## 2 Methodology
### EVD Processor  
#### 3.2.1 Cyclic Jacobi Method  
In this part, we want to present the Cyclic Jacobi method which is often used for the calculation of eigenvalues and vectors. With this method, we try to bring the matrix \(R_{xx}\) to a diagonal shape to find everything. Its eigenvalues and eigenvectors, and this by a sequence of orthogonal transformations [22; 23; 24]. This section describes the basic principle of the Cyclic Jacobi method [25; 26]. It can be implemented with a simple iterative rotation plan process. This method makes it possible to solve problems of symmetric eigenvalues by applying a sequence of orthonormal rotations on the left and right sides of the matrix \(R\)[27]. \[E^{T}RE=D \tag{16}\]  
\[\left[ \begin{array}{c}\ldots E=J_{1}.J_{2}.J_{3}.\ldots\\ J=W_{12}.W_{13}.W_{k-1k}.\ldots\end{array} \right] \tag{17}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Logic Use** & **Used** & **Available** & **Use (\(\%\))** \\ \hline Nb of Slices & 325 & 12,288 & 2 \\ Nb of 4 input LUTs & 648 & 24,576 & 2 \\ Nb of IOBs & 74 & 408 & 18 \\ Nb of GCCLKs & 1 & 4 & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hardware resource utilization of the design (RAM with CORE Generator). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Logic Use** & **Used** & **Available** & **Use (\(\%\))** \\ \hline Nb of Slices & 1337 & 12,288 & 10 \\ Nb of slice Flip Flops & 1219 & 24,576 & 4 \\ Nb of 4 input LUTs & 2523 & 24,576 & 10 \\ Nb of IOBs & 122 & 408 & 29 \\ Nb of GCCLKs & 1 & 4 & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Hardware resource utilization of the design (Multiplier with CORE Generator). With \(J\) is the rotation multiple of cyclically by \(W_{pq}\), which is called a Jacobi sweep, the exponent \({}^{T}\) and \(N\) denote the transpose operation and array length. \[W_{pq}=\left[ \begin{array}{ccccccccc}1&&&\cdot&&&\cdot&&&\\ &\cdot&&&\cdot&&\cdot&0\\ &1&\cdot&&&\cdot&&&\\ \cdot&\cdot&\cdot&\cos\theta&\cdot&\cdot&\sin\theta&\cdot&\cdot&\cdot\\ &&&\cdot&1&&\cdot&\\ &&&\cdot&&\cdot&\cdot&&&\\ &\cdot&\cdot&\cdot&-\sin\theta&\cdot&\cdot&\cos\theta&\cdot&\cdot&\cdot\\ &&&\cdot&&&1&&\cdot&\\ 0&&\cdot&&&\cdot&&&\cdot&\\ &&&\cdot&&&\cdot&1\end{array} \right] \tag{18}\]  
where \(W_{pq}\) is a rotation of an orthonormal plane on an angle \(\theta\) in the plane \((p,q)\) with \(W_{pq}=\cos\theta\)\(W_{pq}=sin\theta\),\(W_{qp}=-sin\theta\) and \(W_{pq}=cos\theta\)\((p>q)\). The symmetric matrix \(R\) is transformed into \(R^{\prime}\) by a rotation is defined by Equation (18) [28,29]. \[R^{\prime}=W_{pq}^{T}RW_{pq} \tag{19}\]  
On the other hand, the execution of a similarity of transformation  
The decomposition of the covariance matrix \(R_{xx}\) is obtained by factorizing the matrix with three matrices \(R_{xx}=Q\wedge Q^{T}\) with \(Q\) is an orthogonal matrix \(QQ^{T}=I\) and \(\wedge\) is a diagonal matrix which contains the values of the matrix \(R_{xx}\). The Jacobi method makes it possible to iteratively calculate the eigenvalues as follows [30,31]:  
\[(R_{xx})_{k+1}=Q_{k}(R_{xx})_{k}Q_{k}^{T} \tag{20}\]  
With \(k=0,1,2\ldots\)  
With \(Q_{k}\) is orthogonal plane of rotation with the angle \(\theta\) in the plane \((i,j)\) and \((q_{ii}=q_{jj}=cos\theta_{k}\) and \(q_{ij}=-q_{ji}=sin\theta_{k})\)  
After running all the pairs \((i,j)\). The matrix \(R_{xx}\)converges to the matrix \(\wedge\) which contains all the eigenvalues \(\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\). \[\lim_{kn\rightarrow\infty}(R_{xx})_{k}=diag[\lambda_{1},\lambda_{2},\ldots, \lambda_{n}] \tag{21}\]  
\[\lim_{kn\rightarrow\infty}(R_{xx})_{k}=\left[ \begin{array}{ccccccccc}\lambda _{1}&0&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&0\\ &\lambda_{2}&&&&\cdot&&\cdot\\ 0&&\cdot&&&&\cdot&\cdot\\ \cdot&&&&\cdot&&\cdot&\cdot\\ \cdot&&&&\cdot&&&\cdot\\ \cdot&&&&\cdot&&&\cdot\\ \cdot&&&&\cdot&&\cdot\\ \cdot&&&&\cdot&&\cdot&\cdot\\ \cdot&&&&\cdot&\cdot&\cdot\\ 0&&&&\cdot&\cdot&\cdot&\cdot&0&\lambda_{n}\end{array}