Topic summary
MINRES
Extracted from the Wikipedia article Minimal residual method.
GMRES vs. MINRES
The GMRES method is essentially a generalization of MINRES for arbitrary matrices. Both minimize the 2-norm of the residual and do the same calculations in exact arithmetic when the matrix is symmetric. MINRES is a short-recurrence method with a constant memory requirement, whereas GMRES requires storing the whole Krylov space, so its memory requirement is roughly proportional to the number of iterations. On the other hand, GMRES tends to suffer less from loss of orthogonality.