# The Convergence Of SOR iteration method

## 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
Homework Helper
Gold Member
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.

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 :tongue: