Symbolics.jl

Symbolics.jl is a fast and modern Computer Algebra System (CAS) for a fast and modern programming language (Julia). The goal is to have a high-performance and parallelized symbolic algebra system that is directly extendable in the same language as that of the users.

Installation

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

using Pkg
Pkg.add("Symbolics")

Citation

If you use Symbolics.jl, please cite this paper

@article{gowda2021high,
  title={High-performance symbolic-numerics via multiple dispatch},
  author={Gowda, Shashi and Ma, Yingbo and Cheli, Alessandro and Gwozdz, Maja and Shah, Viral B and Edelman, Alan and Rackauckas, Christopher},
  journal={arXiv preprint arXiv:2105.03949},
  year={2021}
}

Feature Summary

Because Symbolics.jl is built into the Julia language and works with its dispatches, generic functions in Base Julia will work with symbolic expressions! Make matrices of symbolic expressions and multiply them: it will just work. Take the LU-factorization. Etc. Thus, see the Julia Documentation for a large list of functionality available in Symbolics.jl.

A general list of the features is:

Symbolic arithmetic with type information and multiple dispatch
Symbolic polynomials and trigonometric functions
Pattern matching, simplification and substitution
Differentiation
Symbolic linear algebra (factorizations, inversion, determinants, eigencomputations, etc.)
Discrete math (representations of summations, products, binomial coefficients, etc.)
Logical and Boolean expressions
Symbolic equation solving and conversion to arbitrary precision
Support for non-standard algebras (non-commutative symbols and customizable rulesets)
Special functions (list provided by SpecialFunctions.jl)
Automatic conversion of Julia code to symbolic code
Generation of (high performance and parallel) functions from symbolic expressions
Fast automated sparsity detection and generation of sparse Jacobians and Hessians

and much more.

Extension Packages

Below is a list of known extension packages. If you would like your package to be listed here, feel free to open a pull request!

ModelingToolkit.jl: Symbolic representations of common numerical systems
- Ordinary differential equations
- Stochastic differential equations
- Partial differential equations
- Nonlinear systems
- Optimization problems
- Optimal Control
- Causal and acausal modeling (Simulink/Modelica)
- Automated model transformation, simplification, and composition
Catalyst.jl: Symbolic representations of chemical reactions
- Symbolically build and represent large systems of chemical reactions
- Generate code for ODEs, SDEs, continuous-time Markov Chains, and more
- Simulate the models using the SciML ecosystem with O(1) Gillespie methods
DataDrivenDiffEq.jl: Automatic identification of equations from data
- Automated construction of ODEs and DAEs from data
- Representations of Koopman operators and Dynamic Mode Decomposition (DMD)
SymbolicRegression.jl: Distributed High-Performance symbolic regression
- Parallelized generic algorithms for finding equations from data
- Pareto frontier-based scoring
ReversePropagation.jl: Source-to-source reverse mode automatic differentiation
- Automated tracing of code and construction of backpropagation equations
- Composes with symbolic transformation and simplification functionality

Reproducibility

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

Status `~/work/Symbolics.jl/Symbolics.jl/docs/Project.toml`
  [6e4b80f9] BenchmarkTools v1.5.0
  [e30172f5] Documenter v1.8.0
  [0b43b601] Groebner v0.8.2
  [23fbe1c1] Latexify v0.16.5
⌅ [2edaba10] Nemo v0.47.5
  [1dea7af3] OrdinaryDiffEq v6.90.1
  [91a5bcdd] Plots v1.40.9
  [90137ffa] StaticArrays v1.9.8
  [d1185830] SymbolicUtils v3.7.2
  [0c5d862f] Symbolics v6.22.0 `~/work/Symbolics.jl/Symbolics.jl`
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated`

and using this machine and Julia version.

Julia Version 1.11.2
Commit 5e9a32e7af2 (2024-12-01 20:02 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
  LLVM: libLLVM-16.0.6 (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/Symbolics.jl/Symbolics.jl/docs/Manifest.toml`
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⌅ [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+1
  [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
  [4af54fe1] LazyArtifacts 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
  [1a1011a3] SharedArrays v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [f489334b] StyledStrings v1.11.0
  [4607b0f0] SuiteSparse
  [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 v1.1.1+0
  [781609d7] GMP_jll v6.3.0+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [3a97d323] MPFR_jll v4.2.1+0
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.1+2
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [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 ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`

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