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.
Highlights
Extracting roots or finding solutions to equations is an important quest in mathematical computations
An observation from the table revealed that Trapezoidal – Simpson – Midpoint (TSM) method was unable to obtain solutions for problem four with values equal to 35, 65, 165, 365 665 and 1065, in addition, it recorded the highest number of iterations for most of the problems solved
Trapezoidal – Simpson (TS) and TS – 3/8 methods recorded the lowest number of iterations for all four benchmark problems with each one of them recording the same number of iterations for each problem
Summary
Extracting roots or finding solutions to equations is an important quest in mathematical computations. Many different combinations of the traditional numerical methods and intelligent algorithms are applied to solve systems of nonlinear equations [14, 15], which can overcome the problem of selecting a reasonable initial guess of the solution. In this study the following objectives are achieved: (i) Broyden–like methods is developed using combined weights of the Trapezoidal, Simpson 3/8 and Simpson 1/3 quadrature rules; (ii) The new methods are analyzed 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 a similar way as in the above derivations, a weighted combination of the Trapezoidal – Simpson – 1/3 quadrature rules the the numerical scheme as follows;.
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