Abstract
This paper describes a method for the numerical solution of linear system of equations. The main interest of refinement of accelerated over relaxation (RAOR) method is to minimize the spectral radius of the iteration matrix in order to increase the rate of convergence of the method comparing to the accelerated over relaxation (AOR) method. That is minimizing the spectral radius means increasing the rate of convergence of the method. This motivates us to refine the refinement of accelerated over relaxation method called second refinement of accelerated over relaxation method (SRAOR). In this paper, we proposed a second refinement of accelerated over relaxation method, which decreases the spectral radius of the iteration matrix significantly comparing to that of the refinement of accelerated over relaxation (RAOR) method. The method is a two-parameter generalization of the refinement of accelerated over relaxation methods and the optimal value of each parameter is derived. The third, fourth and in general the kth refinement of accelerated methods are also derived. The spectral radius of the iteration matrix and convergence criteria of the second refinement of accelerated over relaxation (SRAOR) are discussed. Finally a numerical example is given in order to see the efficiency of the proposed method comparing with that of the existing methods.
Highlights
IntroductionA new demand for new means of solving systems of linear equations appeared at the same time as the computing technology emerged which promoted a rapid development of numerical methods for modelling physical processes by sampling (sub-dividing) the calculation range as well as replacing the differential operations by similar algebraic operations
A new demand for new means of solving systems of linear equations appeared at the same time as the computing technology emerged which promoted a rapid development of numerical methods for modelling physical processes by sampling the calculation range as well as replacing the differential operations by similar algebraic operations
With the introduction of new numerical methods, there has been a necessity for solving systems of linear equations with a completely filled matrix and one which does not possess the main diagonal dominance
Summary
A new demand for new means of solving systems of linear equations appeared at the same time as the computing technology emerged which promoted a rapid development of numerical methods for modelling physical processes by sampling (sub-dividing) the calculation range as well as replacing the differential operations by similar algebraic operations. According to the requirements of the final differences, final elements and their modifications, direct and iterative methods for approaching a poorly completed diagonal matrix with a strong main diagonal were developed. Methods for efficient storage of the equation system were developed, taking into account the symmetry of the matrix according to the main diagonal for both direct and iterative methods. With the introduction of new numerical methods (super elements, the method of border elements), there has been a necessity for solving systems of linear equations with a completely filled matrix and one which does not possess the main diagonal dominance. Solving systems of linear equations by iterative methods (such as Gauss Jacobi, Gauss-Seidel, Successive over relaxation (SOR), Accelerated over relaxation (AOR) method) involves the correction of one searched-for unknown value in every step by reducing the difference of a single individual equation; other equations in this process are not used. In order to accelerate the convergence of the iterative process, the methods are complemented by wellness principles which optimize the rate of the variable change in the iterative process [14]
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