I am faced with the practical problem of solving a system $$\rho(\exp(A(t))) = 1$$ numerically, where $\rho$ signifies the spectral radius of the matrix $A(t) = B+ \frac{C}{t},$ $t \in (0, \infty)$. Unfortunately,the matrices $B \in \mathbb{R}^{n \times n}$ and $C \in \mathbb{R}^{n \times n}$ are very complicated and don't have many redeeming qualities. They don't commute, but $B$ is a lower, and $C$ an upper triangular matrix.
To obtain a good convergence order of the numerical solver, I tried to calculate the derivative of $\rho:=\rho(\exp(A(t)))$. Let $v_{\max.}$ be the corresponding eigenvector. \begin{gather*} \frac{\Bbb d}{\Bbb dt} \rho(t) = \frac{\Bbb d}{\Bbb dt}\langle \exp(A)v_{\max.},v_{\max.} \rangle = \left\langle \left(\frac{\Bbb d}{\Bbb dt}\exp(A)\right)v_{\max.}+ \exp(A)\left(\frac{\Bbb d}{\Bbb dt}v_{\max.}\right),v_{\max.} \right\rangle + \left\langle \exp(A)v_{\max.}, \frac{\Bbb d}{\Bbb dt} v_{\max.} \right\rangle \end{gather*}
And I found an explicit formula for the derivative of the matrix exponential in terms of an integral. Now is there any possibility (other than numerical differentiation) to obtain the derivative of $v_{\max.}$? Or some assumptions that allow for a closed form? I can calculate $v_{\max.}$ explicitly at any timepoint.