# How to solve for f(A)x=b without GMRES?

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 $$f$$.

• 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? – smh Feb 28 '20 at 16:30

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.