Gaius.jl

Gaius defines the public functions Gaius.mul and Gaius.mul!. Gaius.mul is to be used like the regular * operator between two matrices whereas Gaius.mul! takes in three matrices C, A, B and stores A*B in C overwriting the contents of C.

The functions Gaius.mul and Gaius.mul! use multithreading. If you want to run the single-threaded variants, use Gais.mul_serial and Gaius.mul_serial! respectively.

julia> using Gaius, BenchmarkTools, LinearAlgebra

julia> A, B, C = rand(104, 104), rand(104, 104), zeros(104, 104);

julia> @btime mul!($C, $A, $B); # from LinearAlgebra
  68.529 μs (0 allocations: 0 bytes)

julia> @btime mul!($C, $A, $B); #from Gaius
  31.220 μs (80 allocations: 10.20 KiB)

julia> using Gaius, BenchmarkTools

julia> A, B = rand(104, 104), rand(104, 104);

julia> @btime $A * $B;
  68.949 μs (2 allocations: 84.58 KiB)

julia> @btime let * = Gaius.mul # Locally use Gaius.mul as * operator.
           $A * $B
       end;
  32.950 μs (82 allocations: 94.78 KiB)

julia> versioninfo()
Julia Version 1.4.0-rc2.0
Commit b99ed72c95* (2020-02-24 16:51 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD Ryzen 5 2600 Six-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-8.0.1 (ORCJIT, znver1)
Environment:
  JULIA_NUM_THREADS = 6

Multi-threading in Gaius works by recursively splitting matrices into sub-blocks to operate on. You can change the matrix sub-block size by calling mul! with the block_size keyword argument. If left unspecified, Gaius will use a (very rough) heuristic to choose a good block size based on the size of the input matrices.

The size heuristics I use are likely not yet optimal for everyone's machines.

Gaius supports the multiplication of matrices of complex numbers, but they must first by converted explicity to structs of arrays using StructArrays.jl (otherwise the multiplication will be done by OpenBLAS):

julia> using Gaius, StructArrays

julia> begin
           n = 150
           A = randn(ComplexF64, n, n)
           B = randn(ComplexF64, n, n)
           C = zeros(ComplexF64, n, n)


           SA =  StructArray(A)
           SB =  StructArray(B)
           SC = StructArray(C)

           @btime mul!($SC, $SA, $SB)
           @btime         mul!($C, $A, $B)
           SC ≈ C
       end
   515.587 μs (80 allocations: 10.53 KiB)
   546.481 μs (0 allocations: 0 bytes)
 true

The following benchmarks were run on this

julia> versioninfo()
Julia Version 1.4.0-rc2.0
Commit b99ed72c95* (2020-02-24 16:51 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD Ryzen 5 2600 Six-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-8.0.1 (ORCJIT, znver1)
Environment:
  JULIA_NUM_THREADS = 6

and compared to OpenBLAS running with 6 threads (BLAS.set_num_threads(6)). I would be keenly interested in seeing analogous benchmarks on a machine with an AVX512 instruction set and/or Intel's MKL.

Note that these are log-log plots.

Gaius outperforms OpenBLAS over a large range of matrix sizes, but does begin to appreciably fall behind around 800 x 800 matrices for Float64 and 650 x 650 matrices for Float32. I believe there is a large amount of performance left on the table in Gaius and I look forward to beating OpenBLAS for more matrix sizes.

Here is Gaius operating on Complex{Float64} structs-of-arrays competeing relatively evenly against OpenBLAS operating on Complex{Float64} arrays-of-structs:

I think with some work, we can do much better.

These benchmarks compare Gaius (on the same machine as above) and compare against Julia's generic matrix multiplication implementation (OpenBLAS does not provide integer mat-mul) which is not multi-threaded.

Note that these are log-log plots.

Benchmarks performed on a machine with the AVX512 instruction set show an even greater performance gain.

If you find yourself in a high performance situation where you want to multiply matrices of integers, I think this provides a compelling use-case for Gaius since it will outperform it's competition at any matrix size and for large matrices will benefit from multi-threading.