rank one update Given a matrix $X$, we can compute its matrix exponential $e^X$. Now one entry of $X$ (say $x_{i,j}$) is changed to $b$, the updated matrix is denoted by $X'$. My problem is how to compute $e^{X'}$ from $e^X$ in a fast way? 
PS: I know that if our goal is to calculate the matrix inversion (not matrix exponential), we can use Sherman–Morrison formula to compute ${X'}^{-1}$ from $X^{-1}$ easily, but currently I have not found a way to deal with the matrix exponential. Hope you can give me a hand. Thanks!

What you are looking for is the Baker–Campbell–Hausdorff formula.
You are trying to compute
$$e^{X+B}$$ where $B=X'-X$ has only one non-vanishing entry. If you are lucky then $X$ and $B$ commute and you can apply the standard formula
$$e^{X+B}=e^X\cdot e^B,$$ where $e^B$ is easy to compute.
In gnereal you will get higher terms which involve (iterated) commutators. It is then up to the form of $X$ and $B$ whether you can find suitable bounds and appoximate the series with a finite sum.
