We have thus $$\hat{A}\hat{q}_k=\beta_{k-1}\hat{q}_{k-1}+\alpha_k\hat{q}_k+\beta_k\hat{q}_{k+1},$$ with $\beta_0\hat{q}_0=0$ for $k=1:n-1$ and $$\alpha_k=\hat{q}_k^T\hat{A}\hat{q}_k.$$ If $$\hat{r}_k=(\hat{A}-\alpha_k\hat{I})\hat{q}_k-\beta_{k-1}\hat{q}_{k-1},$$ is non-zero, then $$\hat{q}_{k+1}=\hat{r}_{k}/\beta_k,$$ with $\beta_k=\pm ||\hat{r}_{k}||_2$.