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Suppose that we apply a Krylov subspace method to the linear system $A x = b$. For example, if $A$ is symmetric positive-definite, then the Conjugate Gradient method may be used.

I remember that the iterations of Krylov subspace methods also reveal information on the spectrum of $A$, at least an estimate for the largest eigenvalue.

I am searching for a detailed derivation of such an estimate that leads to an implementation.

shuhalo
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  • You can use the Rayleigh quotient of the current eigenvector estimate to get an eigenvalue estimate. – Ian Jan 24 '15 at 17:41
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    Good eigenvalue estimates can be obtained from the equivalence between CG and the Lanczos method. See Section 6.7 (namely 6.7.3) here. You just need to reconstruct the Jacobi matrix and compute its eigenvalues (or the maximal eigenvalue). – Algebraic Pavel Jan 24 '15 at 17:56
  • @AlgebraicPavel: Nice reference. – copper.hat Jan 24 '15 at 20:16

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