I have a question about direct and iterative method. Many people including me often say that for very large sparse linear system, Ax=b, an iterative method is necessary because of cpu memory. I also believe that ideas.
But, recent days, I find that MATLAB direct methods command A\b for large sparse system is also fast. So, in my opinion, if an author proposes a new iteration method, he/she must compare the cpu time between the new method and MATLAB A\b. If his method is faster than A\b, then I trust that his new iteration method is better and successful. Otherwise, I can't trust that a new iteration method is a successful one if one method can't beat MATLAB A\b. Actually, in most papers, most people do not compare the CPU time in their numerical examples with A\b. So I still have doubts about their so-called a better method.
So, whether we should compare our new iteration method with matlab A\b the cpu time? Because I think if one can beat matlab function, then it will be more persuasive, right? if not, then we will prefer to matlab built-in function,right? thanks.