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How to solve for $f(A)x=b$? For GMRES, an answer is given in this book chapter: http://link.springer.com/chapter/10.1007%2F978-3-642-58333-9_2. Ungated version: https://www.researchgate.net/profile/Henk_Van_der_Vorst/publication/226001699_Linear_Systems_Eigenvalues_and_Projection/links/02e7e5301220731a71000000/Linear-Systems-Eigenvalues-and-Projection.pdf#page=273

However, that uses GMRES. Are there other methods, e.g., other Krylov methods? (Preferably matrix-free.) Preferably worked through and analyzed, but if there are none, any ideas and expectations?

BTW, for $\sqrt{A}x=b$ some further solutions are given here: https://scicomp.stackexchange.com/a/11400/34228 but I ask for more general $f$.

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    $\begingroup$ look at the conjugate gradient (CG) methods, generalized conjugate residual (GCR) methods, quasi-minimum residual (QMR) methods, etc. Why don't you want to use GMRES? $\endgroup$ – smh Feb 28 at 16:30
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As in smh's comment, you can use pretty much any Krylov subspace method. But you should be careful. For example, standard CG methods require $f(A)$ to be symmetric and positive definite. MINRES relaxes that a little bit and $f(A)$ just has to be symmetric. GMRES should work for general non-singular matrices.

However, the main challenge would not be related to the Krylov subspace method you use but how $f(A)x$ is evaluated. Say for example, $f(A) = e^A$. Then constructing $B=f(A)$ explicitly then multiplying $x$ by $B$ can be very memory consuming. The conference proceeding you linked and the discussion at https://scicomp.stackexchange.com/a/11400/34228 are more about doing it in a graceful and efficient way. That may require some knowledge of the function and the technique may not generalize.

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