Gauss-jacobi Iteration Method

The Jacobi method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal . Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization.

Sometimes, the method is called Jacobi method. We assume that the pivots a_ii ≠ 0, for all i. Write the equations as

The Jacobi iteration method is defined as

.......................1.1

Since, we replace the complete vector x^(k) in the right hand side of (1.1) at the end of each iteration, this method is also called the method of simultaneous displacement.

Note:  

        1. A sufficient condition for convergence of the Jacobi method is that the system of equations is diagonally dominant, that is, the coefficient matrix A is    diagonally dominant. We can verify that

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. However, such manual verification or exchange of equations may not be possible for large systems that we obtain in application problems. 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.

        2. If the system is diagonally dominant, then the iteration converges for any initial solution vector. If no suitable approximation is available, we can choose   x  = 0, that is xi = 0 for all i. Then, the initialapproximation becomes xi = bi /aii, for all i.

Example:    Solve the system of equations

using the Jacobi iteration method. Use the initial approximations as

(i) x_i = 0, i = 1, 2, 3,

Perform five iterations in each case.

Solution :

Note that the given system is diagonally dominant. Jacobi method gives the iterations as