Archive
Special Issues Volume 4, Issue 1, February 2018, Page: 20-23
Numerical Solution of Nonlinear Systems of Algebraic Equations
Kamoh Nathaniel Mahwash, Department of Mathematics and Statistics, Bingham University, Karu, Nigeria
Gyemang Dauda Gyang, Department of Mathematics and Statistics, Plateau State Polytechnic, Barkin Ladi, Nigeria
Received: Jan. 29, 2018;       Accepted: Feb. 27, 2018;       Published: Mar. 23, 2018
DOI: 10.11648/j.ijdsa.20180401.14      View  1449      Downloads  110
Abstract
Considered in this paper are two basic methods of approximating the solutions of nonlinear systems of algebraic equations. The Steepest Descent method was presented as a way of obtaining good and sufficient initial guess (starting value) which is in turn used for the Broyden’s method. Broyden’s method on the other hand replaces the Newton’s method which requires the use of the inverse of the Jacobian matrix at every new step of iteration with a matrix whose inverse is directly determined at each step by up-dating the previous inverse. The result obtained by this method revealed that the setbacks encountered in computing the inverse of the Jacobian matrix at every step number is eliminated hence saving human effort and computer time. The obtained results also showed that the number of steps that is reduced when compared to Newton’s method used on the same problem.
Keywords
Convergent, Jacobian; Matrix, Approximation, Starting Value, Iteration, Nonlinear System
To cite this article
Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang, Numerical Solution of Nonlinear Systems of Algebraic Equations, International Journal of Data Science and Analysis. Vol. 4, No. 1, 2018, pp. 20-23. doi: 10.11648/j.ijdsa.20180401.14
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference

Brazelton, J. (2010) Solving Nonlinear Equations Using Numerical Analysis. Math 451 Seminar, Tuskegee University

Chein-Shan Liu & Satya N. Atluri (2008) A Novel Time Integration Method for Solving A Large System of Non-Linear Algebraic Equations. Technical Science Press CMES, 31(2):71-83

De Cezaro, A (2008): On Steepest-Descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations

Gomes, F. A. M & Martinez, J. M. (1992) Parallel implementations of Broyden's method, Springer link 47(3-4):361-366

Huang, Tsung-Ming (2011): Numerical Solutions of Nonlinear Systems of Equations Department of Mathematics, National Taiwan Normal University, Taiwan E-mail: min@math.ntnu.edu.tw

Gilberto E. Urroz (2004) Solution of non-linear equations

Paul’s online math notes: Algebra-Nonlinear system-Lamar University

Goh, B. S. and McDonald, D. B. (2015) Newton Methods to Solve a System of Nonlinear Algebraic Equations. Journal of Optimization Theory and Application

Vincent, T., Grantham, W. (1997): Nonlinear and Optimal Control Systems. Wiley, New York

Barbashin, E. A. and Krasovskii, N. N. (1952): On the stability of a motion in the large. Dokl. Akad. Nauk. SSR 86, 453–456

Powell, M. J. D. (1986): How Bad are the BFGS Methods when the Objective Function is Quadratic. Mathematics Programme 34, 34–47 8

Goh, B. S. (1994) Global attractivity and stability of a scalar nonlinear difference equation. Computer and Mathematics Application 28, 101–107

Fan, J. Y. and Yuan, Y. X. (2005): On the quadratic convergence of the Levenberg–Marquardt method without non-singularity assumption. Computing 74, 23–39

Powell, M. J. D (1970) A Hybrid Method for Nonlinear Equations, in Numerical Methods for Nonlinear Algebraic Equations, pp. 87–114. Gordon and Breach, London

Byrd, R. H., Marazzi, M. and Nocedal, J. (2004): On the convergence of Newton iterations to non-stationary points. Mathematics Programme 99, 127–148

LaSalle, J. P. (1976): The Stability of Dynamical Systems. SIAM, Philadelphia

Goh, B. S. (2010): Convergence of numerical methods in unconstrained optimization and the solution of nonlinear equations. Journal of Optimization and Theory Application 144, 43–55 