Question

The Gauss-Seidel method as an iterative technique often refers to an improved version of the Jacobi method, since the Gauss-Seidel method generally achieves a faster convergence. Describe the difference between the Gauss-Seidel and Jacobi methods.

Answer 1

The techniques for solving linear systems of equation can be separated into two parts i.e. Direct Methods and Iterative Methods. Direct methods are not suitable for solving large number of equations in a system, mainly when the coefficient matrix is sparse. Iterative methods are very efficient regarding computer storage and time requirements.

The first iterative technique is called the Jacobi method. It is a iterative method of solving system of linear equations. This Jacobi method takes two assumptions. First one, that the system given has a unique solution and second one, that the coefficient matrix has no zeros on its main diagonal. In case if any of the diagonal entries are found zero, then rows or columns have to be interchanged to obtain a coefficient matrix which has nonzero entries on the main diagonal.

The second iterative technique is called Gauss Seidel method, which is nothing but the modification of Jacobi method. This modification increases the efficiency level as it often requires less iteration to produce the same degree of accuracy. In the Jacobi method, the values obtained in the n^th approximations remain unaffected until the entire n^th approximation has been calculated. With the Gauss Seidel method, in contrast, we use the new values of each one, as soon as they are identified. It means, once we have determined values from the very first equation, its value is then used in the second equation to get the new values. Likewise, the new value and the first value are utilized in the third equation to find the new and so onwards.

In other words we can say that , in Jacobi method, each iteration is performed using a set of preceding values, even if new values become available in the mid of the iteration. In Gauss Seidel method, as soon as we have a new iteration for a specific element, we can use it in all further subsequent computations.