Motivations and realizations of Krylov subspace methods for large sparse linear systems

We briefly introduce typical and important direct and iterative methods for solving systems of linear equations, concretely describe their fundamental characteristics in viewpoints of both theory and applications, and clearly clarify the substantial differences among these methods. In particular, the motivations of searching the solution of a linear system in a Krylov subspace are described and the algorithmic realizations of the generalized minimal residual (GMRES) method are shown, and several classes of state-of-the-art algebraic preconditioners are briefly reviewed. All this is useful for correctly, deeply and completely understanding the application scopes, theoretical properties and numerical behaviors of these methods, and is also helpful in designing new methods for solving systems of linear equations.