scholarly journals A family of modified Newton iteration method for solving nonlinear algebraic equations

2017 ◽  
Vol 20 (K2) ◽  
pp. 34-41
Author(s):  
Luc Xuan Nghiem ◽  
Hieu Nhu Nguyen
Keyword(s):  
Iteration Method ◽  
Order Of Convergence ◽  
Approximate Solutions ◽  
Newton Iteration ◽  
Convergence Order ◽  
Correction Function ◽  
Algebraic Equations ◽  
Order Condition ◽  
Nonlinear Algebraic Equations ◽  
Newton Iteration Method

In this study, a modified Newton iteration version for solving nonlinear algebraic equations is formulated using a correction function derived from convergence order condition of iteration. If the second order of convergence is selected, we get a family of the modified Newton iteration method. Several forms of the correction function are proposed in checking the effectiveness and accuracy of the present iteration method. For illustration, approximate solutions of four examples of nonlinear algebraic equations are obtained and then compared with those obtained from the classical Newton iteration method.

scholarly journals On the properties of numerical solutions of dynamical systems obtained using the midpoint method

2019 ◽  
Vol 27 (3) ◽  
pp. 242-262 ◽  
Author(s):  
Vladimir P. Gerdt ◽  
Mikhail D. Malykh ◽  
Leonid A. Sevastianov ◽  
Yu Ying
Keyword(s):  
Difference Scheme ◽  
Coupled Oscillators ◽  
Numerical Solutions ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Integrals Of Motion ◽  
Midpoint Method ◽  
Nonlinear Algebraic Equations ◽  
The Difference

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

scholarly journals Metode Iterasi Tiga Langkah untuk Menyelesaikan Persamaan NonLinear dengan Menggunakan Matlab

2019 ◽  
Vol 4 (2) ◽  
pp. 34
Author(s):  
Deasy Wahyuni ◽  
Elisawati Elisawati
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Equations ◽  
Newton Method ◽  
Iteration Method ◽  
Newton's Method ◽  
Order Of Convergence ◽  
Higher Order ◽  
Convergence Order ◽  
Matlab Program ◽  
Analytical Results

Newton method is one of the most frequently used methods to find solutions to the roots of nonlinear equations. Along with the development of science, Newton's method has undergone various modifications. One of them is the hasanov method and the newton method variant (vmn), with a higher order of convergence. In this journal focuses on the three-step iteration method in which the order of convergence is higher than the three methods. To find the convergence order of the three-step iteration method requires a program that can support the analytical results of both methods. One of them using the help of the matlab program. Which will then be compared with numerical simulations also using the matlab program.  Keywords : newton method, newton method variant, Hasanov Method and order of convergence

Legendre Collocation Method to Solve the Riccati Equations with Functional Arguments

2020 ◽  
Vol 17 (10) ◽  
pp. 2050011
Author(s):  
Şuayip Yüzbaşı ◽  
Gamze Yıldırım
Keyword(s):  
Approximate Solutions ◽  
Coefficient Matrix ◽  
Numerical Results ◽  
Matrix Form ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix ◽  
Legendre Collocation Method

In this study, a method for numerically solving Riccatti type differential equations with functional arguments under the mixed condition is presented. For the method, Legendre polynomials, the solution forms and the required expressions are written in the matrix form and the collocation points are defined. Then, by using the obtained matrix relations and the collocation points, the Riccati problem is reduced to a system of nonlinear algebraic equations. The condition in the problem is written in the matrix form and a new system of the nonlinear algebraic equations is found with the aid of the obtained matrix relation. This system is solved and thus the coefficient matrix is detected. This coefficient matrix is written in the solution form and hence approximate solution is obtained. In addition, by defining the residual function, an error problem is established and approximate solutions which give better numerical results are obtained. To demonstrate that the method is trustworthy and convenient, the presented method and error estimation technique are explicated by numerical examples. Consequently, the numerical results are shown more clearly with the aid of the tables and graphs and also the results are compared with the results of other methods.

scholarly journals Statistical Process Monitoring with Biogeography-Based Optimization Independent Component Analysis

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Xiangshun Li ◽  
Di Wei ◽  
Cheng Lei ◽  
Zhiang Li ◽  
Wenlin Wang
Keyword(s):  
Fault Detection ◽  
Independent Component Analysis ◽  
Process Monitoring ◽  
Iteration Method ◽  
Newton Iteration ◽  
Independent Component ◽  
Statistical Process ◽  
Independent Components ◽  
Newton Iteration Method ◽  
Statistical Process Monitoring

Independent Component Analysis (ICA), a type of typical data-driven fault detection techniques, has been widely applied for monitoring industrial processes. FastICA is a classical algorithm of ICA, which extracts independent components by using the Newton iteration method. However, the choice of the initial iterative point of Newton iteration method is difficult; sometimes, selection of different initial iterative points tends to show completely different effects for fault detection. So far, there is still no good strategy to get an ideal initial iterative point for ICA. To solve this problem, a modified ICA algorithm based on biogeography-based optimization (BBO) called BBO-ICA is proposed for the purpose of multivariate statistical process monitoring. The Newton iteration method is replaced with BBO here for extracting independent components. BBO is a novel and effective optimization method to search extremes or maximums. Comparing with the traditional intelligent optimization algorithm of particle swarm optimization (PSO) and so on, BBO behaves with stronger capability and accuracy of searching for solution space. Moreover, numerical simulations are finished with the platform of DAMADICS. Results demonstrate the practicability and effectiveness of BBO-ICA. The proposed BBO-ICA shows better performance of process monitoring than FastICA and PSO-ICA for DAMADICS.

Shifted Boubaker Lagrangian approach for solving biological systems

2018 ◽  
Vol 11 (03) ◽  
pp. 1850039 ◽  
Author(s):  
Kourosh Parand ◽  
Hossein Yousefi ◽  
Mina Fotouhifar ◽  
Mehdi Delkhosh ◽  
Mehdi Hosseinzadeh
Keyword(s):  
Mathematical Models ◽  
Hantavirus Infection ◽  
Infection Model ◽  
Algebraic Equations ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Immunodeficiency Virus ◽  
Nonlinear Algebraic Equations ◽  
Newton Iteration Method ◽  
Hiv Infection Model

Mathematical models and computer simulations are useful experimental tools for building and testing theories. Many mathematical models in biology can be formulated by a nonlinear system of ordinary differential equations. This work deals with the numerical solution of the hantavirus infection model, the human immunodeficiency virus (HIV) infection model of CD4[Formula: see text]T cells and the susceptible–infected–removed (SIR) epidemic model using a new reliable algorithm based on shifted Boubaker Lagrangian (SBL) method. This method reduces the solution of such system to a system of linear or nonlinear algebraic equations which are solved using the Newton iteration method. The obtained results of the proposed method show highly accurate and valid for an arbitrary finite interval. Also, those are compared with fourth-order Runge–Kutta (RK4) method and with the solutions obtained by some other methods in the literature.

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