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 $f$.