I have been reading the following paper: CG versus MINRES: An empirical comparison. In it a conjugate gradient solver is applied to a system matrix $A$ Jacobi-preconditioned on both sides. Specifically, assume $A$ is spd and we want to solve for $x$:

$$Ax = b.$$

The matrix is Jacobi-preconditioned $d = diag(A), D^{-\frac{1}{2}} = diag(1/\sqrt{d})$, but the right-hand side is also normalized:

$$D^{-\frac{1}{2}}AD^{-\frac{1}{2}}y = \frac{D^{-\frac{1}{2}}b}{\|D^{-\frac{1}{2}}b\|}.$$

I do not understand why the authors normalize the right-hand side - doesn't this change the original system? Is this maybe implicitly taken into account through $x = \|D^{-\frac{1}{2}}b\|D^{-\frac{1}{2}}y$ (this part is not in the paper)? Is there any purpose to such normalization (e.g. faster convergence)?

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    $\begingroup$ Can't really say because they don't explain their reasons. But looking at the content of the paper, I am getting two hints: stopping criteria becomes simpler, and the backward errors become smaller. And usually, more accurate intermediate calculation gets you to the solution faster (except false convergence or divergence cases, which terminate early due to inaccurate calculations) and the solution you obtain is also more accurate. $\endgroup$ Oct 14, 2021 at 4:48


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