PolyChaos

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 nonnegative 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.

Take a look at the documentation.