Finite analytic evaluation from a supplied exact Jordan decomposition

Input a complex matrix $A\in {\mathbb{C}}^{n\times n}$, $n\ge 1$, an open $\mathrm{\Omega }\subseteq \mathbb{C}$ containing $\sigma (A)$ and a holomorphic function $f:\mathrm{\Omega }\to \mathbb{C}$ specified by exact derivative data of that same $f$ on the spectrum, and an exact supplied Jordan certificate $A=ZJ{Z}^{-1}$ with $J=\mathrm{diag}(J_{d_1}({\lambda }_1),\dots ,J_{d_r}({\lambda }_r))$.

1. Verify the dimensions, exact equality $A=ZJ{Z}^{-1}$, invertibility of $Z$, and the stated Jordan block pattern. If any check fails, return invalid-Jordan-certificate.
2. For each block, form the finite sum $F_i={\sum }_{k=0}^{d_i-1}{f}^{(k)}({\lambda }_i)N_{d_i}^k\mathrm{/}k!$ using the supplied exact derivatives. If a required value is absent, return missing-jet.
3. Return $Z\mathrm{diag}(F_1,\dots ,F_r)Z^{-1}$ with status exact.

All loops and matrix products have finite size. This is an exact-data evaluator; it makes no floating-point stability or complexity claim.