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] = solve(ZeroProblem(myfun, u0), levels[i])
    end
    return
end

@btime froots(out, levels, (0, 2))

107.438 ms (0 allocations: 0 bytes)

using BracketingNonlinearSolve, SimpleNonlinearSolve, BenchmarkTools
using BracketingNonlinearSolve: Bisection # Roots also exports Bisection leading to a name conflict

function f(out, levels, u0)
    for i in 1:N
        out[i] = solve(
            IntervalNonlinearProblem{false}(
                IntervalNonlinearFunction{false}(myfun),
                u0, levels[i]
            ),
            ITP()
        ).u
    end
    return
end

function f2(out, levels, u0)
    for i in 1:N
        out[i] = solve(
            IntervalNonlinearProblem{false}(
                IntervalNonlinearFunction{false}(myfun),
                u0, levels[i]
            ),
            Bisection()
        ).u
    end
    return
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
    return
end

@btime f(out, levels, (0.0, 2.0))
@btime f2(out, levels, (0.0, 2.0))
@btime f3(out, levels, 1.0)

46.298 ms (0 allocations: 0 bytes)
  128.755 ms (0 allocations: 0 bytes)
  17.460 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 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 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-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/IntervalNonlinearProblem/Project.toml`
  [6e4b80f9] BenchmarkTools v1.8.0
  [70df07ce] BracketingNonlinearSolve v1.12.6
  [f2b01f46] Roots v3.0.7
⌃ [31c91b34] SciMLBenchmarks v0.1.3
  [727e6d20] SimpleNonlinearSolve v2.14.1
  [10745b16] Statistics v1.11.1
  [de0858da] Printf v1.11.0
  [9a3f8284] Random v1.11.0
Info Packages marked with ⌃ have new versions available and may be upgradable.

And the full manifest:

Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/IntervalNonlinearProblem/Manifest.toml`
  [47edcb42] ADTypes v1.24.0
  [7d9f7c33] Accessors v0.1.45
  [79e6a3ab] Adapt v4.7.0
  [4fba245c] ArrayInterface v7.30.0
  [6e4b80f9] BenchmarkTools v1.8.0
  [70df07ce] BracketingNonlinearSolve v1.12.6
  [38540f10] CommonSolve v0.2.14
  [bbf7d656] CommonSubexpressions v0.3.1
  [34da2185] Compat v4.18.1
  [a33af91c] CompositionsBase v0.1.2
  [2569d6c7] ConcreteStructs v0.2.8
  [8f4d0f93] Conda v1.10.3
  [187b0558] ConstructionBase v1.6.0
  [a8cc5b0e] Crayons v4.2.0
  [9a962f9c] DataAPI v1.16.0
  [e2d170a0] DataValueInterfaces v1.0.0
  [163ba53b] DiffResults v1.1.0
  [b552c78f] DiffRules v1.16.0
  [a0c0ee7d] DifferentiationInterface v0.7.21
  [ffbed154] DocStringExtensions v0.9.5
  [4e289a0a] EnumX v1.0.7
  [f151be2c] EnzymeCore v0.8.21
  [e2ba6199] ExprTools v0.1.11
  [9aa1b823] FastClosures v0.3.2
  [64ca27bc] FindFirstFunctions v3.2.1
  [6a86dc24] FiniteDiff v2.33.0
  [f6369f11] ForwardDiff v1.4.5
  [069b7b12] FunctionWrappers v1.1.3
  [77dc65aa] FunctionWrappersWrappers v1.13.0
  [46192b85] GPUArraysCore v0.2.0
  [d7ba0133] Git v1.5.0
⌅ [eafb193a] Highlights v0.5.3
  [7073ff75] IJulia v1.34.4
  [3587e190] InverseFunctions v0.1.17
  [92d709cd] IrrationalConstants v0.2.6
  [82899510] IteratorInterfaceExtensions v1.0.0
  [692b3bcd] JLLWrappers v1.8.0
⌅ [682c06a0] JSON v0.21.4
  [b964fa9f] LaTeXStrings v1.4.1
  [87fe0de2] LineSearch v0.1.16
  [2ab3a3ac] LogExpFunctions v1.0.1
  [e6f89c97] LoggingExtras v1.2.0
  [1914dd2f] MacroTools v0.5.16
  [bb5d69b7] MaybeInplace v0.1.8
  [ffc61752] Mustache v1.0.21
  [77ba4419] NaNMath v1.1.4
⌃ [be0214bd] NonlinearSolveBase v2.48.0
  [bac558e1] OrderedCollections v2.0.1
⌅ [69de0a69] Parsers v2.8.7
⌃ [d236fae5] PreallocationTools v1.7.0
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.5.2
  [08abe8d2] PrettyTables v3.4.8
  [3cdcf5f2] RecipesBase v1.3.4
  [731186ca] RecursiveArrayTools v4.5.1
  [189a3867] Reexport v1.2.2
  [05181044] RelocatableFolders v1.0.1
  [ae029012] Requires v1.3.1
  [9fe22ead] RespecializeParams v1.3.0
  [f2b01f46] Roots v3.0.7
  [7e49a35a] RuntimeGeneratedFunctions v0.5.25
⌃ [0bca4576] SciMLBase v3.49.2
⌃ [31c91b34] SciMLBenchmarks v0.1.3
  [19f34311] SciMLJacobianOperators v0.1.18
  [a6db7da4] SciMLLogging v2.1.0
  [c0aeaf25] SciMLOperators v1.30.0
  [431bcebd] SciMLPublic v1.3.0
  [53ae85a6] SciMLStructures v1.10.5
  [6c6a2e73] Scratch v1.3.0
  [efcf1570] Setfield v1.1.2
  [727e6d20] SimpleNonlinearSolve v2.14.1
  [276daf66] SpecialFunctions v2.9.0
  [1e83bf80] StaticArraysCore v1.4.4
  [10745b16] Statistics v1.11.1
  [69024149] StringEncodings v0.3.7
  [892a3eda] StringManipulation v0.5.0
  [2efcf032] SymbolicIndexingInterface v0.3.55
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.14.0
  [a759f4b9] TimerOutputs v1.2.0
  [81def892] VersionParsing v1.3.0
  [44d3d7a6] Weave v0.10.12
  [ddb6d928] YAML v0.4.16
  [c2297ded] ZMQ v1.5.1
  [2e619515] Expat_jll v2.8.3+0
  [020c3dae] Git_LFS_jll v3.7.1+0
  [f8c6e375] Git_jll v2.55.0+0
  [94ce4f54] Libiconv_jll v1.18.0+0
  [9bd350c2] OpenSSH_jll v10.5.1+0
  [458c3c95] OpenSSL_jll v3.5.8+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [8f1865be] ZeroMQ_jll v4.3.6+0
  [a9144af2] libsodium_jll v1.0.21+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [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
  [f489334b] StyledStrings v1.11.0
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
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`