Gauss-Turán Quadratures about integrals.jl HOT 8 OPEN

∑wᵢ*∑(aᵢₖ∂ᵏf(xᵢ))

Note that at a high level ∑wᵢ*∑(aᵢₖ∂ᵏf(xᵢ)) = ∑wᵢ* g(xᵢ) so this can be implemented as a meta-algorithm, e.g. GaussTuránRule(sensealg, n, k) == Integrals.ChangeOfVariables((f, domain) -> (gauss_turán_integrand(f, n, k, sensealg), domain), Integrals.QuadratureRule(gauss_turán_weights_nodes, n=n)) where some work would have to be done to implement gauss_turán_weights_nodes (which could simply tabulate wᵢ, xᵢ from the paper) and gauss_turán_integrand which would have to compute gradients of f with an AD backend selected from sensealg (e.g. ForwardDiff.jl) and values of aᵢₖ (which again could be tabulated). Here, Integrals.QuadratureRule is documented and in the public API and Integrals.ChangeOfVariables is internal and recently merged to the main branch, and not actually necessary for this case but a reference as another meta-algorithm.

This would be an interesting contribution for anyone interested in:

• automating the calculation of higher-order derivatives of integrands
• tabulating known Gauss-Turán rules
• optionally implementing a numerical method for the calculation of Gauss-Turán rules

from integrals.jl.