Abstract

CHAPTER 4. THEORY 106
we have at the k-th GMRES that
rk2 ≤p(A)2r02.
The authors in [10] use this fact and the spectrum of A to get a more useful bound
on the norms of the residuals. In [10], the authors show that the convergence rate
of GMRES is determined by how the eigenvalues of A are clustered in the complex
plane; if the eigenvalues are closely clustered together, then GMRES will converge
rapidly. The basic idea comes from using the resolvent integral representation of p(A),
givenbythecomplexintegral
p(A) = 1
(zI − A)
2πi Γ
−1 p(z)dz
Here, Γ is any curve in the complex plane that encloses all the eigenvalues of A.
Therefore, we want to choose the polynomial p so that p is small near all the eigen-
values of A. So if all the eigenvalues are tightly clustered, the degree of the chosen
polynomial p can be smaller than if the eigenvalues were scattered throughout the
complex plane. This means the number of GMRES iterations will be smaller for the
clustered eigenvalue case.
Now, returning to our specific problem, we have
rk2
≤p(A)2 ≤
r02
√ Lp(A)X,