06-10-2016, 03:49 PM
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Abstract–This paper describes an experiment performed to solve a system of linear equations of the form Ax=b. Iterative techniques have become increasingly important since the advent of scientific computing and its use for solving large system of linear equations. Numerical methods for solving differential equations are used in applications ranging from thermal modeling of transistors, structural modeling of bridges, determining the stress profile in airplane components and biomedical devices and have countless other applications. Often these techniques yield very large and sparse matrices for which direct methods may be slow and prone to round off error. In this report we explain what the Jacobi method is and how it is implemented in serial and parallel. We discuss the results by finding out the time taken to implement the Jacobi Algorithm and also the speedup for different number of threads i.e. from 1 to 16. Finally we conclude by discussing other implementations which can be used to solve a system of linear equations.
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I. INTRODUCTION
heJacobi method is a simple iterative method for solving a system of linear equations . The basic idea of the method is to continually solve for each diagonal element until convergence is reached. This method is simple and is designed to work for linear systems that have the corresponding matrix equation Ax=b. The matrix A should be a NxN square for the Jacobi method to work properly. The vectors x and b should have length n that matches the matrix dimensions.
The matrix A must also be diagonally dominant in order to guarantee the convergence of the Jacobi method. Specifically, the Jacobi method is guaranteed to converge of the matrix A has row diagonal dominance. Diagonal dominance means that the absolute value of the diagonal element is greater than the sum of the absolute values of the other terms in the row. For Example the matrix shown below is diagonally dominant.