[trostaft]

I remember being handed this back when I was taking numerical analysis for the first time. It's an old document, but still useful.

IMO the critical pieces of CG that make it a favorable choice for many problems in scientific computing are

1) the fact that it can be performed matrix free

2) its rapid convergence behavior on operators with clusters of eigenvalues (useful for low rank structures)

Thet being said, practically speaking, even if I know my operator is positive semi definite, I often find minres out performing cg. There's a nice paper comparing that, "CG versus MINRES: An Empirical Comparison".

[stabbles]

"minres outperforming cg" likely depends on the stopping criterion, since different norms are used.

[trostaft]

Yes, to (grossly) summarize the conclusion of the paper: minres can outperform CG on the backwards error ||r_k|| / ||x_k|| whereas CG can outperform minres on the absolute error ||x^* - x_k|| and the energy norm ||x^* - x_k||_A.