Question

Difference between direct and indirect method for solving system of equations

Answer 1

Answer:

With a direct method, you have to do a certain amount of work, and then you obtain your solution. For instance, if you're doing LU factorization, you can't stop halfway through and say "alright, that's good enough". (Although there is something called incomplete LU, but it's actually used in iterative methods!) To get a solution, you need to run the algorithm to the end. Typically, the solution you get will then be close to the exact one.

With iterative methods, you always update your old guess and get (hopefully) a bit closer to the true solution. This means that you can decide how much work you want to invest depending on how accurate you need your solution. For instance, CG is known to converge to the exact solution in n steps for a matrix of size n, and was historically first seen as a direct method because of this. However, after a while people figured out that it works really well if you just stop the iteration much earlier - often you will get a very good approximation after much fewer than n steps. In fact, we can analyze how fast CG converges. The end result is that CG is used as an iterative method for large linear systems today.

The second big difference is that, for a direct method, you generally need to have the entire matrix stored in memory. Not so in iterative methods: here, you typically only need a way to compute the application of the matrix to a vector, i.e., the matrix-vector product. So if you have a very large matrix, but you can relatively quickly compute its application to a vector (for instance, because the matrix is very sparse or has some other kind of structure), you can use this to your advantage.

Answer 2

direct you do directly by simultaneous equations..
Indirectly means by using substitution method.