CG search directions

We previously showed that

$$\text{span}\left\{r^{(0)},\dots ,r^{(k)}\right\}=\text{span}\left\{\Delta x^{(0)},\dots ,\Delta x^{(k)}\right\}$$

So we can always write that:

$$\begin{array}{c}\Delta x^{(k)}={\mu }_{k+1}{p}^{(k+1)}\\ r^{(k)}=p^{(k+1)}+\sum _{l=1}^k{\alpha }_l{p}^{(l)}\end{array}$$

We will later see how we can compute ${\mu }_{k+1}$ and ${\alpha }_l$ using the right orthogonality properties.