This post is for those interested high-performance matrix multiplication in BQN (an APL-like array language).The main thing I got out of it was the footnotes, in particular: <a href="https:&#x2F;&#x2F;en.algorithmica.org&#x2F;hpc&#x2F;algorithms&#x2F;matmul&#x2F;" rel="nofollow">https:&#x2F;&#x2F;en.algorithmica.org&#x2F;hpc&#x2F;algorithms&#x2F;matmul&#x2F;</a> is a really nice post on fast matrix multiplication, and is a chapter of what looks like a nice online book.

And the reason +˝ is fairly fast for long rows, despite that page claiming no optimizations, is that ˝ is defined to split its argument into cells, e.g. rows of a matrix, and apply + with those as arguments. So + is able to apply its ordinary vectorization, while it can&#x27;t in some other situations where it&#x27;s applied element-wise. This still doesn&#x27;t make great use of cache and I do have some special code working for floats that does much better with a tiling pattern, but I wanted to improve +˝ for integers along with it and haven&#x27;t finished those (widening on overflow is complicated).

The relevant operations for matrix multiply are leading-axis extension, shown near the end of [0], and Insert +˝ shown in [1]. Both for floats; the leading-axis operation is × but it&#x27;s the same speed as + with floating-point SIMD. We don&#x27;t handle these all that well, with needless copying in × and a lot of per-row overhead in +˝, but of course it&#x27;s way better than scalar evaluation.[0] <a href="https:&#x2F;&#x2F;mlochbaum.github.io&#x2F;bencharray&#x2F;pages&#x2F;arith.html" rel="nofollow">https:&#x2F;&#x2F;mlochbaum.github.io&#x2F;bencharray&#x2F;pages&#x2F;arith.html</a>[1] <a href="https:&#x2F;&#x2F;mlochbaum.github.io&#x2F;bencharray&#x2F;pages&#x2F;fold.html" rel="nofollow">https:&#x2F;&#x2F;mlochbaum.github.io&#x2F;bencharray&#x2F;pages&#x2F;fold.html</a>

Yes; the CBQN interpreter has a number of specialised vectorised codepaths. It picks a good one for the arrays at runtime.

Does BQN just natively vectorize the code? I was surprised not to see anything about that.

Blazing Matrix Products