Nonlinear System of Equations Newton-Raphson and SOR Sucessive Over Relaxatiom

[teknodude]

Homework Statement

3X1 - COS(X2*X3) – 0.5 = 0
X1^2 - 81*(X2 + 0.1)^2 + SIN(X3) + 1.06 = 0
EXP(-X1*X2) + 20*X3 + (10pi – 3)/3 = 0

Homework Equations

Newton Raphsom and Gauss Sceidal with relaxation term

The Attempt at a Solution

I've already solved the above system using Newton Raphsom and got an output of x=[0.5000, 0.0285,-0.5229] using various initial guesses. So far I've only been able to find one solution with my guesses with my Matlab code.

I need some help understanding the method of solving this with SOR. My understanding and approach so far is to use the Newtom Raphsom method to linearize the system. Thus this will generate the delta x (the change in x). From there I can use SOR and input the value found from Newton and iterate until my error is maybe less than 10^-5.

I just can't get it to converge to the answer that I want. One term in x seems to keep increasing and eventually blows up. Is my approach correct?

 

[mighty2000]

Any tips or pointers would be greatly appreciated.


Your approach is on the right track. Here are some tips for using the SOR method to solve this system of equations:

1. Start with a good initial guess for x. This is crucial for the SOR method to converge. You can use the solution you found using Newton Raphson as your initial guess.

2. Choose a value for the relaxation parameter, ω. This parameter controls the speed of convergence and can greatly affect the success of the SOR method. A good starting value to try is ω=1, but you may need to adjust it to find the best value for your system.

3. Set an error tolerance, ε, for your solution. This will be used to determine when the SOR method has converged. A good starting value for ε is 10^-5, but you may need to adjust it depending on the accuracy you require.

4. Use the linearized system from Newton Raphson to set up the iterative formula for the SOR method. This will involve solving for x1, x2, and x3 in terms of the other variables. It may be helpful to write out the formula for each variable to make sure you have it set up correctly.

5. Start with the initial guess for x and use the SOR formula to calculate the new values for each variable. Then use these new values to calculate the error, ε, and check if it is less than your error tolerance. If it is, then you have converged to a solution. If not, continue iterating until ε is less than ε.

6. If the SOR method does not converge, try adjusting the relaxation parameter, ω, and/or the initial guess for x. You may also need to adjust the error tolerance, ε, depending on the accuracy you require.

7. Once you have found a solution, check it against the original system of equations to make sure it satisfies all of them.

Overall, the key to success with the SOR method is to carefully choose your initial guess, relaxation parameter, and error tolerance, and to make sure you have set up the iterative formula correctly. Keep adjusting these parameters until you find a solution that works. Good luck!