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I am interested in cases where Conjugate gradient works much better than GMRES method.

In general, CG is preferable choice in many cases of SPD because it requires less storage and theoretical bound on convergence rate for CG is double of that GMRES. Are there any problems where such rates are actually observed? Is there any characterization of cases where GMRES performs better or comparable to CG for same number of spmvs.

Since Residual history is only available, in many cases to judge how well an algorithm has performed, would GMRES have always lower residual norm than CG in that case?

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    $\begingroup$ This is probably not apt for MO. You should probably try scicomp.stackexchange.com. $\endgroup$
    – user11000
    May 1, 2013 at 4:09
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    $\begingroup$ From the look of it, yes, but there is probably a completely theoretical question in matrix approximation theory hidden behind this one. $\endgroup$
    – Federico Poloni
    May 1, 2013 at 7:14

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