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 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.

A NEW DERIVATIVE-FREE CONJUGATE GRADIENT METHOD FOR LARGE-SCALE NONLINEAR SYSTEMS OF EQUATIONS

2017 ◽  
Vol 95 (3) ◽  
pp. 500-511 ◽  
Author(s):  
XIAOWEI FANG ◽  
QIN NI
Keyword(s):  
Nonlinear Systems ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Large Scale ◽  
Test Problems ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Line Search Method ◽  
Derivative Free

We propose a new derivative-free conjugate gradient method for large-scale nonlinear systems of equations. The method combines the Rivaie–Mustafa–Ismail–Leong conjugate gradient method for unconstrained optimisation problems and a new nonmonotone line-search method. The global convergence of the proposed method is established under some mild assumptions. Numerical results using 104 test problems from the CUTEst test problem library show that the proposed method is promising.

scholarly journals A Derivative-Free Liu–Storey Method for Solving Large-Scale Nonlinear Systems of Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Zhenhua Su ◽  
Min Li
Keyword(s):  
Nonlinear Systems ◽  
Large Scale ◽  
Line Search ◽  
Convergence Property ◽  
Systems Of Equations ◽  
Storage Requirement ◽  
Nonlinear Systems Of Equations ◽  
Large Scale Problems ◽  
Derivative Free ◽  
Global Convergence Property

In this paper, a descent Liu–Storey conjugate gradient method is extended to solve large-scale nonlinear systems of equations. Based on certain assumptions, the global convergence property is obtained with a nonmonotone line search. The proposed method is suitable to solve large-scale problems for the low-storage requirement. Numerical experiment results show that the new method is practically effective.

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 On the global convergence of a new spectral residual algorithm for nonlinear systems of equations

2020 ◽  
Author(s):  
Alessandra Papini ◽  
Margherita Porcelli ◽  
Cristina Sgattoni
Keyword(s):  
Global Convergence ◽  
Systems Of Nonlinear Equations ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Previous Version ◽  
Convergence Results ◽  
Spectral Residual ◽  
Derivative Free ◽  
Derivative Free Method ◽  
Residual Algorithm

AbstractWe present a derivative-free method for solving systems of nonlinear equations that belongs to the class of spectral residual methods. We will show that by endowing a previous version of the algorithm with a suitable new linesearch strategy, standard global convergence results can be attained under mild general assumptions. The robustness of the new method is therefore potentially improved with respect to the previous version as shown by the reported numerical experiments.

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