Is it posible to express a function
$$ Eig_n(A, B) := \{ \text{The } n\text{ eigenvalues of the } n \times n \text{ matrix } A + B \}$$
The input of $Eig$ can be either described as the eigenvalues of A, B (or their characteristic polynomial coefficients), and the orthogonal similarity matrices $S_A, S_B$ that relate the diagonal forms of A and B relative to the diagonal frame of $A+B$.
In recursive terms, based perhaps on $Eig_k(A,B)$ for $0 < k < n$?