Interval Analysis

5.5Iterative methods

One of the earliest references to an iterative method for solving linear systems of equations is contained in a letter by Gauss to his student Christian Ludwig Gerling dated 26 December 1823 in the context of solving least squares problems via the normal equations. After briefly describing his method on a 4 × 4 example, Gauss wrote:

I recommend this method to you for imitation. You will hardly ever again eliminate directly, at least not when you have more than two unknowns. The indirect procedure can be done while half asleep, or while thinking about other things.¹

Although Gauss seems to be a little too optimistic, iterative methods are often applied in numerical analysis, especially for large linear systems. One way to ...