Reference suggestion: eigenvalues of tridiagonal matrices

I would like to ask for a reference on the problem of computing the eigenvalues/eigenvectors of tridiagonal matrices (not necessarily with constant diagonals).

I have seen authors use continued fractions and generating functions. However, I have thus far been unable to really grasp the foundations of this idea. From what I can see, the idea is really to reduce it to a difference equations. Then, perhaps, my request is for a good book on difference equations. Moreover, is there any technique which is really of a broad scope; ie applicable to a broad range of problems.

Thank you all in advance,

Gabieel

Solution 1:

I tried looking up this subject on Google and found this article that seems extremely relevant to your problem: Eigenvalues of Several Tridiagonal Matrices. This article uses symbolic calculus to compute eigenvalues (which I barely know a thing about; I'm not one who works with linear algebra) of multiple tridiagonal matrices. I believe this one should help you, I've skimmed through it.

Solution 2:

If you're looking for numerical computation of the eigenvalues and -vectors, you'll have to look at A Divide-and-Conquer Algorithm for the Symmetric Tridiagonal Eigenproblem by Gu and Eisenstat. It is still considered the gold standard on this topic.