Editing Conjugate gradient method (section)

====Computation of alpha and beta====
In the algorithm, <math>\alpha_k</math> is chosen such that <math>\mathbf{r}_{k+1}</math> is orthogonal to <math>\mathbf{r}_{k}</math>. The denominator is simplified from

:<math>\alpha_k = \frac{\mathbf{r}_{k}^\mathsf{T} \mathbf{r}_{k}}{\mathbf{r}_{k}^\mathsf{T} \mathbf{A} \mathbf{p}_k} = \frac{\mathbf{r}_k^\mathsf{T} \mathbf{r}_k}{\mathbf{p}_k^\mathsf{T} \mathbf{A p}_k} </math>

since <math>\mathbf{r}_{k+1} = \mathbf{p}_{k+1}-\mathbf{\beta}_{k}\mathbf{p}_{k}</math>. The <math>\beta_k</math> is chosen such that <math>\mathbf{p}_{k+1}</math> is conjugate to <math>\mathbf{p}_{k}</math>. Initially, <math>\beta_k</math> is

:<math>\beta_k = - \frac{\mathbf{r}_{k+1}^\mathsf{T} \mathbf{A} \mathbf{p}_k}{\mathbf{p}_k^\mathsf{T} \mathbf{A} \mathbf{p}_k}</math>

using

:<math>\mathbf{r}_{k+1} = \mathbf{r}_{k} - \alpha_{k} \mathbf{A} \mathbf{p}_{k}</math>

and equivalently

<math> \mathbf{A} \mathbf{p}_{k} = \frac{1}{\alpha_{k}} (\mathbf{r}_{k} - \mathbf{r}_{k+1}), </math>

the numerator of <math>\beta_k</math> is rewritten as

:<math> \mathbf{r}_{k+1}^\mathsf{T} \mathbf{A} \mathbf{p}_k = \frac{1}{\alpha_k} \mathbf{r}_{k+1}^\mathsf{T} (\mathbf{r}_k - \mathbf{r}_{k+1}) = - \frac{1}{\alpha_k} \mathbf{r}_{k+1}^\mathsf{T} \mathbf{r}_{k+1} </math>

because <math>\mathbf{r}_{k+1}</math> and <math>\mathbf{r}_{k}</math> are orthogonal by design. The denominator is rewritten as
 
:<math> \mathbf{p}_k^\mathsf{T} \mathbf{A} \mathbf{p}_k = (\mathbf{r}_k + \beta_{k-1} \mathbf{p}_{k-1})^\mathsf{T} \mathbf{A} \mathbf{p}_k = \frac{1}{\alpha_k} \mathbf{r}_k^\mathsf{T} (\mathbf{r}_k - \mathbf{r}_{k+1}) = \frac{1}{\alpha_k} \mathbf{r}_k^\mathsf{T} \mathbf{r}_k </math>

using that the search directions <math>\mathbf{p}_k</math> are conjugated and again that the residuals are orthogonal. This gives the <math>\beta</math> in the algorithm after cancelling <math>\alpha_k</math>.