Editing Conjugate gradient method (section)

====Explicit residual calculation====
The formulas <math>\mathbf{x}_{k+1} := \mathbf{x}_k + \alpha_k \mathbf{p}_k</math> and <math>\mathbf{r}_k := \mathbf{b} - \mathbf{A x}_k</math>, which both hold in exact arithmetic, make the formulas <math>\mathbf{r}_{k+1} := \mathbf{r}_k - \alpha_k \mathbf{A p}_k</math> and <math>\mathbf{r}_{k+1} := \mathbf{b} - \mathbf{A x}_{k+1}</math> mathematically equivalent. The former is used in the algorithm to avoid an extra multiplication by <math>\mathbf{A}</math> since the vector <math>\mathbf{A p}_k</math> is already computed to evaluate <math>\alpha_k</math>. The latter may be more accurate, substituting the explicit calculation <math>\mathbf{r}_{k+1} := \mathbf{b} - \mathbf{A x}_{k+1}</math> for the implicit one by the recursion subject to [[round-off error]] accumulation, and is thus recommended for an occasional evaluation.<ref>{{cite book | first=Jonathan R | last=Shewchuk |title=An Introduction to the Conjugate Gradient Method Without the Agonizing Pain |year=1994 |url=http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf  }}</ref>

A norm of the residual is typically used for stopping criteria. The norm of the explicit residual <math>\mathbf{r}_{k+1} := \mathbf{b} - \mathbf{A x}_{k+1}</math> provides a guaranteed level of accuracy both in exact arithmetic and in the presence of the [[rounding errors]], where convergence naturally stagnates. In contrast, the implicit residual <math>\mathbf{r}_{k+1} := \mathbf{r}_k - \alpha_k \mathbf{A p}_k</math> is known to keep getting smaller in amplitude well below the level of [[rounding errors]] and thus cannot be used to determine the stagnation of convergence.