Article

A Hybrid of the Newton-GMRES and Electromagnetic MetaHeuristic Methods for Solving Systems of Nonlinear Equations

Journal of Mathematical Modelling and Algorithms 01/2009; 8(4):425-443. DOI: 10.1007/s10852-009-9117-1
Source: DBLP

ABSTRACT Solving systems of nonlinear equations is perhaps one of the most difficult problems in all numerical computation. Although
numerous methods have been developed to attack this class of numerical problems, one of the simplest and oldest methods, Newton’s
method is arguably the most commonly used. As is well known, the convergence and performance characteristics of Newton’s method
can be highly sensitive to the initial guess of the solution supplied to the method. In this paper a hybrid scheme is proposed,
in which the Electromagnetic Meta-Heuristic method (EM) is used to supply a good initial guess of the solution to the finite
difference version of the Newton-GMRES method (NG) for solving a system of nonlinear equations. Numerical examples are given
in order to compare the performance of the hybrid of the EM and NG methods. Empirical results show that the proposed method
is an efficient approach for solving systems of nonlinear equations.

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    ABSTRACT: This paper investigates the problems of kinematics, Jacobian, singularity and workspace analysis of a spatial type of 3-PSP parallel manipulator. First, structure and motion variables of the robot are addressed. Two operational modes, non-pure translational and coupled mixed-type are considered. Two inverse kinematics solutions, an analytical and a numerical, for the two operational modes are presented. The direct kinematics of the robot is also solved utilizing a new geometrical approach. It is shown, unlike most parallel robots, the direct kinematics problem of this robot has a unique solution. Next, analytical expressions for the velocity and acceleration relations are derived in invariant form. Auxiliary vectors are introduced to eliminate passive velocity and acceleration vectors. The three types of conventional singularities are analyzed. The notion of non-pure rotational and non-pure translational Jacobian matrices is introduced. The non-pure rotational and non-pure translational Jacobian matrices are combined to form the Jacobian of constraint matrix which is then used to obtain the constraint singularity. Finally, two methods, a discretization method and one based on direct kinematics are presented and robot non-pure translation and coupled mixed-type reachable workspaces are obtained. The influence of tool length on workspace is also studied.
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