## 2 Methodology
### EVD Processor
#### 3.2.5 Systolic Architectural  
* \(P_{ij}\) contains four sub-matrix elements \((R_{xx})^{(k)}_{ij}\in\mathbb{R}^{2\times 2}\)
* From each diagonal processor it is possible to calculate the rotation angles
* Angles are sent to NDPs in the same row and column as DP (used to calculate angles)
* CORDIC is used to calculate the angles of rotation. * Each NDP processor needs two angles \(\alpha_{ii}\) and \(\alpha_{jj}\) (in the case of DP processor it is the same angles \(\alpha_{ii}=\alpha_{jj}\))
* NDP angles \(\alpha_{ii}\) are sent from DPs that are associated with the same line
* Each processor at the angles necessary to perform the double multiplication, to do this we will use the CORDIC architecture. Figure 6: Systolic operating architecture for the decomposition of eigenvalues and vectors. With a single rotation the off-diagonal elements of DP are cancelled on the other hand the off-diagonal elements of NDP are modified. * Even in double rotation the diagonal elements approach zero, but they are not cancelled. **DP:**: Diagonal Processor; **NDP**: Non-Diagonal Processor\[\alpha_{ii}^{(k)}=\frac{1}{2}tan^{-1}\left[\frac{2s_{2l-1,2m}^{(k)}}{2s_{2l,2m}^{(k )}-s_{2l-1,2m-1}^{(k)}}\right] \tag{33}\]  
\[\left(R_{xx}\right)_{i,j}^{(k)}=Q(\alpha_{ij}^{(k)})^{T}.(R_{xx})_{i,j}^{(k)}.Q (\alpha_{ij}^{(k)}) \tag{34}\]  
Figure 9 shows the systolic architecture to obtain eigenvalues. Figure 10 shows the systolic architecture to obtain eigenvectors. Figure 11 shows the material architecture of the rotation angle calculation block using the Jacobi method. The double rotation is carried out for all the processors for the computation of the eigenvalues using the CORDIC. Each processor is defined by these angles \(\alpha_{ii}^{(k)}\),\(\alpha_{ij}^{(k)}\) and these elements (a, b, c and d) in our case the four elements of the \(2\times 2\) sub-matrix [2]. Figure 11: Material architecture. Figure 10: Systolic architecture to obtain eigenvectors. Figure 9: Systolic architecture to obtain eigenvalues. \[\left[ \begin{array}{cc}s^{(k+1)}_{2l+1,2m-1}&s^{(k+1)}_{2l-1,2m}\\ s^{(k+1)}_{2l,2m-1}&s^{(k+1)}_{2l,2m}\end{array} \right]=Q(\alpha^{(k)}_{ii})^{T }.\left[ \begin{array}{cc}s^{(k)}_{2l+1,2m-1}&s^{(k)}_{2l-1,2m}\\ s^{(k)}_{2l,2m-1}&s^{(k)}_{2l,2m}\end{array} \right].Q(\alpha^{(k)}_{ii}) \tag{35}\]  
where  
\[Q(\alpha^{(k)}_{ii})=\left[ \begin{array}{cc}cos\left(\alpha^{(k)}_{ii}\right) &sin\left(\alpha^{(k)}_{ii}\right)\\ -sin\left(\alpha^{(k)}_{ii}\right)&cos\left(\alpha^{(k)}_{ii}\right)\end{array} \right]\]  
\[T^{(k)}_{ij}=\left[ \begin{array}{cc}cos(\alpha^{(k)}_{ii})s^{(k)}_{2l-1,2m-1 }-sin(\alpha^{(k)}_{ii})s^{(k)}_{2l,2m-1}&cos(\alpha^{(k)}_{ii})s^{(k)}_{2l-1, 2m}-sin(\alpha^{(k)}_{ii})s^{(k)}_{2l,2m}\\ sin(\alpha^{(k)}_{ii})s^{(k)}_{2l-1,2m-1}+cos(\alpha^{(k)}_{ii})s^{(k)}_{2l,2 m-1}&sin(\alpha^{(k)}_{ii})s^{(k)}_{2l-1,2m}+cos(\alpha^{(k)}_{ii})s^{(k)}_{2l,2m}\end{array} \right] \tag{36}\]  
The first product of expression (38) is called \(T^{(k)}_{ij}\) (40), we can now express (39) as presented in (41). \[(R_{xx})^{(k+1)}_{ij}=T^{(k)}_{ij}.\left[ \begin{array}{cc}cos\left(\alpha^{ (k)}_{ij}\right)&sin\left(\alpha^{(k)}_{ij}\right)\\ -sin\left(\alpha^{(k)}_{ij}\right)&cos\left(\alpha^{(k)}_{ij}\right)\end{array} \right] \tag{37}\]  
Analysis Equation (33) shows that this expression is equivalent to the coordinate vector rotation \(\left[s^{(k)}_{2l-1,2m-1},s^{(k)}_{2l,2m-1}\right]\). The angle \(\alpha^{(k)}_{ii}\) and a rotation vector \(\left[s^{(k)}_{2l-1,2m-1},s^{(k)}_{2l,2m}\right]\) also an angle \(\alpha^{(k)}_{ii}\). We ask ourselves now:  
\[T^{(k)}_{ij}=\left[ \begin{array}{cc}s^{\prime}_{2l-1,2m-1}{}^{(k)}&s^{\prime }_{2l-1,2m}{}^{(k)}\\ s^{\prime}_{2l,2m-1}{}^{(k)}&s^{\prime}_{2l,2m}{}^{(k)}\end{array} \right] \tag{38}\]  
\[(R_{xx})^{(k+1)}_{ij}=\left[ \begin{array}{cc}s^{\prime}_{2l-1,2m-1}{}^{(k)}& s^{\prime}_{2l-1,2m}{}^{(k)}\\ s^{\prime}_{2l,2m-1}{}^{(k)}&s^{\prime}_{2l,2m}{}^{(k)}\end{array} \right].Q( \alpha^{(k)}_{ii}) \tag{39}\]  
\[(R_{xx})^{(k+1)}_{ij}=\left[