scholarly journals MN-PGSOR Method for Solving Nonlinear Systems with Block Two-by-Two Complex Symmetric Jacobian Matrices

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yu-Ye Feng ◽  
Qing-Biao Wu
Keyword(s):  
Nonlinear Systems ◽  
Iteration Method ◽  
Computing Time ◽  
Coefficient Matrix ◽  
New Method ◽  
Jacobian Matrices ◽  
Modified Newton Method ◽  
Preconditioned Matrix ◽  
Outer Iteration ◽  
Inner Iteration

For solving the large sparse linear systems with 2 × 2 block structure, the generalized successive overrelaxation (GSOR) iteration method is an efficient iteration method. Based on the GSOR method, the PGSOR method introduces a preconditioned matrix with a new parameter for the coefficient matrix which can enhance the efficiency. To solve the nonlinear systems in which the Jacobian matrices are complex and symmetric with the block two-by-two form, we try to use the PGSOR method as an inner iteration, with the help of the modified Newton method as an efficient outer iteration method. This new method is called the modified Newton-PGSOR (MN-PGSOR) method. Local convergence properties of the MN-PGSOR are analyzed under the Hölder condition. Finally, we give the comparison of our new method with some previous methods in the numerical results. The MN-PGSOR method is superior in both iteration steps and computing time.

scholarly journals Two new effective iteration methods for nonlinear systems with complex symmetric Jacobian matrices

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Lv Zhang ◽  
Qing-Biao Wu ◽  
Min-Hong Chen ◽  
Rong-Fei Lin
Keyword(s):  
Nonlinear Systems ◽  
Iterative Methods ◽  
Newton Method ◽  
Convergence Properties ◽  
Jacobian Matrices ◽  
New Methods ◽  
Modified Newton Method ◽  
Iteration Methods ◽  
Complex Symmetric ◽  
Inner Iteration

AbstractIn this paper, we mainly discuss the iterative methods for solving nonlinear systems with complex symmetric Jacobian matrices. By applying an FPAE iteration (a fixed-point iteration adding asymptotical error) as the inner iteration of the Newton method and modified Newton method, we get the so–called Newton-FPAE method and modified Newton-FPAE method. The local and semi-local convergence properties under Lipschitz condition are analyzed. Finally, some numerical examples are given to expound the feasibility and validity of the two new methods by comparing them with some other iterative methods.

scholarly journals A Study on the Convergence Analysis of the Inexact Simplified Jacobi–Davidson Method

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Jutao Zhao ◽  
Pengfei Guo
Keyword(s):  
Convergence Analysis ◽  
Iteration Method ◽  
Eigenvalue Problems ◽  
Rayleigh Quotient ◽  
Convergence Factor ◽  
Cubic Convergence ◽  
Davidson Method ◽  
Outer Iteration ◽  
Rayleigh Quotient Iteration ◽  
Inner Iteration

The Jacobi–Davidson iteration method is very efficient in solving Hermitian eigenvalue problems. If the correction equation involved in the Jacobi–Davidson iteration is solved accurately, the simplified Jacobi–Davidson iteration is equivalent to the Rayleigh quotient iteration which achieves cubic convergence rate locally. When the involved linear system is solved by an iteration method, these two methods are also equivalent. In this paper, we present the convergence analysis of the simplified Jacobi–Davidson method and present the estimate of iteration numbers of the inner correction equation. Furthermore, based on the convergence factor, we can see how the accuracy of the inner iteration controls the outer iteration.

scholarly journals Modeling Design Iteration in Product Design and Development and Its Solution by a Novel Artificial Bee Colony Algorithm

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Tinggui Chen ◽  
Renbin Xiao
Keyword(s):  
Product Design ◽  
Iteration Method ◽  
High Performance ◽  
Artificial Bee Colony ◽  
Market Competition ◽  
Processing System ◽  
Design And Development ◽  
Bee Colony ◽  
Design Iteration ◽  
Inner Iteration

