GitHub

LU Factorization Benchmarks

using BenchmarkTools, Random, VectorizationBase
using LinearAlgebra, LinearSolve, RecursiveFactorization, MKL_jll
# FastLapackInterface backs FastLUFactorization: its `init_cacheval`/`solve!` live in
# LinearSolveFastLapackInterfaceExt, so the algorithm silently has no method unless the
# package is loaded here. Same reason RecursiveFactorization is loaded above (it backs
# both RFLUFactorization and ButterflyFactorization).
using FastLapackInterface
nc = min(Int(VectorizationBase.num_cores()), Threads.nthreads())
BLAS.set_num_threads(nc)
BenchmarkTools.DEFAULT_PARAMETERS.seconds = 0.5

function luflop(m, n = m; innerflop = 2)
    sum(1:min(m, n)) do k
        invflop = 1
        scaleflop = max(m - k, 0)
        updateflop = innerflop * max(m - k, 0) * max(n - k, 0)
        invflop + scaleflop + updateflop
    end
end

# LinearSolve's LoadMKL_JLL preference defaults to false on AMD EPYC (this
# runner's CPU; SciML/LinearSolve.jl#518), and when it is false MKL_jll is not loaded
# at all, so MKLLUFactorization throws instead of running. This folder's
# Project.toml sets the preference to true precisely so MKL IS measured here:
# whether MKL still loses to OpenBLAS and RF on EPYC is a question this
# benchmark should answer, not assume. The probe is a safety net: if MKL
# still cannot load, say so on the page rather than erroring the sweep.
mkl_works = try
    solve(LinearProblem(Matrix(8.0I, 8, 8), ones(8)), MKLLUFactorization())
    true
catch
    false
end
mkl_works || println("MKLLUFactorization excluded: MKL failed to load in this ",
    "environment even with LoadMKL_JLL = true (see SciML/LinearSolve.jl#518).")

algs = [LUFactorization(), GenericLUFactorization(), RFLUFactorization(),
    (mkl_works ? [MKLLUFactorization()] : [])...,
    FastLUFactorization(), SimpleLUFactorization(),
    ButterflyFactorization(Val(true))]
res = [Float64[] for i in 1:length(algs)]

ns = 4:8:500
for i in 1:length(ns)
    n = ns[i]
    @info "$n × $n"
    rng = MersenneTwister(123)
    global A = rand(rng, n, n)
    global b = rand(rng, n)
    global u0 = rand(rng, n)

    for j in 1:length(algs)
        bt = @belapsed solve(prob, $(algs[j])).u setup=(prob = LinearProblem(
            copy(A), copy(b); u0 = copy(u0),
            alias = LinearAliasSpecifier(alias_A = true, alias_b = true)))
        push!(res[j], luflop(n) / bt / 1e9)
    end
end

using Plots
__parameterless_type(T) = Base.typename(T).wrapper
parameterless_type(x) = __parameterless_type(typeof(x))
parameterless_type(::Type{T}) where {T} = __parameterless_type(T)

p = plot(
    ns, res[1]; ylabel = "GFLOPs", xlabel = "N", title = "GFLOPs for NxN LU Factorization",
    label = string(Symbol(parameterless_type(algs[1]))), legend = :outertopright)
for i in 2:length(res)
    plot!(p, ns, res[i]; label = string(Symbol(parameterless_type(algs[i]))))
end
p

Appendix

Appendix

These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this benchmark, do the following commands:

using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/LinearSolve","LUFactorization.jmd")

Computer Information:

Julia Version 1.12.7
Commit 6d172b025e4 (2026-08-15 08:05 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver2)
  GC: Built with stock GC
Threads: 128 default, 1 interactive, 128 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/LinearSolve/Project.toml`
  [6e4b80f9] BenchmarkTools v1.8.0
  [29a986be] FastLapackInterface v2.1.1
⌃ [7ed4a6bd] LinearSolve v5.5.0
  [b51810bb] MatrixDepot v1.1.0
  [46dd5b70] Pardiso v1.1.2
⌃ [91a5bcdd] Plots v1.41.6
⌃ [b7e1f0a2] PureUMFPACK v0.1.4
⌃ [f2c3362d] RecursiveFactorization v0.2.26
⌃ [31c91b34] SciMLBenchmarks v0.1.3 [loaded: v0.1.5]
  [e56a9233] Sparspak v0.3.15
  [10745b16] Statistics v1.11.1
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  [856f044c] MKL_jll v2025.2.0+0
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  [9a3f8284] Random v1.11.0
  [2f01184e] SparseArrays v1.12.0
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

Status `/julia/github-runners/amdci1-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/LinearSolve/Manifest.toml`
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  [44cfe95a] Pkg v1.12.1
  [de0858da] Printf v1.11.0
  [9abbd945] Profile v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.12.0
  [f489334b] StyledStrings v1.11.0
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.3.0+1
  [deac9b47] LibCURL_jll v8.15.0+0
  [e37daf67] LibGit2_jll v1.9.0+0
  [29816b5a] LibSSH2_jll v1.11.3+1
  [14a3606d] MozillaCACerts_jll v2025.11.4
  [4536629a] OpenBLAS_jll v0.3.29+0
  [05823500] OpenLibm_jll v0.8.7+0
  [458c3c95] OpenSSL_jll v3.5.4+0
  [efcefdf7] PCRE2_jll v10.44.0+1
  [bea87d4a] SuiteSparse_jll v7.8.3+2
  [83775a58] Zlib_jll v1.3.1+2
  [8e850b90] libblastrampoline_jll v5.15.0+0
  [8e850ede] nghttp2_jll v1.64.0+1
  [3f19e933] p7zip_jll v17.7.0+0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`