scholarly journals En enhanced matrix-free method via double step length approach for solving systems of nonlinear equations

2017 ◽  
Vol 6 (4) ◽  
pp. 147 ◽  
Author(s):  
Abubakar Sani Halilu ◽  
H. Abdullahi ◽  
Mohammed Yusuf Waziri
Keyword(s):  
Nonlinear Equations ◽  
Large Scale ◽  
Step Length ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
System Of Nonlinear Equations ◽  
Numerical Comparison ◽  
Inexact Line Search ◽  
Derivative Free ◽  
Variant Method

A variant method for solving system of nonlinear equations is presented. This method use the special form of iteration with two step length parameters, we suggest a derivative-free method without computing the Jacobian via acceleration parameter as well as inexact line search procedure. The proposed method is proven to be globally convergent under mild condition. The preliminary numerical comparison reported in this paper using a large scale benchmark test problems show that the proposed method is practically quite effective.

scholarly journals A Derivative-Free Decent Method Via Acceleration Parameter for Solving Systems of Nonlinear Equations

2019 ◽  
Vol 2 (3) ◽  
pp. 1-4
Author(s):  
Abubakar Sani Halilu ◽  
M K Dauda ◽  
M Y Waziri ◽  
M Mamat
Keyword(s):  
Nonlinear Equations ◽  
Large Scale ◽  
Line Search ◽  
Main Idea ◽  
Step Length ◽  
Descent Method ◽  
Systems Of Nonlinear Equations ◽  
Inexact Line Search ◽  
Large Scale Systems ◽  
Derivative Free

An algorithm for solving large-scale systems of nonlinear equations based on the transformation of the Newton method with the line search into a derivative-free descent method is introduced. Main idea used in the algorithm construction is to approximate the Jacobian by an appropriate diagonal matrix. Furthermore, the step length is calculated using inexact line search procedure. Under appropriate conditions, the proposed method is proved to be globally convergent under mild conditions. The numerical results presented show the efficiency of the proposed method.

scholarly journals Inexact Double Step Length Method For Solving Systems Of Nonlinear Equations

2020 ◽  
Vol 8 (1) ◽  
pp. 165-174 ◽  
Author(s):  
Abubakar Sani Halilu ◽  
Mohammed Yusuf Waziri ◽  
Yau Balarabe Musa
Keyword(s):  
Nonlinear Equations ◽  
Large Scale ◽  
Main Idea ◽  
Step Length ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
Double Step ◽  
Inexact Line Search ◽  
Large Scale Problems ◽  
Matrix Free

In this paper, a single direction with double step length method for solving systems of nonlinear equations is presented. Main idea used in the algorithm is to approximate the Jacobian via acceleration parameter. Furthermore, the two step lengths are calculated using inexact line search procedure. This method is matrix-free, and so is advantageous when solving large-scale problems. The proposed method is proven to be globally convergent under appropriate conditions. The preliminary numerical results reported in this paper using a large-scale benchmark test problems show that the proposed method is practically quite effective.

scholarly journals Improved quasi-newton method via SR1 update for solving symmetric systems of nonlinear equations

2019 ◽  
Vol 15 (1) ◽  
pp. 117-120 ◽  
Author(s):  
Muhammad Kabir Dauda ◽  
Mustafa Mamat ◽  
Mohamad Afendee Mohamed ◽  
Mahammad Yusuf Waziri
Keyword(s):  
Nonlinear Equations ◽  
Newton Method ◽  
Large Scale ◽  
Jacobian Matrix ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
Inexact Line Search ◽  
Cpu Time ◽  
Number Of Iterations ◽  
Symmetric Systems

The systems of nonlinear equations emerges from many areas of computing, scientific and engineering research applications. A variety of an iterative methods for solving such systems have been developed, this include the famous Newton method. Unfortunately, the Newton method suffers setback, which includes storing  matrix at each iteration and computing Jacobian matrix, which may be difficult or even impossible to compute. To overcome the drawbacks that bedeviling Newton method, a modification to SR1 update was proposed in this study. With the aid of inexact line search procedure by Li and Fukushima, the modification was achieved by simply approximating the inverse Hessian matrix  with an identity matrix without computing the Jacobian. Unlike the classical SR1 method, the modification neither require storing  matrix at each iteration nor needed to compute the Jacobian matrix. In finding the solution to non-linear problems of the form  40 benchmark test problems were solved. A comparison was made with other two methods based on CPU time and number of iterations. In this study, the proposed method solved 37 problems effectively in terms of number of iterations. In terms of CPU time, the proposed method also outperformed the existing methods. The contribution from the methodology yielded a method that is suitable for solving symmetric systems of nonlinear equations. The derivative-free feature of the proposed method gave its advantage to solve relatively large-scale problems (10,000 variables) compared to the existing methods. From the preliminary numerical results, the proposed method turned out to be significantly faster, effective and suitable for solving large scale symmetric nonlinear equations.

scholarly journals A Derivative-Free Conjugate Gradient Method and Its Global Convergence for Solving Symmetric Nonlinear Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Mohammed Yusuf Waziri ◽  
Jamilu Sabi’u
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
Storage Requirement ◽  
Large Scale Problems ◽  
Derivative Free ◽  
Symmetric Systems

