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Eigenvalues of a tridiagonal matrix

  1. Mar 22, 2012 #1
    Some algorithms of finding the eigenvalues of symmetric matrices first transform the matrix to a tridiagonal matrix which is similar to the original matrix and then find the eigenvalues of the tridiagonal matrix. . Are there special algorithms for a tridiagonal matrix, or do the same algorithms for general matrixes converge faster for a tridiagonal one?

    Your help would be appreciated.
  2. jcsd
  3. Mar 22, 2012 #2


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    A general n by n matrix has [itex]n^2[/itex] entries. An n by n "tri-diagonal" matrix has only 3n- 2 entries (not counting the ones that you know are 0 and don't have to keep track of). There are numerical algorithms for working with tri-diagonal matrices that do NOT "fill in the zeros" so, yes, there is an enormous saving in both time and storage working with tri-diagonal matrices.
  4. Mar 22, 2012 #3
    Thanks HalsofIvy,

    Do you know any of such algorithms?
    The general algorithms that I know, like QR eigenvalue algorithm ,ruin the sparsity of the matrix in the very first iteration. Or maybe I misunderstood it.

    Added: There was a mistake in my code. The QR transformation does preserve the tridiagonality.

    Thanks again.
    Last edited: Mar 22, 2012
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