Research on Solving Systems of Nonlinear Equations Based on Improved PSO

Abstract

Solving systems of nonlinear equations is perhaps one of the most difficult problems in all of numerical computations, especially in a diverse range of engineering applications. The convergence and performance characteristics can be highly sensitive to the initial guess of the solution for most numerical methods such as Newton's method. However, it is very difficult to select reasonable initial guess of the solution for most systems of nonlinear equations. Besides, the computational efficiency is not high enough. Aiming at these problems, an improved particle swarm optimization algorithm (imPSO) is proposed, which can overcome the problem of selecting reasonable initial guess of the solution and improve the computational efficiency. The convergence and performance characteristics of this method are demonstrated through some standard systems. The results show that the improved PSO for solving systems of nonlinear equations has reliable convergence probability, high convergence rate, and solution precision and is a successful approach in solving systems of nonlinear equations.

Probability-Directed Problem Optimization Technique for Solving Systems of Linear and Non-Linear Equations

“…We start our analysisby studying the performance of our algorithm on samples of systems with few but challenging equations obtained from [16] [17]. Table 1 shows these systems and the domains of their variables.…”

“…Researchers have proposed many methods for solving systems of equations [1] [7] [8] (see [5] for a good discussion of these methods). These methods either solve the equations directly by applying numerical methods [4] [9][10] [11] [17] [23] or solve them indirectly by transforming the system of equations into problem optimization [2] [3][4] [5] [12] [16] [19] [22]. This paper offers a technique for solving systems of linear or nonlinear equations.…”

Robust Fractional Order Controllers for Distributed Systems

“…To avoid falling into local optimal value and to ensure a fast convergence speed of optimization, the inertia weight [11] is used:…”

“…where K c is the gain, Once this continuous-time approximation of the fractional order PID controller in (11) has been obtained, the next step is to compute the discrete-time poles and zeros, using the inverse operator of (12):…”

Visual Analysis of Dynamics Behaviour of an Iterative Method Depending on Selected Parameters and Modifications

“…To overcome problems of being trapped in local minima and slow convergence, Jaberipour et al proposed a new way of updating particle’s position [ 17 ]. Another modification of the PSO algorithm for solving systems of non-linear equations was proposed by Li et al in [ 18 ]. The authors proposed: (1) a new way of inertia weight selection, (2) dynamics conditions of stopping iterating, (3) calculation of the standardised number of restarting times based on reliability theory.…”