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 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 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 A new family of conjugate gradient coefficient with application

2018 ◽  
Vol 7 (3.28) ◽  
pp. 36
Author(s):  
Norrlaili Shapiee ◽  
Mohd Rivaie ◽  
Mustafa Mamat ◽  
Puspa Liza Ghazali
Keyword(s):  
Conjugate Gradient ◽  
Large Scale ◽  
Optimization Problems ◽  
Real Life ◽  
Standard Test ◽  
Test Problems ◽  
Inexact Line Search ◽  
New Family ◽  
Unconstrained Optimization Problems ◽  
Large Scale Problems

Conjugate gradient (CG) methods are famous for their utilization in solving unconstrained optimization problems, particularly for large scale problems and have become more intriguing such as in engineering field. In this paper, we propose a new family of CG coefficient and apply in regression analysis. The global convergence is established by using exact and inexact line search. Numerical results are presented based on the number of iterations and CPU time. The findings show that our method is more efficient in comparison to some of the previous CG methods for a given standard test problems and successfully solve the real life problem.  

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 Global Convergence of Schubert’s Method for Solving Sparse Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Huiping Cao
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Large Scale ◽  
Line Search ◽  
Jacobian Matrix ◽  
Nonmonotone Line Search ◽  
Broyden’S Method ◽  
Large Scale Problems ◽  
Broyden's Method

Schubert’s method is an extension of Broyden’s method for solving sparse nonlinear equations, which can preserve the zero-nonzero structure defined by the sparse Jacobian matrix and can retain many good properties of Broyden’s method. In particular, Schubert’s method has been proved to be locally andq-superlinearly convergent. In this paper, we globalize Schubert’s method by using a nonmonotone line search. Under appropriate conditions, we show that the proposed algorithm converges globally and superlinearly. Some preliminary numerical experiments are presented, which demonstrate that our algorithm is effective for large-scale problems.

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 Conjugate Gradient Algorithm under Yuan-Wei-Lu Line Search Technique for Large-Scale Minimization Optimization Models

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Xiangrong Li ◽  
Songhua Wang ◽  
Zhongzhou Jin ◽  
Hongtruong Pham
Keyword(s):  
Conjugate Gradient ◽  
Large Scale ◽  
Line Search ◽  
Optimization Problems ◽  
Conjugate Gradient Algorithm ◽  
Gradient Algorithm ◽  
Search Technique ◽  
Inexact Line Search ◽  
Line Search Technique ◽  
Large Scale Unconstrained Optimization

This paper gives a modified Hestenes and Stiefel (HS) conjugate gradient algorithm under the Yuan-Wei-Lu inexact line search technique for large-scale unconstrained optimization problems, where the proposed algorithm has the following properties: (1) the new search direction possesses not only a sufficient descent property but also a trust region feature; (2) the presented algorithm has global convergence for nonconvex functions; (3) the numerical experiment showed that the new algorithm is more effective than similar algorithms.

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.

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