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.

  • 7
    $\begingroup$ Please see this paper from this question. $\endgroup$ Oct 5, 2012 at 16:47
  • $\begingroup$ What exactly do you mean by the "quality" of the approximation? $\endgroup$
    – Paul
    Oct 5, 2012 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? $\endgroup$ Oct 6, 2012 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, 2012 at 6:33

1 Answer 1


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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