelerated iterative method for weighted Moore-Penrose inverse, Appl. Math. Comput., 222, 365-371, (2013) · Zbl 1329.65073
[14] Soleymani, F; Stanimirović, PS, A higher order iterative method for computing the Drazin inverse, Sci. World J., 2013, 11, (2013)
[15] Tadonki, C.: High performance computing as a combination of machines and methods and programming. Universite Paris Sud (2013) · Zbl 1251.65074
[16] Toutounian, F., Soleymani, F.: An iterative method for computing the approximate inverse of a square matrix and the Moore-Penrose inverse of a non-square matrix. Appl. Math. Comput. 224 671-680 (2013). doi:10.1016/j.amc.2013.08.086 · Zbl 1336.65048
[17] Traub, J.F.: Iterative methods for the solution of equations. Prentice Hall, New York (1964) · Zbl 0121.11204
[18] Tsoulos, IG; Stavrakoudis, A, On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods, Nonlinear Anal. Real World Appl., 11, 2465-2471, (2010) · Zbl 1193.65078
[19] Wagon, S.: Mathematica in action, 3rd edn. Springer, Berlin (2010) · Zbl 1198.65001
[20] Waziri, MY; Leong, WJ; Hassan, MA; Monsi, M, A low memory solver for integral equations of Chandrasekhar type in the radiative transfer problems, Math. Prob. Eng., 2011, 12, (2011) · Zbl 1235.80048
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