## 4 Experiments
Table 3 and Figure 4 present the errors in the eigenpairs, allowing for a comprehensive comparison between our implementations and the MKL and MAGMA implementations. Notably, there is no significant difference in the errors of the eigenpairs obtained from our implementations compared to those from the MKL and MAGMA implementations. Upon reviewing Table 3

\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(n\) & **Dstedc** & **Sequential** & **Dstedx** & **Hybrid** \\ \hline
1000 & 35.31166 & 37.33702 & 348.03122 & 438.65449 \\ \hline
2000 & 102.14143 & 100.76133 & 402.26580 & 484.71602 \\ \hline





The orthogonality error variation between our implementation and current libraries, as shown in Table 4 and Figure 5, is nearly negligible. Therefore, it is difficult to claim that we have significantly improved this aspect. However, it is worth noting that our implementation still achieves high quality in terms of eigenpairs and eigenvectors orthogonality. It is possible to observe orthogonality errors in Table 4 and \[A = \frac{1}{\alpha}B, \tag{11}\]6 We have slightly less error at matrix 1,4 and 7-17 k sizes in the sequential version and at 1,4,7-9 and 12-15 K matrix sizes in the parallel version. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(n\) & **Dstedc** & **Sequential** & **Dstedx** & **Hybrid** \\ \hline



\hline \end{tabular}
\end{table} 
Table 4: Orthogonality error \(\|QQ^{T}-I\|_{F}\)—diagonalization. Figure 5: Orthogonality error \(\|QQ^{T}-I\|_{F}\). In summary, our implementation outperforms state-of-the-art libraries in terms of performance. Additionally, we have achieved this improved performance without compromising accuracy. ## 6 Conclusions

This paper presents a hybrid-heterogeneous parallel implementation of the divide-and-conquer algorithm targeting the symmetric tridiagonal matrix problem. This implementation leverages the computing power of GPUs and CPUs on multiple nodes. By using the multi-node implementation of [PERSON]'s algorithm, the computation time required to diagonalize symmetric tridiagonal matrices is reduced compared to the sequential versions and parallel implementations available in current libraries. An important advantage of this implementation is that it overcomes the memory limitations of many GPUs, which have significantly smaller memory capacity than CPUs. As a result, it is possible to scale this implementation to larger problem sizes that were previously limited by GPU memory constraints. Furthermore, extensive testing has shown that this implementation achieves exceptional accuracy in terms of eigenvector orthogonality and eigenpair quality. The optimization of the implementation time does not compromise the accuracy of the results. These implementation behaviors show the importance of this work in addressing eigenvalue problems that are widespread in scientific and engineering applications. The implications of these results suggest that it could open doors to new scientific discoveries by improving the efficiency and accuracy of diagonalization methods. Finally, it was also shown that on a node with two graphics cards, the speedup was double compared to libraries working with a single graphics card. There are several essential aspects to consider in our work. We first implemented a tridiagonalization algorithm, such as Householder, on multiple nodes to handle dense symmetric matrices efficiently. Secondly, we are exploring low communication cost schemes, like a binary tree, to enable scalability across an arbitrary number of nodes. Additionally, we are investigating alternative methods for solving secular equations to improve accuracy and evaluate their impact on execution time. Comparative studies with existing libraries, such as CULA, will help assess the performance of our algorithm. They will also evaluate energy consumption and leveraging CUDA features like streams, shared memory, and unified memory will aid optimization.