Jacobi, Gauss-Seidel, SOR question

[XodoX]

Homework Statement

Evaluate the number of iterations that are needed to have 10^-9 precision with the Jacobi, Gauss-Seidel, and SOR ( with ω=1.5) methods. Compare these 3 methods for different values of n - for instance 3≤n≤20. Plot the convergence curves for the 3 methods for each n.

Homework Equations

Just the regular formulas for each, I guess.

http://en.wikipedia.org/wiki/Jacobi_method etc.

The Attempt at a Solution

No idea how to do it. Let's start with Jacobi. What is exactly the n here and where do I plug it in? I don't understand it. I guess you just plug in the 3≤n≤20, calculate the result and then plot the curve. For Jacobi, is the Matrix A the n in this case?
Help is very much appreciated!

 

[mighty2000]

Hello,

Great question! The n in this case refers to the size of the matrix A. So for n=3, the matrix A would be a 3x3 matrix, for n=4 it would be a 4x4 matrix, and so on.

To evaluate the number of iterations needed for 10^-9 precision, you can use the formula for the error in the Jacobi method, which is given by ||x^(k) - x^*|| <= (||B||/||A||)^k * ||x^(0) - x^*||, where x^(k) is the approximation at the k-th iteration, x^* is the exact solution, ||B|| is the norm of the matrix B (which is the diagonal elements of A), ||A|| is the norm of A, and ||x^(0) - x^*|| is the initial error.

You can use this formula to calculate the number of iterations needed for different values of n and compare it for the different methods (Jacobi, Gauss-Seidel, and SOR with ω=1.5). You can also plot the convergence curves for each method by plotting the error (||x^(k) - x^*||) against the number of iterations k.

I hope this helps! Let me know if you have any further questions. Good luck!