I have the following question regarding the sensitivity analysis of linear systems.
In my numerical methods course, I have learned theorems which give upper bounds for the relative change in the solution of the linear system Ax=f. For example, if only f is changed, relative change in x = K(A)*relative change in f, where K(A) is the condition number of the matrix A.
Now, for such theorems to be useful in practice, we need the condition number of A which is defined as ||A||*||A_inverse||.
So how does one calculate the condition number of a matrix? Given that real world systems are large, can we precisely calculate K(A)? Or, can we only hope to obtain a upper bound for K(A)?