Vectorization of the Jacobi-type singular value decomposition method

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, with unconditional reproducibility guaranteed for the batched EVD and several others. No avoidable overflow of the results can occur with the proposed EVD or 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.