Simple Interval Rootfinding (NonlinearSolve.jl vs Roots.jl vs MATLAB)
This example comes from MATLAB's documentation showing improved rootfinding performance, and thus can be assumed to be considered optimized from MATLAB's perspective. MATLAB's results are:
In comparison, Roots.jl:
using Roots, BenchmarkTools, Random
Random.seed!(42)
const N = 100_000;
levels = 1.5 .* rand(N);
out = zeros(N);
myfun(x, lv) = x * sin(x) - lv
function froots(out, levels, u0)
for i in 1:N
out[i] = find_zero(myfun, u0, levels[i])
end
end
@btime froots(out, levels, (0, 2))207.226 ms (0 allocations: 0 bytes)using NonlinearSolve, BenchmarkTools
function f(out, levels, u0)
for i in 1:N
out[i] = solve(IntervalNonlinearProblem{false}(IntervalNonlinearFunction{false}(myfun),
u0, levels[i]), ITP()).u
end
end
function f2(out, levels, u0)
for i in 1:N
out[i] = solve(IntervalNonlinearProblem{false}(IntervalNonlinearFunction{false}(myfun),
u0, levels[i]), NonlinearSolve.Bisection()).u
end
end
function f3(out, levels, u0)
for i in 1:N
out[i] = solve(NonlinearProblem{false}(NonlinearFunction{false}(myfun),
u0, levels[i]), SimpleNewtonRaphson()).u
end
end
@btime f(out, levels, (0.0, 2.0))
@btime f2(out, levels, (0.0, 2.0))
@btime f3(out, levels, 1.0)33.746 ms (0 allocations: 0 bytes)
97.575 ms (0 allocations: 0 bytes)
12.851 ms (0 allocations: 0 bytes)MATLAB 2022a reportedly achieves 1.66s. Try this code yourself: we receive ~0.05 seconds, or a 33x speedup.
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/IntervalNonlinearProblem","simpleintervalrootfind.jmd")Computer Information:
Julia Version 1.10.10
Commit 95f30e51f41 (2025-06-27 09:51 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
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, znver2)
Threads: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
JULIA_CPU_THREADS = 128
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953
Package Information:
Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/IntervalNonlinearProblem/Project.toml`
[6e4b80f9] BenchmarkTools v1.6.0
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⌃ [f2b01f46] Roots v2.2.7
[31c91b34] SciMLBenchmarks v0.1.3 `../..`
⌃ [727e6d20] SimpleNonlinearSolve v2.3.0
[de0858da] Printf
[9a3f8284] Random
[10745b16] Statistics v1.10.0
Info Packages marked with ⌃ have new versions available and may be upgradable.
Warning The project dependencies or compat requirements have changed since the manifest was last resolved. It is recommended to `Pkg.resolve()` or consider `Pkg.update()` if necessary.And the full manifest:
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