Study of conjugate gradient method for system of linear equations using preconditioning

System of linear equations are solved either by direct methods which yield exact solution or by iterative methods which gives an approximation solution. In computational mathematics, iterative proccess plays a vital role that uses an initial guess to determine the approximation solution of any given problem. [7, 9, 10] There are several methods to solve system of linear and nonlinear equations, such as iterative methods, approximation methods, elimination methods and interpolation methods [13, 16, 17]. In this paper we use Conjugate Gradient Method to solve system of linear equations, as this is one of the most popular and well known iterative techniques for solving sparse symmetric positive definite systems of linear equations [24]. In particular, we discuss about pure conjugate gradient method and preconditioned conjugate gradient method. Numerical examples are shown for each method and comparison of these method is done based on the number of iterations and faster convergence. Thus we observe that the preconditioned conjugate gradient method is more preferred when the system is large because it gives less number of iterations and converge faster.