A Comparison of three Iterative Methods for the Solution of Linear Equations

Abstract

This paper presents a survey of three iterative methods for the solution of linear equations has been evaluated in this work. The result shows that the Successive Over-Relaxation method is more efficient than the other two iterative methods, considering their performance, using parameters as time to converge, number of iterations required to converge, storage and level of accuracy. This research will enable analyst to appreciate the use of iterative techniques for understanding linear equations. @ JASEM The direct methods of solving linear equations are known to have their difficulties. For example the problem with Gauss elimination approach lies in control of the accumulation of rounding errors Turner, (1989). This has encouraged many authors like Rajase Keran (1992), Fridburd et al (1989), Turner (1994) Hageman et al (1998) and Forsyth et al (1999) to investigate the solutions of linear equations by direct and indirect methods. Systems of linear equations arise in a large number of areas both directly in modeling physical situations and indirectly in the numerical solutions of the other mathematical models. These application occur in virtually all areas of the physical, biological and social science. Linear systems are in the numerical solution of optimization problems, system of non linear equations and partial differential equations etc. The most common type of problem is to solve a square linear system AX = b-(1) of moderate order with coefficient that are mostly non zero, such linear system of any order are called dense since the coefficient matrix A is generally stored in the main memory of the computer in order to efficiently solve the linear system, memory storage limitations in most computer will limit the system to be less than 100 to 200 depending on the computer. The efficiency of any method will be judged by two criteria Viz: i) How fast it is? That is how many operations are involved. ii) How accurate is the computer solution. Because of the formidable amount of computations required to linear equation for large system, the need to answer the first questions is clear. The need to answer the second, arise because small round off errors may cause errors in the computer solution out of all proportion to their size. Furthermore, because of the large number of operations involved in solving high-order system, the potential round off errors could cause substantial loss of accuracy. Generally, the matrices of coefficient that occur in practice fall into one of two categories. a. Filled but not large:-This means that there are few zero elements, but not large, that is to say a matrix of order less than 100. Such matrices occur in a wide variety of problems e.g. engineering are statistics etc. b. Sparse and perhaps very large:-In contrast to the above a sparse matrix has few non zero elements, very large matrix of order say one thousand. Such matrices arise commonly in the numerical solution of partial differential equations.