Suppose we want to find $x$ such that:
$$x=b+Ax$$
where $A$ is a large sparse square matrix with eigenvalues in the unit circle.
There are two representations of the solution:
1) $$x=(I-A)^{-1}b,$$
2) $$x=\sum_{k=0}^\infty{A^k b}.$$
Since we cannot practically store powers of the matrix $A$ (due to memory limits: indeed in practice, we just have a function that computes $Ax$ given $x$), using the representation 2) means evaluating the recurrence:
$$x_t=b+A x_{t-1}.$$
However, since $A$ may have eigenvalues close to $1$ in magnitude, this will converge slowly.
Is there a multivariate series acceleration method which would handle such recursions? (E.g. a multivariate version of the Aitken "delta-squared" process/Steffensen's method.)
Alternatively, is there a technique for solving 1) directly that is guaranteed to have good performance? Do our assumptions on $A$ imply anything about the performance of GMRES / CGS / BICGSTAB(L) / QMR etc.?
Note: This question: "Is there a faster method to compute the geometric series of a matrix?" is similar but seems to be interested in finite sums of small dense matrices, rather than infinite sums of large sparse matrices.