Due to fierce market competition, how to improve product quality and reduce development cost determines the core competitiveness of enterprises. However, design iteration generally causes increases of product cost and delays of development time as well, so how to identify and model couplings among tasks in product design and development has become an important issue for enterprises to settle. In this paper, the shortcomings existing in WTM model are discussed and tearing approach as well as inner iteration method is used to complement the classic WTM model. In addition, the ABC algorithm is also introduced to find out the optimal decoupling schemes. In this paper, firstly, tearing approach and inner iteration method are analyzed for solving coupled sets. Secondly, a hybrid iteration model combining these two technologies is set up. Thirdly, a high-performance swarm intelligence algorithm, artificial bee colony, is adopted to realize problem-solving. Finally, an engineering design of a chemical processing system is given in order to verify its reasonability and effectiveness.

scholarly journals Asymptotic Solution for a Class of Epidemic Contagion Ecological Nonlinear Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Yi-Hu Feng ◽  
Lei Hou
Keyword(s):  
Approximate Solution ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Asymptotic Solution ◽  
Iteration Method ◽  
Ecological Model ◽  
Virus Transmission ◽  
Practical Application ◽  
Generalized Variation ◽  
Variation Iteration Method

In this paper, a class of systems for epidemic contagion is considered. An epidemic virus ecological model is described. Using the generalized variation iteration method, the corresponding approximate solution to the nonlinear system is obtained and the method for this approximate solution is pointed out. The accuracy of approximate solution is discussed, and it can control the epidemic virus transmission by using the parameters of the system. Thus, it has the value for practical application.

scholarly journals A New Method to Non-Fragile Piecewise H ∞ Control for Networked Nonlinear Systems With Packet Dropouts

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 196102-196111
Author(s):  
Lingchun Li ◽  
Guangming Zhang ◽  
Meiying Ou
Keyword(s):  
Nonlinear Systems ◽  
New Method ◽  
Packet Dropouts

Application of an Implicit Relaxation Method Solving the Euler Equations for Time-Accurate Unsteady Problems

1990 ◽  
Vol 112 (4) ◽  
pp. 510-520 ◽  
Author(s):  
A. Brenneis ◽  
A. Eberle
Keyword(s):  
Euler Equations ◽  
Computing Time ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Relaxation Method ◽  
Coefficient Matrix ◽  
Viscous Effects ◽  
Left Hand ◽  
Implicit Equations ◽  
The Right

A numerical procedure is presented for computing time-accurate solutions of flows about two and three-dimensional configurations using the Euler equations in conservative form. A nonlinear Newton method is applied to solve the unfactored implicit equations. Relaxation is performed with a point Gauss-Seidel algorithm ensuring a high degree of vectorization by employing the so-called checkerboard scheme. The fundamental feature of the Euler solver is a characteristic variable splitting scheme (Godunov-type averaging procedure, linear locally one-dimensional Riemann solver) based on an eigenvalue analysis for the calculation of the fluxes. The true Jacobians of the fluxes on the right-hand side are used on the left-hand side of the first order in time-discretized Euler equations. A simple matrix conditioning needing only few operations is employed to evade singular behavior of the coefficient matrix. Numerical results are presented for transonic flows about harmonically pitching airfoils and wings. Comparisons with experiments show good agreement except in regions where viscous effects are evident.

Singularity Analysis of a Plane-Symmetry 3-RPS Parallel Robot Based on Translational/Rotational Jacobian Matrices

2011 ◽  
Vol 121-126 ◽  
pp. 1590-1594
Author(s):  
Yan Shi ◽  
Hong Xin Yue ◽  
Yi Lu ◽  
Lian He Guo
Keyword(s):  
Jacobian Matrix ◽  
Parallel Robot ◽  
Singularity Analysis ◽  
Parallel Robots ◽  
New Method ◽  
Plane Symmetry ◽  
Jacobian Matrices ◽  
Different Types

Firstly, 3-DOF parallel robots were classified into different types from the view of moving form. A new method of analyzing the singularity of 3-DOF parallel robots was introduced, which is based on translational Jacobian matrix and rotational Jacobian matrix. The singularity of parallel robots with pure translational form and pure rotational form was introduced summarily. Secondly, the process of solving the plane-symmetry 3-RPS parallel robot with combined moving forms was focused on, through which translational Jacobian matrix and rotational Jacobian matrix were adopted. Finally, the solving results were compared with the axis-symmetry 3-RPS parallel robot, which showed more general singularity can be solved through the new method.

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