The Convergence Of SOR iteration method

In summary, the SOR iteration method converges for the system when the coefficient matrix has positive eigenvalues and when the value of $$x^tAx$$ is calculated correctly. Other conditions for convergence include a positive definite matrix and 0≤ω≤2, or if $$ρ(T_{ω})$$≥|ω-1|, or if the norm of $$T_{ω}$$ is less than 1.
  • #1
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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 :blushing:
 
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  • #2
sigh1342 said:
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.
 
  • #3
fzero said:
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 . :frown:
Thanks you so much :tongue:
 

1. What is the Convergence of SOR iteration method?

The Convergence of SOR (Successive Over-Relaxation) iteration method is a numerical method used to solve linear systems of equations. It is an iterative method that updates the solution at each step by using a relaxation factor to improve the convergence rate.

2. How does the SOR iteration method work?

The SOR iteration method works by starting with an initial guess for the solution and then using a relaxation factor to update the solution at each step. This process continues until the solution converges to a desired accuracy. The relaxation factor is a parameter that controls the rate of convergence and must be carefully chosen for optimal results.

3. What is the advantage of using the SOR iteration method?

The SOR iteration method has several advantages over other numerical methods for solving linear systems of equations. It is relatively simple to implement and can be used to solve large systems of equations efficiently. Additionally, the convergence rate of the SOR method can be faster than other methods, especially for systems with a dominant diagonal.

4. How do I know if the SOR iteration method has converged?

The SOR iteration method has converged when the difference between the current solution and the previous solution is less than a specified tolerance. This means that the solution has reached a steady state and will no longer change significantly with further iterations. The tolerance can be adjusted depending on the desired accuracy of the solution.

5. What are some applications of the SOR iteration method?

The SOR iteration method is commonly used in scientific and engineering fields to solve linear systems of equations. It has applications in areas such as fluid dynamics, electromagnetism, and structural analysis. It can also be used to solve optimization problems and to simulate dynamic systems.

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