I have a system of equations $$   S_{m}(\xi) +P_{m}(\xi)=f(\xi)$$where $\xi$ can be choosen arbitrary in some domain in $\mathbb{C}$, $f$ is known, $P_m$ is a polynomial of degree at most $m$. Here $S_{m}$ is a generic function from $\mathbb{C}$ to $\mathbb{C}$ but for me it is sufficient to know it's values in only $(m+1)$ mutually distinct points.Taking $(m+1)$ mutually distinct points $\xi_0, \ldots, \xi_m$ we can invert the Vandermonde matrix and hence write coefficients of $P_m$ as linear combination of $S_m(\xi_k)$,$k=0,\ldots,m$. Introducing this result back in equations we get a linear system on $S_{m}(\xi_k)$. So theoretically we can find $S_{m}(\xi_k)$. But is there some way to do it practically?
How to solve numerically such system of equations
linear algebra;numerics