Is there some easy way to calculate an element $b_{ij}$ for fixed $i$ and $j$ of the matrix exponential $(b_{ij})=B=\exp(A)$? ($A$ is symmetric real matrix)
I know that I could diagonalize $A$ and then calculate $B$ using that. However, this is quite slow when the matrix $A$ is very big. Because I'm only interested in one element of $B$, it makes me think this might not be necessary. Is there some trick that makes it faster to find one element of the matrix exponential than diagonalizing $A$ (or calculating the whole matrix B by some other method)?