I am working with a very large linear system, using a flexible pre-conditioned conjugate gradient solver. The system is (B + C)x = b, with B and C symmetric and positive definite, and I know the condition number of C, but not B. I have two approximations to C, C1 and C2, which have the same condition number but different eigenspectra. (C1 uses Steinian linear shrinkage, C2 uses the Ky-Fan p-k norm (essentially raises all of the smallest eigenvalues until the desired condition number is reached)). The convergence characteristics differ markedly. My understanding is that the condition number gives an upper bound on convergence, but that the structure of the eigenspectrum also affects convergence. I would like to find a reference that I can use to understand this effect and cite it in papers.
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Preconditioning is more likely to be effective if the condition number is large solely because of relatively few very large/small eigenvalues. The actual convergence rate without preconditioning is pretty much exactly that given by the standard estimate, unless your right hand side is (nearly) an eigenvector. – Ian Oct 11 '16 at 22:58