The Jacobi Iterative method question

[Spectre Moncy]

Homework Statement

(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.

Homework 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.

 

[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. :-)

 

[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 :(