The Convergence Of SOR iteration method

[sigh1342]

Homework Statement

show SOR iteration method converges for the system.
$$6x+4y+2z=11$$
$$4x+7y+4z=3$$
$$2x+4y+5=-3$$

Homework Equations

if the coeff. matrix is positive definite matrix and 0≤ω≤2. Then SOR converge for any initial guess.
Or if $$ρ(T_{ω})$$≥|ω-1|, then SOR converge for any initial guess.ρ(T) means the largest magnitude of all eigenvalue of T.$$T_{ω}=(I − ωL)^{-1} ((1 − ω)I + ωU)$$
Or any norm of $$T_{ω} <1 $$ Then SOR converge for any initial guess

The Attempt at a Solution

I found that the coeff. matrix is not positive definite matrix . and the ρ(T) is hard to find .
Any other method ? Or what I miss. Thanks you

 

[fzero]

I found that the coeff. matrix is not positive definite matrix

Can you explain how you drew that conclusion? The coefficient matrix has positive eigenvalues.

 

[sigh1342]

Can you explain how you drew that conclusion? The coefficient matrix has positive eigenvalues.

Oh I find that the value $$x^tAx$$ that I was computed is wrong .
Thanks you so much