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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

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    @Elias Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-22

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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.

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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.

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A continuous version of a problem may be the Sturm-Liouville problem: https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory

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I strongly recommend the paper "Eigenvalue computation in the 20th century" by Golub & van der Vorst and the references therein.