关于对称三对角Jacobi矩阵特征值、特征向量求解及矩阵指数计算的资料问询

Hey folks, I'm currently working with a real symmetric tridiagonal Jacobi matrix and hoping someone can point me to useful papers or resources for solving its eigenvalues and eigenvectors.

First off, here's the matrix I'm dealing with:
$$
\begin{pmatrix}
a & z & 0 & 0 & 0 \
z & b & z & 0 & 0 \
0 & z & 0 & z & 0 \
0 & 0 & z & c & z \
0 & 0 & 0 & z & d \
\end{pmatrix}
$$

I already know its determinant can be calculated using the continuant of its elements, but what I really need is a clear path to finding its eigenvalues and eigenvectors. Are there any specialized references that cover this specific type of symmetric tridiagonal matrix?

My end goal is to compute the matrix exponential of this matrix, which is why diagonalizing it (via finding eigenvalues and eigenvectors) is my top priority right now.

备注：内容来源于stack exchange，提问作者clement3337