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My question is, should I be using these libraries to compute e.g. dot products, matrix products etc.? Is that there ultimate intended use? If so, is OpenBLAS a reasonable choice for this? Or should I use something like usimd to compute these using SIMD instructions directly myself?
For graphics DX math is a very good library.
Picking what size you are targeting is really important, though. Could the matrices you are working with realistically be bigger than say 32x32? BLAS is good for big matrices. It's not as great for small matrices. Eigen or Fastor will do better for these smaller problems. And for various common operations on sizes 2, 3, and 4, hand coded graphics-oriented libraries might outperform those.