I'm looking for the eigenvalues of the following symmetric tridiagonal matrix
\begin{pmatrix} a & z & 0 & 0 & 0 \\ z & b & z & 0 & 0 \\ 0 & z & 0 & z & 0 \\ 0 & 0 & z & c & z \\ 0 & 0 & 0 & z & d \\ \end{pmatrix}
which is a real symmetric Jacobi matrix. I know that its determinant is given by the continuant of its elements but I was wondering if any of you knows some papers or links in which I can find the eigenvalues and eigenvectors of this matrix.
My goal is to compute the exponential of this matrix, which is why I'm trying to diagonalize it.