Divide and Conquer Linear Algebra
Author MasonProtter
90 Stars
Updated Last
1 Year Ago
Started In
February 2020


Because Caesar.jl was taken

Continuous Integration Continuous Integration (Julia nightly) Code Coverage

Gaius.jl is a multi-threaded BLAS-like library using a divide-and-conquer strategy to parallelism, and built on top of the fantastic LoopVectorization.jl. Gaius spawns threads using Julia's depth first parallel task runtime and so Gaius's routines may be fearlessly nested inside multi-threaded Julia programs.

Gaius is not stable or well tested. Only use it if you're adventurous.

Note: Gaius is not actively maintained and I do not anticipate doing further work on it. However, you may find it useful as a relatively simple playground for learning about the implementation of linear algebra routines.

There are other, more promising projects that may result in a scalable, multi-threaded pure Julia BLAS library such as:

  1. Tullio.jl
  2. Octavian.jl

In general:

  • Octavian is the most performant.
  • Tullio is the most flexible.

Quick Start

julia> using Gaius

julia> Gaius.mul!(C, A, B) # (multi-threaded) multiply A×B and store the result in C (overwriting the contents of C)

julia> Gaius.mul(A, B) # (multi-threaded) multiply A×B and return the result

julia> Gaius.mul_serial!(C, A, B) # (single-threaded) multiply A×B and store the result in C (overwriting the contents of C)

julia> Gaius.mul_serial(A, B) # (single-threaded) multiply A×B and return the result

Remember to start Julia with multiple threads with e.g. one of the following:

  • julia -t auto
  • julia -t 4
  • Set the JULIA_NUM_THREADS environment variable to 4 before starting Julia

The functions in this list are part of the public API of Gaius:

  • Gaius.mul!
  • Gaius.mul
  • Gaius.mul_serial!
  • Gaius.mul_serial

All other functions are internal (private).

Matrix Multiplication

Currently, fast, native matrix-multiplication is only implemented between matrices of types Matrix{<:Union{Float64, Float32, Int64, Int32, Int16}}, and StructArray{Complex}. Support for other other commonly encountered numeric struct types such as Rational and Dual numbers is planned.

Using Gaius

Click to expand:

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
  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
  LIBM: libopenlibm
  LLVM: libLLVM-8.0.1 (ORCJIT, znver1)

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.

Complex Numbers

Click to expand:

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
   515.587 μs (80 allocations: 10.53 KiB)
   546.481 μs (0 allocations: 0 bytes)


Floating Point Performance

Click to expand:

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
  LIBM: libopenlibm
  LLVM: libLLVM-8.0.1 (ORCJIT, znver1)

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.

Float64 Matrix Multiplication

Float32 Matrix Multiplication

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.

Complex Floating Point Performance

Click to expand:

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

Complex{Float64} Matrix Multiplication

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

Integer Performance

Click to expand:

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.

Int64 Matrix Multiplication

Int32 Matrix Multiplication

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.

Other BLAS Routines

I have not yet worked on implementing other standard BLAS routines with this strategy, but doing so should be relatively straightforward.


If you must break the law, do it to seize power; in all other cases observe it.

-Gaius Julius Caesar

If you use only the functions Gaius.mul!, Gaius.mul, Gaius.mul_serial!, and Gaius.mul_serial, automatic array size-checking will occur before the matrix multiplication begins. This can be turned off in mul! by calling Gaius.mul!(C, A, B; sizecheck=false), in which case no sizechecks will occur on the arrays before the matrix multiplication occurs and all sorts of bad, segfaulty things can happen.

All other functions in this package are to be considered internal and should not be expected to check for safety or obey the law. The functions Gaius.gemm_kernel! and Gaius.add_gemm_kernel! may be of utility, but be warned that they do not check array sizes.

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