Journal of Mathematics and Statistics 6 (3): 246-252, 2010 ISSN 1549-3644 © 2010 Science Publications Corresponding Author: M.Y. Waziri, Department of Mathematics, Faculty of Science, University Putra Malaysia 43400 Serdang, Malaysia 246 A New Newton’s Method with Diagonal Jacobian Approximation for Systems of Nonlinear Equations M.Y. Waziri, W.J. Leong, M.A. Hassan and M. Monsi Department of Mathematics, Faculty of Science, University Putra Malaysia 43400 Serdang, Malaysia Abstract: Problem statement: The major weaknesses of Newton method for nonlinear equations entail computation of Jacobian matrix and solving systems of n linear equations in each of the iterations. Approach: In some extent function derivatives are quit costly and Jacobian is computationally expensive which requires evaluation (storage) of n×n matrix in every iteration. Results: This storage requirement became unrealistic when n becomes large. We proposed a new method that approximates Jacobian into diagonal matrix which aims at reducing the storage requirement, computational cost and CPU time, as well as avoiding solving n linear equations in each iterations. Conclusion/Recommendations: The proposed method is significantly cheaper than Newton’s method and very much faster than fixed Newton’s method also suitable for small, medium and large scale nonlinear systems with dense or sparse Jacobian. Numerical experiments were carried out which shows that, the proposed method is very encouraging. Key words: Nonlinear equations, large scale systems, Newton’s method, diagonal updating, Jacobian approximation INTRODUCTION Consider the system of nonlinear equations: F(x) = 0 (1) where, F(x) : R n → R n with the following properties: • There exist x* with F(x*) = 0 • F is continuously differentiable in a neighbourhood of x* • F F'(x*) = J (x*) 0 ≠ The most well-known method for solving (1), is the classical Newton’s method. However, the Newton’s method for nonlinear equations has the following general form: Given an initial point x 0 , we compute a sequence of corrections {s k } and iterates {x k } as follows: Algorithm CN (Newton’s method): where, k = 0, 1, 2... and J F (x k ) is the Jacobian matrix of F, then: Stage 1: Solve F k J (x ) s k = -F(x k ) Stage 2: Update x k+1 = x k + s k Stage 3: Repeat 1-2 until converges. The convergence of Algorithm CN is attractive. However, the method depends on a good starting point (Dennis, 1983). Newton’s method will converges to x* provided the initial guess x 0 is sufficiently close to the x* and J F (x*) ≠ 0 with J F (x) Lipchitz continuous and the rate is quadratic (Dennis, 1983), i.e.: k1 k x x* hx x* + - ≤ - (2) For some h. Even though it has good qualities, CN method has some major shortfalls as the dimension of the systems increases which includes (Dennis, 1983) for details): • Computation and storage of Jacobian in each iteration • Solving system of n linear equations in each iteration • More CPU time consumption as the equations dimension increases There are several strategies to overcome the above drawbacks. The first is fixed Newton method, i.e., by setting J F (x k ) ≡ J F (x 0 ) for k>0. Fixed Newton is the easiest and simplest strategy to overcome the shortfalls