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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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    $\begingroup$ A few notes: if the products of the corresponding subdiagonal and superdiagonal elements are positive, you can, through a similarity transformation, always reduce your tridiagonal eigenproblem to a symmetric tridiagonal eigenproblem, which is very much well studied. The unsymmetric eigenproblems that do not yield to this treatment are said to be a difficult case for the QR algorithm and related methods. $\endgroup$ Aug 20, 2012 at 3:27
  • $\begingroup$ See Horn and Johnson: Topics in Matrix Analysis. This book is reference in matrix analysis. $\endgroup$ Nov 15, 2012 at 20:10
  • $\begingroup$ @Elias Great. Thank you very much. $\endgroup$ Nov 17, 2012 at 2:19
  • $\begingroup$ @Elias Please consider converting your comment into an answer, so that this question gets removed from the unanswered tab. If you do so, it is helpful to post it to this chat room to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see here, here or here. $\endgroup$ Jun 22, 2013 at 8:53

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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.

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