JuliaLang / julia / #37899

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Commit Message

Avoid materializing arrays in bidiag matmul (#55450)

Currently, small `Bidiagonal`/`Tridiagonal` matrices are materialized in
matrix multiplications, but this is wasteful and unnecessary. This PR
changes this to use a naive matrix multiplication for small matrices,
and fall back to the banded multiplication for larger ones.
Multiplication by a `Bidiagonal` falls back to a banded matrix
multiplication for all sizes in the current implementation, and iterates
in a cache-friendly manner for the non-`Bidiagonal` matrix.

In certain cases, the matrices were being materialized if the
non-structured matrix was small, even if the structured matrix was
large. This is changed as well in this PR.

Some improvements in performance:
```julia
julia> B = Bidiagonal(rand(3), rand(2), :U); A = rand(size(B)...); C = similar(A);

julia> @btime mul!($C, $A, $B);
  193.152 ns (6 allocations: 352 bytes) # nightly v"1.12.0-DEV.1034"
  18.826 ns (0 allocations: 0 bytes) # This PR

julia> T = Tridiagonal(rand(99), rand(100), rand(99)); A = rand(2, size(T,2)); C = similar(A);

julia> @btime mul!($C, $A, $T);
  9.398 μs (8 allocations: 79.94 KiB) # nightly
  416.407 ns (0 allocations: 0 bytes) # This PR

julia> B = Bidiagonal(rand(300), rand(299), :U); A = rand(20000, size(B,2)); C = similar(A);

julia> @btime mul!($C, $A, $B);
  33.395 ms (0 allocations: 0 bytes) # nightly
  6.695 ms (0 allocations: 0 bytes) # This PR (cache-friendly)
```

Closes https://github.com/JuliaLang/julia/pull/55414

---------

Co-authored-by: Daniel Karrasch <daniel.karrasch@posteo.de>

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