scholarly journals On the derivative-free quasi-Newton-type algorithm for separable systems of nonlinear equations

2021 ◽  
Author(s):  
Hassan Mohammad ◽  
Aliyu Muhammed Awwal ◽  
Auwal Bala Abubakar ◽  
Ahmad Salihu Ben Musa
Keyword(s):  
Nonlinear Equations ◽  
Large Scale ◽  
Numerical Test ◽  
Linear Convergence ◽  
Diagonal Matrix ◽  
Systems Of Nonlinear Equations ◽  
Residual Vector ◽  
Derivative Free ◽  
Type Algorithm ◽  
Quasi Newton

A derivative-free quasi-Newton-type algorithm in which its search direction is a product of a positive definite diagonal matrix and a residual vector is presented. The algorithm is simple to implement and has the ability to solve large-scale nonlinear systems of equations with separable functions. The diagonal matrix is simply obtained in a quasi-Newton manner at each iteration. Under some suitable conditions, the global and R-linear convergence result of the algorithm are presented. Numerical test on some benchmark separable nonlinear equations problems reveal the robustness and efficiency of the algorithm.

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.

MODIFICATION OF PSB QUASI-NEWTON UPDATE AND ITS GLOBAL CONVERGENCE FOR SOLVING SYSTEMS OF NONLINEAR EQUATIONS

2021 ◽  
Vol 4 (4) ◽  
pp. 382-390
Author(s):  
Muhammad Kabir Dauda
Keyword(s):  
Large Scale ◽  
Nonlinear Problems ◽  
Benchmark Problems ◽  
Systems Of Nonlinear Equations ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Derivative Free ◽  
Broyden Update ◽  
Quasi Newton ◽  
Jacobian Inverse

Nonlinear problems mostly emanate from the work of engineers, physicists, mathematicians and many other scientists. A variety of iterative methods have been developed for solving large scale nonlinear systems of equations. A prominent method for solving such equations is the classical Newton’s method, but it has many shortcomings that include computing Jacobian inverse that sometimes fails. To overcome such drawbacks, an approximation with derivative free line is used on an existing method. The method uses PSB (Powell-Symmetric Broyden) update. The efficiency of the proposed method has been improved in terms of number of iteration and CPU time, hence the aim of this research. The preliminary numerical results show that the proposed method is practically efficient when applied on some benchmark problems.

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>

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 An Improved Diagonal Jacobian Approximation via a New Quasi-Cauchy Condition for Solving Large-Scale Systems of Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Mohammed Yusuf Waziri ◽  
Zanariah Abdul Majid
Keyword(s):  
Nonlinear Equations ◽  
Large Scale ◽  
Systems Of Nonlinear Equations ◽  
Numerical Comparison ◽  
Large Scale Systems ◽  
Additional Information ◽  
Cauchy Relation ◽  
Diagonal Updating ◽  
Quasi Newton

We present a new diagonal quasi-Newton update with an improved diagonal Jacobian approximation for solving large-scale systems of nonlinear equations. In this approach, the Jacobian approximation is derived based on the quasi-Cauchy condition. The anticipation has been to further improve the performance of diagonal updating, by modifying the quasi-Cauchy relation so as to carry some additional information from the functions. The effectiveness of our proposed scheme is appraised through numerical comparison with some well-known Newton-like methods.

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 An efficient three-term conjugate gradient-type algorithm for monotone nonlinear equations

2020 ◽  
Author(s):  
Jamilu Sabi'u ◽  
Abdullah Shah
Keyword(s):  
Global Convergence ◽  
Nonlinear Equations ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Numerical Results ◽  
Search Direction ◽  
Projection Technique ◽  
Type Algorithm ◽  
Gradient Type

In this article, we proposed two Conjugate Gradient (CG) parameters using the modified Dai-{L}iao condition and the descent three-term CG search direction. Both parameters are incorporated with the projection technique for solving large-scale monotone nonlinear equations. Using the Lipschitz and monotone assumptions, the global convergence of methods has been proved. Finally, numerical results are provided to illustrate the robustness of the proposed methods.

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