Abstract
The search for a more efficient and robust numerical method for solving problems have become an interesting area for many researchers as most problems resulting into nonlinear system of equations would require a very good numerical method for its computation. The introduction of the Broyden method has served as the foundation to developing several others, which are referred to as Broyden – like methods by some authors. These methods, in most cases, have proven to be superior to the original classical Broyden method in terms of the number of iterations needed and the CPU time required to reach a solution. This research sought to develop new Broyden – like methods using weighted combinations of quadrature rules (i.e., Simpson -1/3 and Simpson -3/8 rules against Midpoint, Trapezoidal, and Simpson quadrature rules). The weighted combination of the quadrature rules in the development of the new methods led to the discovery of several new methods. Some of which have proven to be more efficient and robust when compared with some existing methods. A comparison of these newly developed methods with the classical Broyden method together with some existing improved Broyden method revealed that, one of the newly developed methods namely, Midpoint–Simpson-3/8 (MS–3/8) method, outperformed all the others, with the MS–3/8 method giving the best of numerical results in all the benchmark problems considered in the study.
Highlights
Finding solutions to equations is an important quest in mathematical computations
In this study the following objectives are achieved: (i) Development of five new Broyden – like methods using combined weights of the quadrature rules; (ii) The new methods are analysed by comparing the number of iterations and the CPU time with the existing Broyden–like methods using selected systems of nonlinear equations as test problems
In order to evaluate the performance of the new methods, they were tested, together with four other existing methods (i.e. Classical Broyden Method (CB), Trapezoidal–Simpson Method (TS), Midpoint-Trapezoidal (MT), Trapezoidal Simpson Midpoint Method (TSMM) method, on four benchmark problems (Osinuga et al, 2018), using a set of seven dimensions ranging from 5 to 1065 variables
Summary
Finding solutions to equations is an important quest in mathematical computations. The roots of equations provide answers to many practical problems under study. Many different combinations of the traditional numerical methods and intelligent algorithms are applied to solve systems of nonlinear equations [15, 17], which can overcome the problem of selecting a reasonable initial guess of the solution. A related work by the authors [10] constructed a Broyden– like method called the Trapezoidal–Simpson’s 3/8 method with the weighted combination of the Trapezoidal and Simpson’s 3/8 quadrature rules Though this method performed better in some bench mark problems than the other Broyden–like methods, there were instances where some other methods had a better result. In this study the following objectives are achieved: (i) Development of five new Broyden – like methods using combined weights of the quadrature rules; (ii) The new methods are analysed by comparing the number of iterations and the CPU time with the existing Broyden–like methods using selected systems of nonlinear equations as test problems.
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