Overview

PolyChaos is a collection of numerical routines for orthogonal polynomials, written in the Julia programming language. Starting from some non-negative weight (aka an absolutely continuous non-negative measure), PolyChaos allows

to compute the coefficients for the monic three-term recurrence relation,
to evaluate the orthogonal polynomials at arbitrary points,
to compute the quadrature rule,
to compute tensors of scalar products,
to do all of the above in a multivariate setting (aka product measures).

If the weight function is a probability density function, PolyChaos further provides routines to compute polynomial chaos expansions (PCEs) of random variables with this very density function. These routines allow

to compute affine PCE coefficients for arbitrary densities,
to compute moments,
to compute the tensors of scalar products.

PolyChaos contains several canonical orthogonal polynomials, such as Jacobi or Hermite polynomials. For these, closed-form expressions and state-of-the art quadrature rules are used whenever possible. However, a cornerstone principle of PolyChaos is to provide all the functionality for user-specific, arbitrary weights.

Note

What PolyChaos is not (at least currently):

a self-contained introduction to orthogonal polynomials, quadrature rules and polynomial chaos expansions. We assume the user brings some experience to the table. However, over time we will focus on strengthening the tutorial character of the package.
a symbolic toolbox
a replacement for FastGaussQuadrature.jl

Installation

The package requires Julia 1.3 or newer. To install PolyChaos.jl, use the Julia package manager:

using Pkg
Pkg.add("PolyChaos")

This will install PolyChaos and its dependencies. Once that is done, load the package:

using PolyChaos

That's it.

Let's take a look at a simple example. We would like to solve the integral

\[\int_0^1 6 x^5 \mathrm{d}x.\]

Exploiting the underlying uniform measure, the integration can be done exactly with a 3-point quadrature rule.

julia> using PolyChaos

julia> opq = Uniform01OrthoPoly(3, addQuadrature = true);

julia> isapprox(integrate(x -> 6x^5, opq), 1.0)
true

julia> size(coeffs(opq))
(4, 2)

To get going with PolyChaos check out the tutorials such as the one on numerical integration. In case you are unfamiliar with orthogonal polynomials, perhaps this background information is of help.

References

The code base of PolyChaos is partially based on Walter Gautschi's OPQ Matlab suite of programs for generating orthogonal polynomials and related quadrature rules, with much of the theory presented in his book Orthogonal Polynomials: Computation and Approximation published in 2004 by the Oxford University Press.

For the theory of polynomial chaos expansion, we mainly consulted T. J. Sullivan. Introduction to Uncertainty Quantification. Springer International Publishing Switzerland. 2015.

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

Citing

If you found the software useful and applied it to your own research, we'd appreciate a citation. Add the following to your BibTeX file

@ARTICLE{2020arXiv200403970M,
       author = {{M{\"u}hlpfordt}, Tillmann and {Zahn}, Frederik and {Hagenmeyer}, Veit and
         {Faulwasser}, Timm},
        title = "{PolyChaos.jl -- A Julia Package for Polynomial Chaos in Systems and Control}",
      journal = {arXiv e-prints},
     keywords = {Electrical Engineering and Systems Science - Systems and Control, Mathematics - Numerical Analysis, Mathematics - Optimization and Control},
         year = 2020,
        month = apr,
          eid = {arXiv:2004.03970},
        pages = {arXiv:2004.03970},
archivePrefix = {arXiv},
       eprint = {2004.03970},
}

Of course, you are more than welcome to partake in GitHub's gamification: starring and forking is much appreciated.

Enjoy.

Reproducibility

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

Status `~/work/PolyChaos.jl/PolyChaos.jl/docs/Project.toml`
  [1520ce14] AbstractTrees v0.4.5
  [0c46a032] DifferentialEquations v8.1.1
  [e30172f5] Documenter v1.19.0
  [28b8d3ca] GR v0.73.27
  [b964fa9f] LaTeXStrings v1.4.1
  [91a5bcdd] Plots v1.41.7
  [8d666b04] PolyChaos v2.1.2 `~/work/PolyChaos.jl/PolyChaos.jl`
  [1fd47b50] QuadGK v2.11.3
  [276daf66] SpecialFunctions v2.9.0
  [37e2e46d] LinearAlgebra v1.13.0

and using this machine and Julia version.

Julia Version 1.13.0
Commit d1c37793dd2 (2026-09-09 19:00 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × Intel(R) Xeon(R) 6973P-C
  WORD_SIZE: 64
  LLVM: libLLVM-20.1.8 (ORCJIT, graniterapids)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 4 virtual cores)

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

Status `~/work/PolyChaos.jl/PolyChaos.jl/docs/Manifest.toml`
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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.