I have a Hermitian matrix of the type
$$H = H(c_1, c_2, \dots, c_n)$$
where $c_i$'s are some complex parameters. I need to find the derivatives
$$\frac{\partial}{\partial c_j}\text{exp}(H)$$
for each $c_j$. I don't think it will be of the form $\text{exp}(H)\frac{dH}{dc_j}$ because $H$ and $\frac{dH}{dc_j}$ won't necessarily commute. Does anyone have an analytical/numerical suggestion for this problem?
This looks promising: Derivative of matrix exponential - but would be very complicated to implement numerically. Are there any other methods for this?
Edit: Will the fact that $H$ is a sparse matrix help?