The Jacobi Iterative method question

1. Mar 19, 2012

Spectre Moncy

1. The problem statement, all variables and given/known data

(Ax = B)

A:

3.1410 -2.7180 1.4140 -1.7321
9.8690 2.7180 -7.3890 0.4280
2.2360 -2.4490 1.0000 -1.4140
31.0060 7.3890 -2.6450 0.1110

B:

3.316
0
3.141
1.414

The question in my Numerical Methods assignment asks to use the Jacobi Iterative method to solve the system.

2. Relevant equations

The Jacobi Iterative method works ONLY IF a matrix is diagonally dominant. It's not mentioned in my Numerical Analysis textbook. I have only found out about this on wiki.org.

I have found out that the matrix A (See above) is not diagonally dominant. So the method will not work on this system (Ax = B).

Did I miss something? What should I do? I tried the method very carefully. The solution set doesn't make sense.

2. Mar 19, 2012

clamtrox

Jacobi method can converge even if the matrix is not diagonally dominant. However, you notice immediately from the iteration formula
$x_i^{n+1} = \frac{b_i - \sum_{j \neq i} a_{ij} x^{n}_j}{a_{ii}}$
that if the matrix is not diagonally dominant,
$\frac{\sum_{j \neq i} a_{ij}}{a_{ii}} \gt 1$
then the convergence depends on the initial value you choose for x. Perhaps you can make a better guess for the initial x, or if that fails, look up the correct value from wolfram alpha and adjust your guess accordingly. :-)

3. Mar 19, 2012

Spectre Moncy

The initial x^(0) (provided by the assignment question paper) is x^(0) = (3, 0, 3, 1).

I tried this. It converged to some solution set but the problem is that this solution set is extremely inaccurate.

I have no problem finding the right solution set when applying Gaussian Elimination (with partial pivoting) and LU Decomposition method on this system (Ax = b).

I can't say the same for the Jacobi method :(