Iterative Methods for Linear Problems

Abstract

In this chapter the field of scalars is K = ℝ or ℂ.

We have seen in the previous Chapter a few direct methods for solving a linear system Ax = b, when $A \in M_n \left( K \right)$ is invertible. For example, if A admits an LU factorization, the successive resolution of Ly = b, Ux = y is called the Gauss method. When a leading principal minor of A vanishes, a permutation of the columns allows us to return to the generic case. More generally, the Gauss method with pivoting consists in permuting the columns at each step of the factorization in such a way as to limit the magnitude of round-off errors and that of the conditioning number of the matrices L, U.

The direct computation of the solution of a Cramer’s linear system Ax = b, by the Gauss method or by any other direct method, is rather costly, on the order of n3 operations. It also presents several inconveniences. On the one hand, it does not exploit completely the sparse shape of many matrices A; in numerical analysis it happens frequently that an n x n matrix has only O(n) nonzero entries, instead of O(n2). On the other hand, the computation of an LU factorization is rather unstable, because the round-off errors produced by the computer are amplified at each step of the computation.