CG may be used to estimate the extremal eigenvalues of a SPD matrix (by computing eigenvalues of tridiagonal matrix associated with the Lanczos algorithm). After a few iterations the largest eigenvalue is generally well approximated whereas the smallest is generally overestimated.

Is there a way to have some estimate of the quality of those approximate eigenvalues?

Thank you.

  • 6
    $\begingroup$ Please see this paper from this question. $\endgroup$ – Jack Poulson Oct 5 '12 at 16:47
  • $\begingroup$ What exactly do you mean by the "quality" of the approximation? $\endgroup$ – Paul Oct 5 '12 at 17:33
  • $\begingroup$ @JackPoulson - I'm okay with letting this stand as a separate question, do you mind moving your comment to an answer? $\endgroup$ – Aron Ahmadia Oct 6 '12 at 3:42
  • $\begingroup$ I would like to have some lower and upper bounds on the approximation I get. I'll have a look to the paper, thank you ! $\endgroup$ – Tom Oct 8 '12 at 6:33

This question is very related to another SE question on condition number estimates which contains relevant materials.

As @Jack_Poulson mentioned, the following paper contains a detailed discussion on the asked topic:

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