# Is there general formula for the exponential of a tridiagonal matrix?

For an arbitrary tridiagonal matrix of the form

$$A = \begin{pmatrix} b_1 & c_1 & 0 & 0 & ... \\ a_2 & b_2 & c_2 & 0 & ... \\ 0 & a_3 & b_3 & c_3 & ... \\ \vdots &&\ddots&\ddots&\ddots\end{pmatrix}$$

is there a formula to calculate $\exp(A)$? Or at least for some special tridiagonal matrices?

The special case I am most interested in is a $(2n+1)^2$ matrix with $b_k = i(k-n-1)$ and $c_k = (a_{2n+2-k})^*$, i.e.

$$\begin{pmatrix} -in & c_1 & 0 & \\ c_{2n}^* & -i(n-1) & c_2 & \\ 0 & c_{2n-1}^* & -i(n-2) & \ddots \\ &&\ddots&\ddots \end{pmatrix}$$

• A closed form for that exponential would entail finding a closed form for the characteristic polynomial of the tridiagonal matrix, since the eigenvectors can be expressed in terms of derivatives of the characteristic polynomial evaluated at appropriate values... – J. M. is a poor mathematician Aug 10 '11 at 8:53
• Did you ever find a solution to your problem? – John Salvatier Jan 10 '13 at 16:55
• @JohnSalvatier I'm afraid not :-/ – Tobias Kienzler Jan 10 '13 at 17:00
• I'm looking for a way to compute exp(At)*x_0 cheaply when A's a symmetric tridiagonal matrix. I think I may just have to eigen-decompose A and do it that way. Luckily I only have to decompose A once, and then it's O(n**2), which I guess is okay. Since you should be able to compute Ax_0 in O(n) steps since its tridiagonal, I was hoping for something better, but maybe that's not possible. – John Salvatier Jan 10 '13 at 20:26