  Current dense symmetric eigenvalue (EIG) and singular value decomposition
(SVD) implementations may suffer from the lack of concurrency during the
tridiagonal and bidiagonal reductions, respectively. This performance
bottleneck is typical for the two-sided transformations due to the Level-2 BLAS
memory-bound calls. Therefore, the current state-of-the-art EIG and SVD
implementations may achieve only a small fraction of the system's sustained
peak performance. The QR-based Dynamically Weighted Halley (QDWH) algorithm may
be used as a pre-processing step toward the EIG and SVD solvers, while
mitigating the aforementioned bottleneck. QDWH-EIG and QDWH-SVD expose more
parallelism, while relying on compute-bound matrix operations. Both run closer
to the sustained peak performance of the system, but at the expense of
performing more FLOPS than the standard EIG and SVD algorithms. In this paper,
we introduce a new QDWH-based solver for computing the partial spectrum for EIG
(QDWHpartial-EIG) and SVD (QDWHpartial-SVD) problems. By optimizing the
rational function underlying the algorithms only in the desired part of the
spectrum, QDWHpartial-EIG and QDWHpartial-SVD algorithms efficiently compute a
fraction (say 1-20%) of the corresponding spectrum. We develop high-performance
implementations of QDWHpartial-EIG and QDWHpartial-SVD on distributed-memory
anymore systems and demonstrate their numerical robustness. Experimental
results using up to 36K MPI processes show performance speedups for
QDWHpartial-SVD up to 6X and 2X against PDGESVD from ScaLAPACK and KSVD,
respectively. QDWHpartial-EIG outperforms PDSYEVD from ScaLAPACK up to 3.5X but
remains slower compared to ELPA. QDWHpartial-EIG achieves, however, a better
occupancy of the underlying hardware by extracting higher sustained peak
performance than ELPA, which is critical moving forward with accelerator-based
supercomputers.
