SKEDSOFT
This is the modification from of gauss jacobi method .This modification is no more difficult to use than the Jacobi method, and it often requires fewer iterations to produce the same degree of accuracy.
we use the updated values of x1, x2,..., xi-1 in computing
the value of the variable xi. We assume that the pivots aii ≠ 0, for all i. We write the equations as
The Gauss-Seidel iteration method is defined as
..........................1.1
This method is also called the method of successive displacement.
We observe that (1.1) is same as writing the given system as
...................1.2
Note:
A sufficient condition for convergence of the Gauss-Seidel method is that the system of equations is diagonally dominant, that is, the coefficient matrix A is diagonally dominant.This implies that convergence may be obtained even if the system is not diagonally dominant. If the system is not diagonally dominant, we may exchange the equations, if possible, such that the new system is diagonally dominant and convergence is guaranteed. The necessary and sufficient condition for convergence is that the spectral radius of the iteration matrix H is less than one unit, that is, ρ(H) < 1, where ρ(H) is the largest eigen value in
magnitude of H.
If both the Gauss-Jacobi and Gauss-Seidel methods converge, then Gauss-Seidel method converges at least two times faster than the Gauss-Jacobi method.
Example: Find the solution of the system of equations.
correct to three decimal places, using the Gauss-Seidel iteration method.
Solution : The given system of equations is strongly diagonally dominant. Hence, we can expect fast convergence. Gauss-Seidel method gives the iteration
Starting with x1(0) = 0, x2(0)= 0, x3(0) = 0, we get the following results.
First iteration
