NonlinearSolve.jl: High-Performance Unified Nonlinear Solvers

NonlinearSolve.jl is a unified interface for the nonlinear solving packages of Julia. The package includes its own high-performance nonlinear solvers which include the ability to swap out to fast direct and iterative linear solvers, along with the ability to use sparse automatic differentiation for Jacobian construction and Jacobian-vector products. NonlinearSolve.jl interfaces with other packages of the Julia ecosystem to make it easy to test alternative solver packages and pass small types to control algorithm swapping. It also interfaces with the ModelingToolkit.jl world of symbolic modeling to allow for automatically generating high-performance code.

Performance is key: the current methods are made to be highly performant on scalar and statically sized small problems, with options for large-scale systems. If you run into any performance issues, please file an issue. Consult the NonlinearSystemSolvers page for information on how to import solvers from different packages.

Installation

To install NonlinearSolve.jl, use the Julia package manager:

using Pkg
Pkg.add("NonlinearSolve")

Contributing

Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to SciML.
See the SciML Style Guide for common coding practices and other style decisions.
There are a few community forums:
- The #diffeq-bridged and #sciml-bridged channels in the Julia Slack
- The #diffeq-bridged and #sciml-bridged channels in the Julia Zulip
- On the Julia Discourse forums
- See also SciML Community page

Reproducibility

The documentation of this SciML package was built using these direct dependencies,

Status `~/work/NonlinearSolve.jl/NonlinearSolve.jl/docs/Project.toml`
  [47edcb42] ADTypes v1.14.0
  [2169fc97] AlgebraicMultigrid v1.0.0
  [4fba245c] ArrayInterface v7.18.0
  [6e4b80f9] BenchmarkTools v1.6.0
  [70df07ce] BracketingNonlinearSolve v1.1.3 `../lib/BracketingNonlinearSolve`
  [2b5f629d] DiffEqBase v6.168.0
  [a0c0ee7d] DifferentiationInterface v0.6.52
  [e30172f5] Documenter v1.10.1
  [daee34ce] DocumenterCitations v1.3.7
  [d12716ef] DocumenterInterLinks v1.0.0
  [40713840] IncompleteLU v0.2.1
  [7ed4a6bd] LinearSolve v3.8.0
  [8913a72c] NonlinearSolve v4.7.0 `..`
  [be0214bd] NonlinearSolveBase v1.5.3 `../lib/NonlinearSolveBase`
  [5959db7a] NonlinearSolveFirstOrder v1.3.1 `../lib/NonlinearSolveFirstOrder`
  [2ac3b008] NonlinearSolveHomotopyContinuation v0.1.1 `../lib/NonlinearSolveHomotopyContinuation`
  [9a2c21bd] NonlinearSolveQuasiNewton v1.2.1 `../lib/NonlinearSolveQuasiNewton`
  [26075421] NonlinearSolveSpectralMethods v1.1.1 `../lib/NonlinearSolveSpectralMethods`
  [b1df2697] OrdinaryDiffEqTsit5 v1.1.0
  [ace2c81b] PETSc v0.3.1
  [91a5bcdd] Plots v1.40.13
  [0bca4576] SciMLBase v2.85.0
  [19f34311] SciMLJacobianOperators v0.1.2 `../lib/SciMLJacobianOperators`
  [727e6d20] SimpleNonlinearSolve v2.2.2 `../lib/SimpleNonlinearSolve`
  [9f842d2f] SparseConnectivityTracer v0.6.17
  [90137ffa] StaticArrays v1.9.13
  [9672c7b4] SteadyStateDiffEq v2.5.0
  [c3572dad] Sundials v4.28.0
  [b77e0a4c] InteractiveUtils
  [9a3f8284] Random

and using this machine and Julia version.

Julia Version 1.10.9
Commit 5595d20a287 (2025-03-10 12:51 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_DEBUG = Documenter

A more complete overview of all dependencies and their versions is also provided.

Status `~/work/NonlinearSolve.jl/NonlinearSolve.jl/docs/Manifest.toml`
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⌅ [f1be7e48] Bibliography v0.2.20
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  [2ab3a3ac] LogExpFunctions v0.3.29
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  [8913a72c] NonlinearSolve v4.7.0 `..`
  [be0214bd] NonlinearSolveBase v1.5.3 `../lib/NonlinearSolveBase`
  [5959db7a] NonlinearSolveFirstOrder v1.3.1 `../lib/NonlinearSolveFirstOrder`
  [2ac3b008] NonlinearSolveHomotopyContinuation v0.1.1 `../lib/NonlinearSolveHomotopyContinuation`
  [9a2c21bd] NonlinearSolveQuasiNewton v1.2.1 `../lib/NonlinearSolveQuasiNewton`
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  [35ca27e7] eudev_jll v3.2.9+0
  [214eeab7] fzf_jll v0.61.1+0
  [1a1c6b14] gperf_jll v3.1.1+1
  [a4ae2306] libaom_jll v3.11.0+0
  [0ac62f75] libass_jll v0.15.2+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [2db6ffa8] libevdev_jll v1.11.0+0
  [f638f0a6] libfdk_aac_jll v2.0.3+0
  [36db933b] libinput_jll v1.18.0+0
  [b53b4c65] libpng_jll v1.6.47+0
  [f27f6e37] libvorbis_jll v1.3.7+2
  [009596ad] mtdev_jll v1.1.6+0
  [1317d2d5] oneTBB_jll v2022.0.0+0
⌅ [1270edf5] x264_jll v2021.5.5+0
⌅ [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+2
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.10.0
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [6462fe0b] Sockets
  [2f01184e] SparseArrays v1.10.0
  [10745b16] Statistics v1.10.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.1.1+0
  [781609d7] GMP_jll v6.2.1+6
  [deac9b47] LibCURL_jll v8.4.0+0
  [e37daf67] LibGit2_jll v1.6.4+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [3a97d323] MPFR_jll v4.2.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.2+1
  [14a3606d] MozillaCACerts_jll v2023.1.10
  [4536629a] OpenBLAS_jll v0.3.23+4
  [05823500] OpenLibm_jll v0.8.1+4
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.2.1+1
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.52.0+1
  [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`

You can also download the manifest file and the project file.