What is the recomputation of residuals in the deepest descent method?

Question

From Jonathan Richard Shewchuk:

By using equation (13), we have lost track of $x_(i)$ in the process of iteration. How is it possible to recompute the correct residual using equation (10) ever again?

Answer 1

Of course, the algorithm computes both $x_i$ and $r_i$ in each step. The calculation (13) is only done to save one additional matrix vector multiplication: $Ar_i$ has to be computed to get the step size $\alpha_i$.