  The eigenvalue decomposition (EVD) of (a batch of) Hermitian matrices of
order two has a role in many numerical algorithms, of which the one-sided
Jacobi method for the singular value decomposition (SVD) is the prime example.
In this paper the batched EVD is vectorized, with a vector-friendly data layout
and the AVX-512 SIMD instructions of Intel CPUs, alongside other key components
of a real and a complex OpenMP-parallel Jacobi-type SVD method, inspired by the
sequential xGESVJ routines from LAPACK. These vectorized building blocks should
be portable to other platforms that support similar vector operations.
Unconditional numerical reproducibility is guaranteed for the batched EVD,
sequential or threaded, and for the column transformations, that are, like the
scaled dot-products, presently sequential but can be threaded if nested
parallelism is desired. No avoidable overflow of the results can occur with the
proposed EVD or the whole SVD. The measured accuracy of the proposed EVD often
surpasses that of the xLAEV2 routines from LAPACK. While the batched EVD
outperforms the matching sequence of xLAEV2 calls, speedup of the parallel SVD
is modest but can be improved and is already beneficial with enough threads.
Regardless of their number, the proposed SVD method gives identical results,
but of somewhat lower accuracy than xGESVJ.
