<s>[INST] The enhanced Krylov subspace method (EKSM) uses the following iterative formula:\[\textbf{x}_{k+1}=\textbf{x}_{k}+\sum_{i=1}^{m}\gamma_{i}^{(k)}\textbf{p}_{i}^{(k) }.\]
where:
\(\textbf{x}_{k}\in\mathbb{R}^{n}\), is the current estimate of the solution of a linear system \(\textbf{A}\textbf{x}=\textbf{b}\).
\(\gamma_{i}^{(k)}\) is a parameter that we can choose.
\(\left\{\textbf{p}_{1}^{(k)},\ldots,\textbf{p}_{m}^{(k)}\right\}\) forms an orthonormal basis of the enhanced Krylov subspace \(\text{span}\left\{\textbf{r}_{k},\textbf{A}\textbf{r}_{k},\ldots,\textbf{A} ^{m-1}\textbf{r}_{k}\right\}\), where \(\textbf{r}_{k}=\textbf{b}-\textbf{A}\textbf{x}_{k}\) which is the residual.
How do we form the columns of a matrix \(\boldsymbol{P}_{k}\) whose columns are the vectors \(\textbf{p}_{1}^{(k)},\ldots,\textbf{p}_{m}^{(k)}\)?[/INST]