Solving systems of nonlinear equations using decomposition technique

Journal of Linear and Topological Algebra

مقاله 4، دوره 05، شماره 03، آذر 2016، صفحه 187-198   

نوع مقاله: Research Paper

نویسندگان

M. Nili Ahmadabadi^{* 1}؛ F. Ahmad²؛ G. Yuan³؛ X. Li⁴

¹Department of Mathematics, Najafabad Branch, Islamic Azad University, Najafabad, Iran

²Departament de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Comte d'Urgell 187, 08036 Barcelona, Spain

³College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, P.R. China

⁴School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin, Guangxi, China

چکیده

A systematic way is presented for the construction of multi-step iterative method with frozen Jacobian. The inclusion of an auxiliary function is discussed. The presented analysis shows that how to incorporate auxiliary function in a way that we can keep the order of convergence and computational cost of Newton multi-step method. The auxiliary function provides us the way to overcome the singularity and ill-conditioning of the Jacobian. The order of convergence of proposed p-step iterative method is p + 1. Only one Jacobian inversion in the form of LU-factorization is required for a single iteration of the iterative method and in this way, it offers an efficient scheme. For the construction of our proposed iterative method, we used a decomposition technique that naturally provides different iterative schemes. We also computed the computational convergence order that confirms the claimed theoretical order of convergence. The developed iterative scheme is applied to large scale problems, and numerical results show that our iterative scheme is promising.

کلیدواژه‌ها

systems of nonlinear equations؛ Decomposition؛ Order of convergence؛ Higher order methods؛ Computational efficiency

مراجع

[1] J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, NewYork, 1970.

[2] C. T. Kelley, Solving Nonlinear Equations with Newtons Method, SIAM, Philadelphia, 2003.

[3] F. Ahmad, E. Tohidi, J. A. Carrasco, A parameterized multi-step Newton method for solving systems of nonlinear equations. Numerical Algorithms, (2015), 1017-1398.

[4] F. Ahmad, E. Tohidi, M. Z. Ullah, J. A. Carrasco, Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: Application to PDEs and ODEs, Computers & Mathematics with Applications, 70 (4) (2015), 624-636.

[5] M. Z. Ullah, S. Serra-Capizzano, F. Ahmad, An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs, Applied Mathematics and Computation, 250 (2015), 249-259. 

[6] E. S. Alaidarous, M. Z. Ullah, F. Ahmad, A.S. Al-Fhaid, An Efficient Higher-Order Quasilinearization Method for Solving Nonlinear BVPs, Journal of Applied Mathematics, vol. 2013 (2013), Article ID 259371, 11 pages.

[7] F. Ahmed Shah, M, Aslam Noor, Some numerical methods for solving nonlinear equations by using decomposition technique, Appl. Math. Comput. 251 (2015) 378-386.

[8] J. M. Gutirrez, M. A. Hernndez, A family of Chebyshev-Halley type methods in banach spaces, Bull. Aust. Math. Soc. 55 (1997) 113-130.

[9] M. Palacios, Kepler equation and accelerated Newton method, J. Comput. Appl. Math. 138 (2002) 335-346.

[10] S. Amat, S. Busquier, J. M. Gutierrez, Geometrical constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157 (2003) 197-205.

[11] M. Frontini, E. Sormani, Some variant of Newtons method with third-order convergence, Appl. Math. Comput. 140 (2003) 419-426.

[12] M. Frontini, E. Sormani, Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math. Comput. 149 (2004) 771-782.

[13] H. H. H. Homeier, A modified Newton method with cubic convergence: the multivariable case. J. Comput. Appl. Math. 169 (2004) 161-169.

[14] M.T. Darvishi, A. Barati, A fourth-order method from quadrature formulae to solve systems of nonlinear equations. Appl. Math. Comput. 188 (2007) 257-261.

[15] A. Cordero, J.R. Torregrosa, Variants of Newtons method using fifth-order quadrature formulas. Appl. Math. Comput. 190 (2007) 686-698.

[16] M. A. Noor, M. Waseem, Some iterative methods for solving a system of nonlinear equations. Comput. Math. Appl. 57 (2009) 101-106.

[17] M. Grau-Sanchez, A. Grau, M. Noguera, Ostrowski type methods for solving systems of nonlinear equations, Appl. Math. Comput. 218 (2011) 2377-2385.

[18] J.A. Ezquerro, M.Grau-Sanchez, A. Grau, M.A. Hernandez, M. Noguera, N. Romero, On iterative methods with accelerated convergence for solving systems of nonlinear equations, J. Optim. Theory Appl. 151 (2011) 163-174.

[19] A. Cordero, J.L. Hueso, E. Martinez, J.R. Torregrosa, Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett. 25 (2012) 2369-2374.

[20] J.R. Sharma, R.K. Guha, R. Sharma, An efficient fourth order weighted-Newton method for systems of nonlinear equations, Numer. Algorithms 62 (2013) 307-323.

[21] X. Xiao, H. Yin, A new class of methods with higher order of convergence for solving systems of nonlinear equations, Appl. Math. Comput. 264 (2015) 300-309.

[22] E. D. Dolan and J. J. More, Benchmarking optimization software with performance profiles, Mathematical Programming, 91(2002), 201-213.

[23] V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 (2006) 753-763.

[24] J. H. He, Variational iteration method-some recent results and new interpretations, J. Comput. Appl. Math. 207 (2007) 3-17.

[25] Sergio Amat, Sonia Busquier, ngela Grau, Miquel Grau-Sanchez, Maximum efficiency for a family of Newton-like methods with frozen derivatives and some applications, Applied Mathematics and Computation 219 (2013) 7954-7963.

[26] Xinyuan Wu, Note on the improvement of Newtons method for system of nonlinear equations, Applied Mathematics and Computation. 189 (2007) 1476-1479.