Second Degree Generalized Successive Over Relaxation Method for Solving System of Linear Equations

Momona Ethiopian Journal of Science (MEJS), V12(1):60-71,2020 ©CNCS, Mekelle University,ISSN:2220-184X Submitted on: 24-10-2018 Revised and Accepted on: 11-04-2020 Second Degree Generalized Successive Over Relaxation Method for Solving System of Linear Equations Firew Hailu, Genanew Gofe Gonfa and Hailu Muleta Chemeda* Department of Mathematics, Bonga College of Teachers’ Education, Bonga, Ethiopia Department of Mathematics, College of Natural Sciences, Salale University, Salale, Ethiopia Department of Mathematics, College of Natural Sciences, Jimma University, Jimma, Ethiopia (* muletah@gmail.com).


Consider a system of linear equations
(1) Where, is an nonsingular coefficient matrix, is a column vector and is solution vector to be determined.
Splitting the matrix as in Young (1972); and Genanew Gofe (2016), Where, be a banded matrix with band length 2m+1 is defined as and are strictly lower and upper triangular parts of , respectively, and they are defined as follows: The linear stationary first degree iteration method defined by Young (1972) is given by (3) Here, is an iteration matrix and is the corresponding column vector.
The first degree Jacobi, Gauss Seidel and Successive over relaxation methods for solving equation (1) can be obtained in the form of , respectively, when .
The First Degree Generalized Successive over relaxation for solving equation (1) reformulated by Davod (2007) is given by (4) where,  Firew, H., Genanew, G. G and Hailu, M. C (MEJS)  converges with any initial guess . The detail of the proof is given by Davod (2007).
Theorem2. Let and A be IWDD matrix and be irreducible matrix. Then the associated GSOR method is convergent for every initial guess . One can refer Davod (2007) for the detail of the proof.

SECOND DEGREE GENERALIZED SUCCESSIVE OVER RELAXATION METHOD
Tesfaye Kebede (2016) defined the linear stationary second degree method as Substituting equation (3) into the right hand side of equation (7), we obtain  , which is completely consistent for any constants and such that ≠ 0.
The Second Degree Generalized SOR iteration method is denoted by (10) is the iteration matrix of generalized successive over relaxation iterative method and is its corresponding column vector. Hence , and .
Using the idea of Golub and Varga (1961), equation (8) can be written in the matrix form as The necessary and sufficient conditions for convergence of the method is that the spectral radius of must be less than unity in magnitude for any and , Where, is the second degree iteration matrix.
Let be the spectral radius of .
If is the eigenvalue of and is the eigenvalue of , then As discussed in Young (1972), if we let (15) The Spectral radius of .
So substituting equation (15) into equation (14), we obtain After simplifying and collecting like terms, we get ( Adding equations (17) and (18) Where, and (21) The corresponding values of the spectral radius of ̂ is (22) Thus, with these choices of and , we have Hence, equation (8) becomes (23) According to Young (1972); and David (1994), if is symmetric positive definite (SPD) matrix, then has real nonnegative eigenvalues and we can apply the second degree iterative method using and .
Where, ̅ is the spectral radius of the Generalized SOR iteration matrix.
From equation (20), we have Therefore, the Second Degree Generalized SOR, according to equation (8), is given by Letting and , we get Where,

Numerical Examples
To illustrate the feasibility and efficiency of the present method when employed to solve system of linear equations, we used two systems of linear equations. All the numerical experiments presented in this section are made in the same condition, using the same problems, using the same processor, memory size and operating system. The processor used is Intel(R) core (TM) i3-  31110M CPU @2.40GHz 2.40GHz with 4GM memory (RAM) with 64 bits operating system (Window 7 home premium). The language program used is MATLAB version 7.60(R2008a). The major factors considered in comparing different numerical methods are i) the accuracy of the numerical solutions, and ii) its computational time (Bedet et al., 1975;and Gananew Gofe, 2016).
It should also be noted that there are other factors to be considered such as stability, versatility, proof against run-time error, and so on which are being considered in most of the MATLAB built in routines (Yang et al., 2005;and Kalambi, 2008). Data about iteration number and computational time (in seconds) obtained using SOR, GSOR and SDGSOR is used for analysis of the result.
1. Solve the system of linear equations considered by Noreen (2012) by SOR, GSOR and SDGSOR iterative methods.

Solve the system of linear equations considered by Noreen (2012) by SOR, GSOR and SDGSOR iterative methods
The above examples are solved numerically by using SOR, GSOR and SGSOR and the results are presented in tables 1, 2, 3 and 4.   Table 3. Spectral radii of SOR, GSOR and SDGSOR iterative methods when = 1.

DISCUSSION
In this paper, a Second Degree Generalized Successive over Relaxation Iterative method for solving large system of Linear Equations is presented. Two practical examples (4 4, and 9 9 system of linear equations) are considered. The initial approximation for both systems is taken as all zero vectors. The stopping criterion is used. We let m=1 and in this case is a tri-diagonal matrix. A simple experimental determination of is used to find the optimum relaxation factor. We tried different values of and compared the rates of convergence and continued the experiment with the value of which gives better approximation. The results presented in tables 1, 2, 3 and 4 show that the Second Degree Generalized Successive over relaxation requires less computational running time, less number of iterations and approximates the exact solution better than the other methods used for comparison. Hailu Muleta and Genanew Gofe (2018) stated that method which registers small number of iterations demands less computer storage to store its data. As a result, the Second Degree Generalized Successive over relaxation demands less computer storage to store its data compared to the methods considered for comparison.

ACKNOWLEDGMENTS
We, the authors, would like to express our heartfelt thanks to Jimma University for funding the paper partially so that it became reality.

CONFLICT OF INTEREST
There are no conflicts of interest.