A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations

Abstract

Since its introduction, the Broyden method has been used as the foundation to develop several other Broyden-like methods (or hybrid Broyden methods) which in many cases have turned out to be improved forms of the original method. The modified classical Broyden methods developed by many authors to solve system of nonlinear equations have been effective in overcoming the deficiency of the classical Newton Raphson method, however there are new trends of methods proposed by authors, which have proven to be more efficient than some already existing ones. This work introduces two Broyden-like method developed from a weighted combination of quadrature rules, namely the Trapizoidal, Simpson 3/8 and Simpson 1/3 quadrature rules. Hence the new Broyden-like methods named by the authors as TS-3/8 and TS -1/3 methods have been developed from these rules. After subjecting the proposed methods together with some other existing Broyden-like methods to solve four bench-mark problems, the results of numerical test confirm that the TS-3/8 method is promising (in terms of speed and in most cases accuracy) when compared with other proposed Broyden-like methods. Results gathered after the comparison of TS -3/8 with the other methods revealed that TS -3/8 method performed better than all the methods in terms of speed and the number of iterations needed to reach a solution. On the other hand, TS -1/3 method yielded results for all the benchmark problems but with a relatively higher number of iterations compared with the other methods selected for comparison.

The Solution of Non-Linear Equations System Containing Interpolation Functions by Relaxing the Newton Method

“…According to Dwail and Shiker (2021), the Trust Region Method (TRM) is combined with Broyden-Fletcher-Goldfarb-Shanno (BFGS) technique to find the solution to the non-linear systems. Next, two Broyden-like methods are developed from a weighted combination of quadrature rules, namely the Trapezoidal, Simpson 3/8, and Simpson 1/3 quadrature rules (Isaac et al, 2021).…”