I am using GMRES to approximate the solution of a system of equations $Ax=b$, I am using a preconditioner $P$ to make GMRES converge faster.
My question is how do I know if the preconditioner I am using is "good", I mean, I have read that the condition number is defined as $\kappa(A) = ||A^{-1}|| \cdot ||A||$, the closer to one the better, but that involves to compute $A^{-1}$. Is there a practical (not costly) way to know if the preconditioner $P$ is "good". For example, in the number of iterations taken by the preconditioned GMRES, how many iterations should it take to say that it is good?. Let say that the non preconditioned GMRES takes 1000 iterations to converge, and that the preconditioned GMRES takes 100, could we just conclude that the preconditioner $P$ is "good" or how can I measure how "good" (or bad) is $P$. Please help!!!