Why Is Gauss-Seidel Method's Convergence Rate Double That of Jacobi's?

[abrowaqas]

How we prove that rate of convergence of gauss-Seidel method is approximately twice that of Jacobi iterative method without doing an example itself ?

What's the general proof of this statement ? I didn't fin in any book ?
Can anyone please help me ?

 

[AlephZero]

Your statement about the convergence speeds is not true in general, though it seems to be widely believed (or at least, it's easy to find assertions that it is true, but without any proof, on the web!).

For example see the first few lines of http://www.mit.edu/~jnt/Papers/J025-89-Jacobi_GS.pdf

The convergence rate of Gauss-Siedel is dependent on the ordering of the matrix, and in some cases it may converge for some orderings but not for others. So proving any general result that GS has better convergence than Jacobi is not going to be easy, even if it is true.

Finding an counter-example might be an easier task.

 

[abrowaqas]

Thanks Alphazero . I was also thinking of that .but in many books I have found thy statement . That's why i raise this question.