Matrix elements of exponential of tridiagonal matrices 
Is there a way to compute one matrix element of the exponential of a tridiagonal matrix without having to compute the rest of the elements?

Motivation: I'm trying to find the first passage time distribution from a master equation. I can impose an absorbing boundary at the threshold $n$, and the master equation with the new boundary condition is of the form $\frac{dp}{dt}=A p$ for a tridiagonal $n\times n$ matrix $A$. Then the first passage time distribution can be written as a particular matrix element of $\exp(A\,t)$. It takes forever for Mathematica to compute $\exp(A\,t)$ for large $n$, so I was wondering if there is a way to compute only the desired matrix element and not the whole matrix.
 A: Yes! Most methods to compute exponentials of large sparse matrices are based on computing directly $\exp(A)b$ for a given vector $b$ rather than the full matrix $\exp(A)$. Just take $b$ as a vector of the canonical basis, and you're set.
The basic idea is that these algorithms are based on approximating the exponential with a rational function which is then expanded into a sum of partial fractions
$$\exp(A)b \approx q(A)^{-1}p(A)b = \sum_{i=1}^k \omega_k (A-\tau_k I)^{-1} b.$$
Hence the problem is reduced to solving several shifted linear systems of the form $(A-\tau_k I)x_k=b$; this should work particularly well for a tridiagonal matrix, for which linear systems can be solved cheaply. 
This technique is often combined with ideas from Krylov subspace methods, especially rational Krylov methods.
See for instance a chapter in the review  https://doi.org/10.1017/S0962492910000036,  https://doi.org/10.1137/100788860, or http://onlinelibrary.wiley.com/doi/10.1002/gamm.201310002/full, as well as the Matlab code in Stefan Güttel's webpage which you can use to compute it.
All these algorithms are approximate (but in the end, what is not approximate on a computer?), so you may encounter trouble if the element that you wish to compute is many orders of magnitudes smaller than $\|\exp(A)\|$. In that case, you may be interested in a bound on the decay rate of off-diagonal entries in https://arxiv.org/abs/1501.07376.
