As we saw in the previous chapter, there exist two types of methods to solve systems of equations with short recurrence relations: (1) minimization algorithms, such as the conjugate gradient type algorithms, and (2) Lanczos-type algorithms. The short recurrence and the minimization properties have been shown to hold for the conjugate gradient methods for matrices that are selfadjoint and positive definite w.r.t. to the inner product used in the algorithm. The short recurrence holds for the biconjugate gradient-type Lanczos algorithms, also for nonsymmetric matrices, but these algorithms can break down when the matrix is indefinite.

In this chapter, it will first be shown that such short recurrence relations for the conjugate gradient minimization type algorithms exist for a broader class of matrices, the H-normal class w.r.t. the initial vector. This extends the applicability of these methods. However, many matrices occurring in practice belong to still more general classes. On the other hand, the Lanczos-type algorithms do not have a minimization property as a rule and, as just said, can even break down.

We shall also analyze a general class of methods based on minimizing the least square norm of the residual (w.r.t. an inner product) and using a set of search directions (orthogonal w.r.t. another inner product, in general).