\[fig:all\_summit\], all eigenvalues and eigenvectors are computed. In applications such as materials simulations based on KS-DFT, only a portion of the eigenspectrum, e.g., typically 20% to 60% for LCAO basis sets, is of interest. In this case, the advantage of ELPA2 over ELPA1 should be more significant, as the computational complexity of the back-transformation is proportional to the number of eigenvectors to be calculated. This is demonstrated in Fig. \[fig:decomp3\_summit\], where the total timings (red circles) of the GPU-ELPA1 (solid) and GPU-ELPA2 (dashed) are decomposed into the forward tridiagonalization (blue squares), the solution of the tridiagonal problem (yellow diamonds), and the backward transformation (violet triangles). Two test cases, namely $N$ = 40,000 real and $N$ = 100,000 real, are shown as examples for a single node on Summit, i.e., the worst-case scenario for ELPA2. The two-stage tridiagonalization in ELPA2 is always faster than the one-stage tridiagonalization in ELPA1 by a factor of approximately three. The back-transformation accounts for a small fraction of the total time in ELPA1, but is the most time-consuming part in ELPA2 when all the eigenvectors are computed. When computing fewer eigenvectors, the burden of the two-stage back-transformation in ELPA2 can be greatly alleviated, making ELPA2 more 