Abstract
In mathematical term, the method of solving models and finding the best alternatives is known as optimization. Conjugate gradient (CG) method is an evolution of computational method in solving optimization problems. In this article, an alternative modified conjugate gradient coefficient for solving large-scale nonlinear system of equations is presented. The method is an improved version of the Rivaie et el conjugate gradient method for unconstrained optimization problems. The new CG is tested on a set of test functions under exact line search. The approach is easy to implement due to its derivative-free nature and has been proven to be effective in solving real-life application. Under some mild assumptions, the global convergence of the proposed method is established. The new CG coefficient also retains the sufficient descent condition. The performance of the new method is compared to the well-known previous PRP CG methods based on number of iterations and CPU time. Numerical results using some benchmark problems show that the proposed method is promising and has the best efficiency amongst all the methods tested.
Highlights
IntroductionThe following nonlinear system of equations is considered
In this paper, the following nonlinear system of equations is considered.F(x) = 0, x ∈ Rn; (Eq 1)where F: Rn → Rn is continuously differentiable
Considering the benchmark problems above with their respective initial points, the new Conjugate gradient (CG), DSHM has been proven to be the best method when compared to standard PRP CG methods
Summary
The following nonlinear system of equations is considered. Newton and quasi-Newton methods are the most widely used methods to solve such problems because they have very attractive convergence properties and practical application (see [1,2,3,4]). They are not usually suitable for large-scale nonlinear systems of equations because they require Jacobian matrix, or an approximation to it, at every iteration while solving optimization problems. The iterative method used to solve (Eq 1) is formed by xk+1 = xk + αkdk, k = 0,1,2,,
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