zbMATH — the first resource for mathematics

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.

65H10 Numerical computation of solutions to systems of equations
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
Full Text: DOI
[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.