Superior accuracy of Ostrowski homotopy continuation method with quadratic Bezier homotopy and linear fixed point functions for nonlinear equations

2014 ◽  
Author(s):  
Hafizudin Mohamad Nor ◽  
Amirah Rahman ◽  
Ahmad Izani Md. Ismail ◽  
Ahmad Abd. Majid
Keyword(s):  
Fixed Point ◽  
Nonlinear Equations ◽  
Continuation Method ◽  
Homotopy Continuation ◽  
Homotopy Continuation Method

scholarly journals A Parameter Perturbation Homotopy Continuation Method for Solving Fixed Point Problems with Both Inequality and Equality Constraints

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Menglong Su ◽  
Yufeng Shang ◽  
Wenzhuang Zhu
Keyword(s):  
Fixed Point ◽  
Global Convergence ◽  
Continuation Method ◽  
Equality Constraints ◽  
Fixed Point Problems ◽  
Homotopy Continuation ◽  
Parameter Perturbation ◽  
Numerical Examples ◽  
Homotopy Continuation Method ◽  
Nonconvex Sets

In this paper, we propose a parameter perturbation homotopy continuation method for solving fixed point problems on more general nonconvex sets with both inequality and equality constraints. By adopting appropriate techniques, we make the initial points not certainly in the set consisting of the equality constraints. This point can improve the computational efficiency greatly when the equality constraints are complex. In addition, we also weaken the assumptions of the previous results in the literature so that the method proposed in this paper can be applied to solve fixed point problems in more general nonconvex sets. Under suitable conditions, we obtain the global convergence of this homotopy continuation method. Moreover, we provide several numerical examples to illustrate the results of this paper.

scholarly journals New collocation path-following approach for the optimal shape parameter using Kernel method

2021 ◽  
Vol 3 (2) ◽  
Author(s):  
Zouhair Saffah ◽  
Abdelaziz Timesli ◽  
Hassane Lahmam ◽  
Abderrahim Azouani ◽  
Mohamed Amdi
Keyword(s):  
Shape Parameter ◽  
Kernel Method ◽  
Continuation Method ◽  
Path Following ◽  
High Order ◽  
Homotopy Continuation ◽  
Homotopy Continuation Method ◽  
Optimal Value ◽  
Rbf Kernel ◽  
Order Algorithm

AbstractThe goal of this work is to develop a numerical method combining Radial Basic Functions (RBF) kernel and a high order algorithm based on Taylor series and homotopy continuation method. The local RBF approximation applied in strong form allows us to overcome the difficulties of numerical integration and to treat problems of large deformations. Furthermore, the high order algorithm enables to transform the nonlinear problem to a set of linear problems. Determining the optimal value of the shape parameter in RBF kernel is still an outstanding research topic. This optimal value depends on density and distribution of points and the considered problem for e.g. boundary value problems, integral equations, delay-differential equations etc. These have been extensively attempts in literature which end up choosing this optimal value by tests and error or some other ad-hoc means. Our contribution in this paper is to suggest a new strategy using radial basis functions kernel with an automatic reasonable choice of the shape parameter in the nonlinear case which depends on the accuracy and stability of the results. The computational experiments tested on some examples in structural analysis are performed and the comparison with respect to the state of art algorithms from the literature is given.

Sign in / Sign up

Export Citation Format

Share Document