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On Broyden-like update via some quadratures for solving nonlinear systems of equations

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In this work, we propose a new alternative approximation based on the quasi-Newton approach for solving systems of nonlinear equations using the average of midpoint and Simpson’s quadrature. Our goal is to enhance the efficiency of the method (Broyden’s method) by reducing the number of iterations it takes to reach a solution. | Turk J Math (2015) 39: 335 – 345 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1404-41 Research Article On Broyden-like update via some quadratures for solving nonlinear systems of equations Hassan MOHAMMAD∗, Muhammad Yusuf WAZIRI Department of Mathematical Sciences, Faculty of Sciences, Bayero University, Kano, Kano State, Nigeria Received: 14.04.2014 • Accepted/Published Online: 14.01.2015 • Printed: 29.05.2015 Abstract: In this work, we propose a new alternative approximation based on the quasi-Newton approach for solving systems of nonlinear equations using the average of midpoint and Simpson’s quadrature. Our goal is to enhance the efficiency of the method (Broyden’s method) by reducing the number of iterations it takes to reach a solution. Local convergence analysis and computational results showing the relative efficiency of the proposed method are given. Key words: Broyden’s method, superlinear convergence, quadrature formulae, predictor-corrector, nonlinear systems 1. Introduction Consider the numerical solution of the systems of nonlinear equations of the form F (x) = 0. (1) One of the methods used to solve (1) is Newton’s method. It is famous with quadratic order of convergence under some mild assumptions [13]. Despite its good convergence property, Newton’s method has some shortcomings, such as computing and storing Jacobian matrices, solving systems of linear equation in every iteration, and inefficiency in handling large-scale systems. In an attempt to reduce the computational cost of Newton’s method, quasi-Newton methods have been introduced [2]. These methods approximate the Jacobian matrix or its inverse using a derivative-free matrix that is updated in each iteration, and its order of convergence was proven to be superlinear [8]. The most successful and simplest quasi-Newton method for solving nonlinear systems of equations is the Broyden method. Broyden’s method is given by xk+1 =