A New Improved Classical Iterative Algorithm for Solving System of Linear Equations

Abstract

The fundamental problem of linear algebra is to solve the system of linear equations (SOLE’s). To solve SOLE’s, is one of the most crucial topics in iterative methods. The SOLE’s occurs throughout the natural sciences, social sciences, engineering, medicine and business. For the most part, iterative methods are used for solving sparse SOLE’s. In this research, an improved iterative scheme namely, ‘’a new improved classical iterative algorithm (NICA)’’ has been developed. The proposed iterative method is valid when the co-efficient matrix of SOLE’s is strictly diagonally dominant (SDD), irreducibly diagonally dominant (IDD), M-matrix, Symmetric positive definite with some conditions and H-matrix. Such types of SOLE’s does arise usually from ordinary differential equations (ODE’s) and partial differential equations (PDE’s). The proposed method reduces the number of iterations, decreases spectral radius and increases the rate of convergence. Some numerical examples are utilized to demonstrate the effectiveness of NICA over Jacobi (J), Gauss Siedel (GS), Successive Over Relaxation (SOR), Refinement of Jacobi (RJ), Second Refinement of Jacobi (SRJ), Generalized Jacobi (GJ) and Refinement of Generalized Jacobi (RGJ) methods.

An Improved Iterative Scheme using Successive Over-relaxation for Solution of Linear System of Equations

“…The below example 1, 2, 3 and 4 are referred from previously conducted studies [6][7][8][9][10][11][12][13]. Also, the results are depicted in tables 1-4 and figure 1 and 2.…”

“…Improvements in the classical iterative techniques have been done earlier [11], however since the SOR outperforms all the other iterative techniques, an improvement in the SOR scheme will be highly effective. In this research work an iterative scheme namely "improved iterative scheme using successive overrelaxation (IIS)" is developed.…”

An Accelerated Iterative Technique: Third Refinement of Gauss–Seidel Algorithm for Linear Systems

“…Iterative approaches are unquestionably the most effective approach to employ when solving huge sparse linear systems. However, such an approach may require several rounds to converge, which may reduce computer storage and computing performance [12]. In such cases, it is necessary to modify or accelerate existing methods in order to achieve approximate answers with rapid convergence.…”