scaling of CPU-ELPA2. (2) For the same number of nodes, the speedup becomes larger as the matrix size increases. (3) For the same node count and the same matrix size, the speedup is larger for complex matrices than for real matrices.

![image](fig/all_talos.png){width="99.00000%"}

The timing experiment in Fig. \[fig:all\_talos\] is repeated on the Summit supercomputer. The results are shown in Fig. \[fig:all\_summit\]. CPU-ELPA1 is omitted for simplicity. On Summit, the GPU-accelerated solvers GPU-ELPA1 and GPU-ELPA2 are always faster than the CPU-ELPA2 solver, with a maximum speedup of over 20x. The speedup of GPU-ELPA2 over CPU-ELPA2 remains 2.2x ($N$ = 40,000 real) to 6.7x ($N$ = 100,000 complex) even for 64 nodes. For the same matrix size and the same node count, the speedup on Summit appears greater than on Talos, which can be partially attributed to the difference in hardware. Summit has 6 GPUs per node, whereas Talos only has two GPUs per node. Data transfers on Summit take advantage of the NVLink technology [@nvlink_foley_2017] for high-bandwidth interconnect between CPUs and GPUs. Besides, a high-performance CPU kernel for the tridiagonal-to-banded back-transformation, written in AVX512 instructions [@elpa_kus_2019b], is employed for the Intel Xeon Gold CPUs on Talos, rendering better performance of CPU-ELPA2 