scholarly journals An Inexact Optimal Hybrid Conjugate Gradient Method for Solving Symmetric Nonlinear Equations

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1829
Author(s):  
Jamilu Sabi’u ◽  
Kanikar Muangchoo ◽  
Abdullah Shah ◽  
Auwal Bala Abubakar ◽  
Kazeem Olalekan Aremu
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Convex Combination ◽  
Iteration Process ◽  
Nonlinear Problems ◽  
Conjugacy Condition ◽  
Derivative Free ◽  
Numerical Performance ◽  
Hybrid Conjugate Gradient Method ◽  
Newton Direction

This article presents an inexact optimal hybrid conjugate gradient (CG) method for solving symmetric nonlinear systems. The method is a convex combination of the optimal Dai–Liao (DL) and the extended three-term Polak–Ribiére–Polyak (PRP) CG methods. However, two different formulas for selecting the convex parameter are derived by using the conjugacy condition and also by combining the proposed direction with the default Newton direction. The proposed method is again derivative-free, therefore the Jacobian information is not required throughout the iteration process. Furthermore, the global convergence of the proposed method is shown using some appropriate assumptions. Finally, the numerical performance of the method is demonstrated by solving some examples of symmetric nonlinear problems and comparing them with some existing symmetric nonlinear equations CG solvers.

scholarly journals A Modified PRP-CG Type Derivative-Free Algorithm with Optimal Choices for Solving Large-Scale Nonlinear Symmetric Equations

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 234
Author(s):  
Jamilu Sabi’u ◽  
Kanikar Muangchoo ◽  
Abdullah Shah ◽  
Auwal Bala Abubakar ◽  
Lateef Olakunle Jolaoso
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Search Direction ◽  
Large Scale Systems ◽  
Measure Function ◽  
Derivative Free ◽  
Newton Direction

Inspired by the large number of applications for symmetric nonlinear equations, this article will suggest two optimal choices for the modified Polak–Ribiére–Polyak (PRP) conjugate gradient (CG) method by minimizing the measure function of the search direction matrix and combining the proposed direction with the default Newton direction. In addition, the corresponding PRP parameters are incorporated with the Li and Fukushima approximate gradient to propose two robust CG-type algorithms for finding solutions for large-scale systems of symmetric nonlinear equations. We have also demonstrated the global convergence of the suggested algorithms using some classical assumptions. Finally, we demonstrated the numerical advantages of the proposed algorithms compared to some of the existing methods for nonlinear symmetric equations.

scholarly journals A comparison on classical-hybrid conjugate gradient method under exact line search

2019 ◽  
Vol 5 (2) ◽  
pp. 150
Author(s):  
Nur Syarafina Mohamed ◽  
Mustafa Mamat ◽  
Mohd Rivaie ◽  
Shazlyn Milleana Shaharudin
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Line Search ◽  
Convex Combination ◽  
Processing Unit ◽  
Central Processing ◽  
Exact Line Search ◽  
Hybrid Conjugate Gradient Method ◽  
Number Of Iterations

One of the popular approaches in modifying the Conjugate Gradient (CG) Method is hybridization. In this paper, a new hybrid CG is introduced and its performance is compared to the classical CG method which are Rivaie-Mustafa-Ismail-Leong (RMIL) and Syarafina-Mustafa-Rivaie (SMR) methods. The proposed hybrid CG is evaluated as a convex combination of RMIL and SMR method. Their performance are analyzed under the exact line search. The comparison performance showed that the hybrid CG is promising and has outperformed the classical CG of RMIL and SMR in terms of the number of iterations and central processing unit per time.

scholarly journals Hybrid conjugate gradient parameter for solving symmetric systems of nonlinear equations

2019 ◽  
Vol 16 (1) ◽  
pp. 539
Author(s):  
M. K. Dauda ◽  
Mustafa Mamat ◽  
Mohamad A. Mohamed ◽  
Nor Shamsidah Amir Hamzah
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Numerical Solutions ◽  
Graphical Representation ◽  
Systems Of Nonlinear Equations ◽  
Inexact Line Search ◽  
Derivative Free ◽  
Gradient Parameter ◽  
The Given ◽  
Symmetric Systems

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.

scholarly journals Spectral CG Algorithm for Solving Fuzzy Nonlinear Equations

2022 ◽  
pp. 1-10
Author(s):  
Mezher M. Abed ◽  
Ufuk Öztürk ◽  
Hisham M. Khudhur
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Conjugate Gradient Algorithm ◽  
Gradient Algorithm ◽  
Conjugacy Condition ◽  
Wide Range ◽  
Non Linear ◽  
Fuzzy Equations

The nonlinear conjugate gradient method is an effective technique for solving large-scale minimizations problems, and has a wide range of applications in various fields, such as mathematics, chemistry, physics, engineering and medicine. This study presents a novel spectral conjugate gradient algorithm (non-linear conjugate gradient algorithm), which is derived based on the Hisham–Khalil (KH) and Newton algorithms. Based on pure conjugacy condition The importance of this research lies in finding an appropriate method to solve all types of linear and non-linear fuzzy equations because the Buckley and Qu method is ineffective in solving fuzzy equations. Moreover, the conjugate gradient method does not need a Hessian matrix (second partial derivatives of functions) in the solution. The descent property of the proposed method is shown provided that the step size at meets the strong Wolfe conditions. In numerous circumstances, numerical results demonstrate that the proposed technique is more efficient than the Fletcher–Reeves and KH algorithms in solving fuzzy nonlinear equations.

scholarly journals A Globally Convergent Hybrid Conjugate Gradient Method and Its Numerical Behaviors

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Yuan-Yuan Huang ◽  
San-Yang Liu ◽  
Xue-Wu Du ◽  
Xiao-Liang Dong
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Globally Convergent ◽  
Numerical Performance ◽  
Hybrid Conjugate Gradient Method

We consider a hybrid Dai-Yuan conjugate gradient method. We confirm that its numerical performance can be improved provided that this method uses a practical steplength rule developed by Dong, and the associated convergence is analyzed as well.

scholarly journals New hybrid conjugate gradient method as a convex combination of FR and PRP methods

Filomat ◽  
2016 ◽  
Vol 30 (11) ◽  
pp. 3083-3100 ◽  
Author(s):  
Snezana Djordjevic
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Convex Combination ◽  
Conjugate Gradient Algorithm ◽  
Gradient Algorithm ◽  
Numerical Comparisons ◽  
Hybrid Conjugate Gradient Method ◽  
Better Than

We consider a newhybrid conjugate gradient algorithm,which is obtained fromthe algorithmof Fletcher-Reeves, and the algorithmof Polak-Ribi?re-Polyak. Numerical comparisons show that the present hybrid conjugate gradient algorithm often behaves better than some known algorithms.

scholarly journals Derivative-free SMR conjugate gradient method for constraint nonlinear equations

2021 ◽  
Vol 24 (02) ◽  
pp. 147-164
Author(s):  
Abdulkarim Hassan Ibrahim ◽  
Kanikar Muangchoo ◽  
Nur Syarafina Mohamed ◽  
Auwal Bala Abubakar
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Derivative Free

scholarly journals Comments on ”New hybrid conjugate gradient method as a convex combination of FR and PRP methods”

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4573-4574
Author(s):  
Chenna Nasreddine ◽  
Sellami Badreddine
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Convex Combination ◽  
Hybrid Conjugate Gradient Method

In this note, we present a new theory as a modification and an alternative to S.Djordjevic?s Theorem (2.2), Here we rephrase the text of theory (2.2) by deleting condition (2.16), Notations and equation numbers as in S.Djordjevic.

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