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6

Wolfgang Bangerth's answer already says almost everything, but another subtle detail is that GMRES/MINRES minimize the norm of the residual, i.e., $\|Ax_k-b\|$, while CG minimizes the (A-)norm of the error, i.e., $\|x_*-x_k\|_A$. In most cases what you really want to minimize is the error, and the residual serves as an imperfect proxy: recall that by the ...


5

In some sense it doesn't matter: In finite dimensions, all norms are equivalent, so if an algorithm conveniently has the property that proving convergence in one norm is easy, then that's what people will choose. In practice, one oftentimes wants to reduce the norm of the residual by a certain factor, and from a practical perspective, a certain reduction in ...


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