We suggest a conjugate gradient (CG) method for solving symmetric systems of nonlinear equations without computing Jacobian and gradient via the special structure of the underlying function. This derivative-free feature of the proposed method gives it advantage to solve relatively large-scale problems (500,000 variables) with lower storage requirement compared to some existing methods. Under appropriate conditions, the global convergence of our method is reported. Numerical results on some benchmark test problems show that the proposed method is practically effective.

scholarly journals Efficient matrix-free direction method with line search for solving large-scale system of nonlinear equations

2020 ◽  
Vol 30 (4) ◽  
pp. 399-412
Author(s):  
Abubakar Halilu ◽  
Mohammed Waziri ◽  
Ibrahim Yusuf
Keyword(s):  
Nonlinear Equations ◽  
Large Scale ◽  
Line Search ◽  
Test Problems ◽  
System Of Nonlinear Equations ◽  
Inexact Line Search ◽  
Large Scale Problems ◽  
Scale Test ◽  
Line Search Technique ◽  
Matrix Free

We proposed a matrix-free direction with an inexact line search technique to solve system of nonlinear equations by using double direction approach. In this article, we approximated the Jacobian matrix by appropriately constructed matrix-free method via acceleration parameter. The global convergence of our method is established under mild conditions. Numerical comparisons reported in this paper are based on a set of large-scale test problems and show that the proposed method is efficient for large-scale problems.

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 Modified Conjugate Gradient Method via DFP Update for Solving Systems of Nonlinear Equations

2020 ◽  
Vol 3 (1) ◽  
pp. 43-49
Author(s):  
M K Dauda
Keyword(s):  
Large Scale ◽  
Memory Storage ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Derivative Free ◽  
Scale Test ◽  
Derivative Free Method ◽  
Modified Conjugate Gradient Method

In this study, a fully derivative-free method for solving large scale nonlinear systems of equations via memoryless DFP update is presented. The new proposed method is an enhanced DFP (Davidon-FletcherPowell) update which is matrix and derivative free thereby require low memory storage. Under suitable conditions, the proposed method converges globally. Numerical comparisons using a set of large-scale test problems showed that the proposed method is efficient.

scholarly journals A Projection Hestenes–Stiefel Method with Spectral Parameter for Nonlinear Monotone Equations and Signal Processing

2020 ◽  
Vol 25 (2) ◽  
pp. 27
Author(s):  
Aliyu Muhammed Awwal ◽  
Lin Wang ◽  
Poom Kumam ◽  
Hassan Mohammad ◽  
Wiboonsak Watthayu
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Gradient Methods ◽  
System Of Nonlinear Equations ◽  
Numerical Comparison ◽  
Sufficient Descent Condition ◽  
Monotone Equations ◽  
Matrix Free ◽  
Lipschitzian Continuous

A number of practical problems in science and engineering can be converted into a system of nonlinear equations and therefore, it is imperative to develop efficient methods for solving such equations. Due to their nice convergence properties and low storage requirements, conjugate gradient methods are considered among the most efficient for solving large-scale nonlinear equations. In this paper, a modified conjugate gradient method is proposed based on a projection technique and a suitable line search strategy. The proposed method is matrix-free and its sequence of search directions satisfies sufficient descent condition. Under the assumption that the underlying function is monotone and Lipschitzian continuous, the global convergence of the proposed method is established. The method is applied to solve some benchmark monotone nonlinear equations and also extended to solve ℓ 1 -norm regularized problems to reconstruct a sparse signal in compressive sensing. Numerical comparison with some existing methods shows that the proposed method is competitive, efficient and promising.

scholarly journals Modifications of the continuation method for the solution of systems of nonlinear equations

1979 ◽  
Vol 2 (2) ◽  
pp. 299-308
Author(s):  
G. R. Lindfield ◽  
D. C. Simpson
Keyword(s):  
Nonlinear Equations ◽  
Continuation Method ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
System Of Nonlinear Equations ◽  
Modified Method ◽  
Wide Range

Modifications are proposed to the Davidenko-Broyden algorithm for the solution of a system of nonlinear equations. The aim of the modifications is to reduce the overall number of function evaluations by limiting the number of function evaluations for any one subproblem. To do this alterations are made to the strategy used in determining the subproblems to be solved. The modifications are compared with other methods for a wide range of test problems, and are shown to significantly reduce the number of function evaluations for the difficult problems. For the easier problems the modified method is equivalent to the Davidenko-Broyden algorithm.

scholarly journals A Simple Three-term Conjugate Gradient Algorithm for Solving Symmetric Systems of Nonlinear Equations

2016 ◽  
Vol 5 (3) ◽  
pp. 118 ◽  
Author(s):  
M. Y. Waziri ◽  
L. Muhammad ◽  
J. Sabi’u
Keyword(s):  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Conjugate Gradient Algorithm ◽  
Gradient Algorithm ◽  
Systems Of Nonlinear Equations ◽  
Large Scale Systems ◽  
Large Scale Problems ◽  
Derivative Free ◽  
Symmetric Systems

<p>This paper presents a simple three-terms Conjugate Gradient algorithm for solving Large-Scale systems of nonlinear equations without computing Jacobian and gradient via the special structure of the underlying function. This three term CG of the proposed method has an advantage of solving relatively large-scale problems, with lower storage requirement compared to some existing methods. By incoporating the Powel restart approach in to the algorithm, we prove the global convergence of the proposed method with a derivative free line search under suitable assumtions. The numerical results are presented which show that the proposed method is promising.</p>

Sign in / Sign up

Export Citation Format

Share Document