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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.

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

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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. 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

I don't know about closed formulas, but there are several ways to find good approximations of the matrix see e.g.

Most of these aproaches are using Padè approximations of the exponential function of the matrix. These consist of a simple fracture, and don't need much terms to be near machine precision.

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Thanks for the interesting link; however link-only answers are strongly discouraged here since links can go stale (even to papers), so at least a short summary would be helpful... – Tobias Kienzler Nov 11 '13 at 8:46

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