Introduction

Abstract

After some historical comments in Section 1.1, we introduce a model problem (Section 1.2) serving as a first test example of the various iterative methods. Deliberately, a simply structured problem is chosen since this allows us to determine all required quantities explicitly. The role of the ordering of the unknowns is explained. Section 1.3 introduces the notation for vectors and matrices. Besides the behaviour of an iterative method for a single system, one is often more interested in the behaviour with respect to a whole family of systems (see Section 1.4). In Section 1.5, the cost of the direct solution by the Gauss elimination is determined. This cost can be compared with the cost of the iterative methods introduced later. In Section 1.6, the Gauss–Seidel and SOR iteration are presented as first examples of linear iterations. Finally, in Section 1.7, sparsity of the underlying matrix discussed.

I recommend this [iterative] method to you for imitation. You will hardly ever again eliminate directly, at least not when you have more than 2 unknowns. The indirect procedure can be done while half asleep, or while thinking about other things.

(C.F. Gauss in a letter to Gerling [148], Dec. 1823).