Effective Quadratures

Effective Quadratures is an open-source library for uncertainty quantification, machine learning, optimisation, numerical integration and dimension reduction -- all using orthogonal polynomials. It is particularly useful for models / problems where output quantities of interest are smooth and continuous; to this extent it has found widespread applications in computational engineering models (finite elements, computational fluid dynamics, etc). It is built on the latest research within these areas and has both deterministic and randomized algorithms. Effective Quadratures is actively being developed by researchers at the University of Cambridge, Imperial College London, Stanford University, The University of Utah, The Alan Turing Institute and the University of Cagliari. Effective Quadratures is a NumFOCUS affiliated project.

Key words associated with this code: polynomial surrogates, polynomial chaos, polynomial variable projection, Gaussian quadrature, Clenshaw Curtis, polynomial least squares, compressed sensing, gradient-enhanced surrogates, supervised learning.


The latest version of the code is version 9.0 and was released in August 2020.

To download and install the code please use the python package index command:

pip install equadratures

or if you are using python3, then

pip3 install equadratures

Alternatively you can click either on the Fork Code button or Clone. For issues with the code, please do raise an issue on our Github page; do make sure to add the relevant bits of code and specifics on package version numbers. We welcome contributions and suggestions from both users and folks interested in developing the code further.

Our code is designed to require minimal dependencies; current package requirements include numpy, scipy and matplotlib.

Documentation, tutorials and the blog

Code documentation and details on the syntax can be found here.

Code tutorials can be found here.

We've recently started an EQ-Blog! Check it out here.

Code objectives

Specific goals of this code include:

  • probability distributions and orthogonal polynomials
  • supervised machine learning: regression and compressive sensing
  • numerical quadrature and high-dimensional sampling
  • transforms for correlated parameters
  • computing moments from models and data-sets
  • sensitivity analysis and Sobol' indices
  • data-driven dimension reduction
  • ridge approximations and neural networks
  • surrogate-based design optimisation

Papers (theory and applications)

  • Wong, C. Y., Seshadri, P., Parks, G. T., (2019) Extremum Global Sensitivity Analysis with Least Squares Polynomials and their Ridges. Preprint.

  • Wong, C. Y., Seshadri, P., Parks, G. T., Girolami, M., (2019) Embedded Ridge Approximations: Constructing Ridge Approximations Over Localized Scalar Fields For Improved Simulation-Centric Dimension Reduction. Preprint.

  • Seshadri, P., Iaccarino, G., Ghisu, T., (2019) Quadrature Strategies for Constructing Polynomial Approximations. Uncertainty Modeling for Engineering Applications. Springer, Cham, 2019. 1-25. Paper. Preprint.

  • Seshadri, P., Narayan, A., Sankaran M., (2017) Effectively Subsampled Quadratures for Least Squares Polynomial Approximations." SIAM/ASA Journal on Uncertainty Quantification 5.1 : 1003-1023. Paper.

  • Seshadri, P., Parks, G. T., (2017) Effective-Quadratures (EQ): Polynomials for Computational Engineering Studies, Journal of Open Source Software, 2(11), 166, Paper.

  • Kalra, T. S., Aretxabaleta, A., Seshadri, P., Ganju, N. K., Beudin, A. (2017). Sensitivity Analysis of a Coupled Hydrodynamic-Vegetation Model Using the Effectively Subsampled Quadratures Method (ESQM v5. 2). Geoscientific Model Development, 10(12), 4511. Paper.

Get in touch

Feel free to follow us via Twitter or email us at

Community guidelines

If you have contributions, questions, or feedback use either the Github repository, or get in touch. We welcome contributions to our code. In this respect, we follow the NumFOCUS code of conduct.


This work was supported by wave 1 of The UKRI Strategic Priorities Fund under the EPSRC grant EP/T001569/1, particularly the Digital Twins in Aeronautics theme within that grant, and The Alan Turing Institute.