TABLE OF CONTENTS The next two chapters explore a few methods that are considered currently to be among the most important iterative techniques available for solving large linear systems. These techniques are based on projection processes, both orthogonal and oblique, onto Krylov subspaces, which are subspaces spanned by vectors of the form p(A)v, where p is a polynomial. In short, these techniques approximate A ?1 b by p(A)b, where p is a "good" polynomial. This chapter covers methods derived from, or related to, the Arnoldi orthogonalization. The next chapter covers methods based on Lanczos biorthogonalization.
Recall from the previous chapter that a general projection method for solving the linear system
extracts an approximate solution x m from an affine subspace 
of dimension m by imposing the Petrov-Galerkin condition
where 
is another subspace of dimension m. Here, x 0 represents an arbitrary initial guess to the solution. A Krylov subspace method is a method for which the subspace 
is the Krylov subspace
where r 0 = b ? Ax 0. When there is no ambiguity, 
( A, r 0) will be denoted by 
. The different versions of Krylov subspace methods arise from different choices of the subspace 
and from the ways in which the system is preconditioned, a topic that will be covered in detail in later chapters.
Viewed from the angle of approximation theory, it is clear that the approximations obtained...
