Suppose we know, that the domain containing the spectrum of a matrix we want to approximate is compact and enclosed by a polygon.
Given the SC-disk-to-exterior-map from Discroll&Trefethen's book we can compute the map from the exterior of the unit disc to the complement of a compact domain under the assumption that it is simply connected as:
$$ f(u) = A - C \int^u w^{-2}(1-\frac{\omega}{u_k})^{1-\alpha_k} d\omega.$$
Numerically we can then determine the Laurent expansion coefficients $c_0 ... c_m$ of said map as in this paper and plug them into the Faber polynomial recurrence relation $$\phi_{m+1}=\frac{1}{c}(z \phi_m-m c_m-\sum_{m=0}^{M}c_m \phi_{M-m} \ )$$ The initial values are given as $\phi_0 = 1$ and $\phi_1 = \frac{1}{c}(z-c_0)$, where $c$ denotes the logarithmic capacity of the domain whose exterior we are mapping to.
Let's just set $c$ to $1$.
Let's assume that I am trying to solve a differential equation with the solution $$z = exp(At)z_0$$ and approximate the exponential with Faber polynomials like so: $$exp(At)z_0 = \sum_{m=0}^{M}a_m(t)\phi_m(A)z_0$$ where $a_m$ denotes the time dependent Faber coefficients. As visible from the post above the recurrence relation of the Faber polynomials has to be matrix valued.
Am I correct in assuming the recurrence relation will look something like this?
$$\phi_{m+1}(A)=(A*\phi_m(A)-m c_m-\sum_{m=0}^{M}c_m \phi_{M-m}(A) \ )$$
Given a function $g(z)$ analytic in our domain, to compute the coefficients $a_m$ we have to solve $$a_m = \frac{1}{2\pi}\int{\frac{g(\Psi (\omega))}{\omega^{m+1}}}d\omega$$ with $\Psi$ denoting the exterior map. This is the part that confuses me. How can I compute that integral, when what I want to approximate is the matrix A?
I am an engineering undergraduate. Sorry if this sounds trivial.