Abstract
In this paper, a second degree generalized successive over relaxation iterative method for solving system of linear equations based on the decomposition 
 A= Dm+Lm+Um is presented and the convergence properties of the proposed method are discussed. Two numerical examples are considered to show the efficiency of the proposed method. The results presented in tables show that the Second Degree Generalized Successive Over Relaxation Iterative method is more efficient than the other methods considered based on number of iterations, computational running time and accuracy. Keywords: Second Degree, Generalized Gauss Seidel, Successive over relaxation, Convergence.
Highlights
Consider a system of linear equations Ax b (1)Where, A is an nxn nonsingular coefficient matrix, b is a column vector and x is solution vector to be determined.Splitting the matrix A as in Young (1972); and Genanew Gofe (2016),A Dm Lm Um (2)Where, Dm dij be a banded matrix with band length 2m+1 is defined as dij aij, j i m 0; othetwiseLm and Um are strictly lower and upper triangular parts of A Dm, respectively, and they are defined as follows: Momona Ethiopian Journal of Science (MEJS), V12(1):60-71,2020 ©CNCS, Mekelle University,ISSN:2220-184XSubmitted on: 24-10-2018Revised and Accepted on: 11-04-2020
The Second Degree Generalized SOR iteration method is denoted by x(k 1) Gx k Hx(k 1) l
The necessary and sufficient conditions for convergence of the method is that the spectral radius of G must be less than unity in magnitude for any x(0) and x(1), Where, G
Summary
A is an nxn nonsingular coefficient matrix, b is a column vector and x is solution vector to be determined. G1 is an iteration matrix and C is the corresponding column vector. Equation (5) is the Generalized Successive over relaxation iteration matrix and equation (6) is its iteration vector. PAPT is a block upper triangular matrix; otherwise it is an irreducible. A matrix A is said to be strictly diagonally dominant (SDD) if n aii aij , i 1, 2,..., n. A matrix A is said to be irreducibly weakly diagonally dominant (IWDD) if A is WDD and irreducible. A real matrix A is said to be positive definite or positive real if (Ax, x) 0 ,. The detail of the proof is given by Davod (2007). Let 0 1 and A be IWDD matrix and Dm be irreducible matrix. One can refer Davod (2007) for the detail of the proof
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