Convergence of GMRES

Recall that we are minimizing the following norm

$$\Vert r^{(k)}{\Vert }_2=\Vert b-A{Q}_{k}y{\Vert }_2=\Vert (I-Aq(A))b{\Vert }_2$$

where $q$ is a polynomial of degree $k-1$. To derive an error bound, we should search for a polynomial $p(x)$ of degree $k$ such that $p(0)=1$ and which minimizes $\Vert p(A)b{\Vert }_2$.

Theorem. Suppose that $A$ has an eigenvalue decomposition $A=X\Lambda X^{-1}$. Then the GMRES error can be upper-bounded by

$$\Vert r^{(k)}{\Vert }_2\le \Vert b{\Vert }_2\kappa (X)\underset{\begin{array}{c}p\text{ of degree k }\\ p(0)=1\end{array}}{\min }\underset{\begin{array}{c}\lambda \text{ eigenvalue}\\ \text{of A}\end{array}}{\max }|p(\lambda )|$$

where $\kappa (X)=\Vert X{\Vert }_2\Vert X^{-1}{\Vert }_2$ is the condition number of $X$.

The take-home key ideas to understand the convergence of GMRES are:

1. Distribution of eigenvalues: are they distributed in a few compact clusters or uniformly distributed around 0? Are they accumulating near 0?
2. Condition number of the eigenvector basis: is it small or large?

Failure case

A challenging scenario arises when the eigenvalues of a matrix $A$ are uniformly distributed on the unit circle, which is the circle with a radius of 1 centered at the origin.

Consider, for instance, if $A$ is a permutation matrix that represents a permutation $\pi :\{1,\dots ,n\}\to \{1,\dots ,n\}$, and our goal is to solve $Ax=e_1.$ The solution in this case is the standard basis vector $e_r$, where $\pi (r)=1.$ In this context, $\mathcal{K}(A,e_1,k)$ represents a coordinate subspace, spanned by $\{e_1,e_{\pi (1)},e_{{\pi }^2(1)},\dots ,e_{{\pi }^{k-1}(1)}\}.$ Convergence is achieved only when $k$ is such that $e_r$ is within $\mathcal{K}(A,e_1,k),$ which occurs if and only if ${\pi }^k(1)=1.$ This might require $k$ to be as large as $n,$ especially in cases where $\pi$ is a cyclic permutation like $\pi (i)-1=i+1$ mod $n.$ We also notice that, in this case, the solution remains orthogonal to the Krylov subspace until $k=n$.