## A parameterized multi-step Newton method for solving systems of nonlinear equations.(English)Zbl 1350.65046

The authors introduce a new multi-step method solving systems nonlinear equations $$\mathbf{F}(\mathbf{x})=0$$, where $$\mathbf{F}: \Gamma \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^r$$ is Fréchet differentiable at $$\mathbf{x}\in\mathrm{interior}(\Gamma)$$ with $$\mathbf{F}(\mathbf{x^\ast})=0$$ and $$\det(\mathbf{F}'(x^\ast))\neq 0$$. They prove that the method needs $$m$$ steps to obtain $$m+1$$ convergence order. The method is a generalization of the multi-step Newton method based on a parameter $$\theta$$. Applying the method for solving the nonlinear complex Zakharov system [A. H. Bhrawy, Appl. Math. Comput. 247, 30–46 (2014; Zbl 1339.65188)], the authors show that the appropriate choice of $$\theta$$ leads to faster convergence and larger radius of convergence.

### MSC:

 65H10 Numerical computation of solutions to systems of equations 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs

Zbl 1339.65188
Full Text:

### References:

 [1] Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964) · Zbl 0121.11204 [2] Cruz, WL; Martinez, JM; Raydan, M, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comput., 75, 1429-1448, (2006) · Zbl 1122.65049 [3] An, HB; Bai, ZZ, A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations, Appl. Numer. Math., 57, 235-252, (2007) · Zbl 1123.65040 [4] Zak, MK; Toutounian, F, Nested splitting conjugate gradient method for matrix equation $$A$$$$X$$$$B$$ = $$C$$ and preconditioning, Comput. Math. Appl., 66, 269-278, (2013) · Zbl 1347.65078 [5] Cordero, A; Hueso, JL; Martinez, E; Torregrosa, JR; modified, A, Newton-jarratts composition, Numer. Algoritm., 55, 87-99, (2010) · Zbl 1251.65074 [6] Sharma, JR; Arora, H, Efficient jarratt-like methods for solving systems of nonlinear equations, Calcolo, 51, 193-210, (2014) · Zbl 1311.65052 [7] Sharma, JR; Guha, RK; Sharma, R, An efficient fourth order weighted-Newton method for systems of nonlinear equations, Numer. Algoritm., 62, 307-323, (2013) · Zbl 1283.65051 [8] Soleymani, F; Lotfi, T; Bakhtiari, P, A multi-step class of iterative methods for nonlinear systems, Optim. Lett., 8, 1001-1015, (2014) · Zbl 1286.93068 [9] Ullah, MZ; Serra-Capizzano, S; Ahmad, F, An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with odes, Appl. Math. Comput., 250, 249-259, (2015) · Zbl 1328.65156 [10] Ortega, J.M., Rheinbodt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, United Kingdom (1970) · Zbl 0241.65046 [11] Budzko, D., Cordero, A., Torregrosa, J.R.: Modifications of Newtons method to extend the convergence domain. Bolet. Sociedad. Espanola. Mat. Aplicada. doi:10.1007/s40324-014-0020-y · Zbl 1308.65072 [12] Alaidarous, ES; Zaka Ullah, M; Ahmad, F; Al-Fhaid, AS, An efficient higher-order quasilinearization method for solving nonlinear BVPs, J. Appl. Math., 2013, 11, (2013) · Zbl 1397.34046 [13] Motsa, SS; Shateyi, S, New analytic solution to the Lane-Emden equation of index 2, Math. Probl. Eng., 2012, 19, (2012) · Zbl 1264.65130 [14] Ladeia, C.A., Romeiro, N.M.L.: Numerical Solutions of the 1D Convection-Diffusion-Reaction and the Burgers Equation using Implicit Multi-stage and Finite Element Methods, Integral Methods in Science and Engineering, pp 205-216 (2013) · Zbl 1279.65107 [15] Jang, An integral equation formalism for solving the nonlinear Klein-Gordon equation, Appl. Math. Comput., 243, 322-338, (2014) · Zbl 1335.35159 [16] Gurarie, D; Chow, KW, Vortex arrays for sinh-Poisson equation of two-dimensional fluids: equilibria and stability, Phys. Fluids, 16, 9, (2004) · Zbl 1187.76196 [17] Averick, B.M., Ortega, J.M.: fast solution of nonlinear Poisson-type equations, Argonne national laboratory, 9700 South Cass Avenue (1991) · Zbl 0755.65101 [18] Bhrawy, AH, An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system, Appl. Math. Comput, 247, 30-46, (2014) · Zbl 1339.65188 [19] Abbasbandy, S; Babolian, E; Ashtiani, M, Numerical solution of the generalized Zakharov equation by homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 14, 4114-4121, (2009) · Zbl 1221.65269 [20] Chang, Q; Jiang, H, A conservative difference scheme for the Zakharov equations, J. Comput. Phys., 113, 309, (1994) · Zbl 0807.76050 [21] Chang, Q; Guo, B; Jiang, H, Finite difference method for generalized Zakharov equations, Math. Comput., 64, 537, (1995) · Zbl 0827.65138 [22] Javidi, M; Golbabai, A, Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method, Chaos Solitons Fract., 36, 309-313, (2008) · Zbl 1350.35182 [23] Bao, W; Sun, F; Wei, GW, Numerical methods for the generalized Zakharov system, J. Comput. Phys., 190, 201-228, (2003) · Zbl 1236.76043 [24] Bao, W; Sun, F, Efficient and stable numerical methods for the generalized and vector Zakharov system, SIAM, J. Sci. Comput., 26, 1057-1088, (2005) · Zbl 1076.35114 [25] Dehghan, M; Izadi, FF, The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves, Math. Comput. Model, 5, 1865-1877, (2011) · Zbl 1219.65106 [26] Doha, EH; Bhrawy, AH; Ezz-Eldien, SS, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model, 35, 5662-5672, (2011) · Zbl 1228.65126 [27] Doha, EH; Bhrawy, AH; Hafez, RM, On shifted Jacobi spectral method for high-order multi-point boundary value problems, Commun. Nonlinear Sci. Numer. Simul., 17, 3802-3810, (2012) · Zbl 1251.65112 [28] Samadi, ORN; Tohidi, E, The spectral method for solving systems of Volterra integral equations, J. Appl. Math. Comput., 40, 477-497, (2012) · Zbl 1295.65128 [29] Tohidi, E; Samadi, ORN, Optimal control of nonlinear Volterra integral equations via Legendre polynomials, IMA J. Math. Control. Inf., 30, 67-83, (2013) · Zbl 1275.49056 [30] Tohidi, E; Lotfi Noghabi, S, An efficient Legendre pseudospectral method for solving nonlinear quasi bang-bang optimal control problems, J. Appl. Math. Stat. Inform., 8, 73-85, (2012) · Zbl 1277.65104 [31] Wang, Z; Guo, B-Y, Jacobi rational approximation spectral method for differential equations of degenerate type, Math. Comput., 77, 883-907, (2008) · Zbl 1132.41315
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