When does an iterative method converge?

whenever you want ;-)

(Assuming a well posed and convergent problem...)

Since iterative methods are based on minimizing residuals the user must choose which residual is acceptable or not...

to put it in very simple terms,when the results are not changing with the iterations.. the method is said to be converged. Ideally it is not possible to get the results exactly same after 2 iterations. It depends on the order of accuracy that you are looking for. I mean what order of error is acceptable to you.. Hope this will clarify your doubt. Cheers Rags

I read something about the eigvalues of the matrix C=M^(-1)*N which is the matrix of error; if E is the error and n the number of iterations then E(n+1)=C*E(n) I read that eigenvalues (L) of C must be lower than 1 and that L(k)<L(k-1)<...<L(1)<1 where [k x k] is C dimension. Why do we need this condition? Thank you very much!

Are you asking when you should stop your iteration because your solution is good enough? That's the question folks are answering.

Or are you asking how to evaluate an iterative method before you use it to be sure you have a convergent method? Your comment about eigenvalues suggests that's what you mean.

Maybe the linear algebra experts reading these posts will have a good answer for you.

Good luck.

an itertion is a process of steps until u get a satisfied answer.bcoz when applying numerical methods we have error in our solution, for every iteration this error is getting reduced, finally error reaches nearly zero that showr our rsults are converged, if the errors are amplified our results become unstable(refer stability analysis)