Hybrid conjugate gradient parameter for solving symmetric systems of nonlinear equations

Abstract

Mathematical models from recent research are mostly nonlinear equations in nature. Numerical solutions to such systems are widely needed and applied in those areas of mathematics. Although, in recent years, this field received serious attentions and new approach were discovered, but yet the efficiency of the previous versions suffers setback. This article gives a new hybrid conjugate gradient parameter, the method is derivative-free and analyzed with an effective inexact line search in a given conditions. Theoretical proofs show that the proposed method retains the sufficient descent and global convergence properties of the original CG methods. The proposed method is tested on a set of test functions, then compared to the two previous classical CG-parameter that resulted the given method, and its performance is given based on number of iterations and CPU time. The numerical results show that the new proposed method is efficient and effective amongst all the methods tested. The graphical representation of the result justify our findings. The computational result indicates that the new hybrid conjugate gradient parameter is suitable and capable for solving symmetric systems of nonlinear equations.

A new kind of parameter conjugate gradient for unconstrained optimization

“…The test functions and their primary values are wan from [10]. Furthermore, Optimization problems used in many papers for example, see [11][12][13][14][15][16][17][18]. In addition to these functions, there are various other functions that havebeen used for testing in the folloowing research [19][20][21][22][23][24][25].…”

Extended modified three-term conjugate gradient method for large-scale nonlinear equations

“…They also provided theoretical analysis to support the global convergence of their method. For further convergence results concerning the CG method, please consult [18]- [23].…”

Interaction between updated FR-CG algorithms with optimal Cuckoo algorithm

“…Step 2: The best amount of step to algorithm results: set from SWC in (3), calculate the parameters used by the new 1 search direction (12)(13).…”

“…Zoutendijk [2] and Al-Baali [9]. Based on these theories in the previous article, many researchers have developed and updated the formulas of the correlation parameter as in [10][11][12][13][14][15][16][17][18]. "…”