This package enables globally h-adaptive quadrature for smooth functions and functions with endpoint singularities. The usage is similar to QuadGK.jl, however, instead of passing an interval with breakpoints, the caller will pass multiple function spaces, each with an interval and the kinds of functions to expect on that interval.

For example, the behavior of QuadGK.jl can be reproduced as follows

using FunQuad
funquadh(x -> sin(10x), FunSpace(0, 1))

The default functions for a FunSpace are GaussKronrod, which are polynomials orthogonal w.r.t. the Legendre weight function. To specify other kinds of functions, pass a function kind to the first argument of FunSpace. For example, integrating logarithmic singularities as follows

funquadh(x -> log(x) *sin(8x) + x^3, FunSpace(LogAtMin(), 0, 1))

The caller should add breakpoints to the initial domain so that there is at most one kind of singularity in any interval. For example (not tested)

funquadh(x -> log(x) *sin(8x) + x^3*log(1-x), FunSpace(LogAtMin(), 0, 0.5), FunSpace(LogAtMax(), 0.5, 1))

This package will attempt to evaluate the integrand in the interior of the interval, not at the endpoints. If available, it will use Kronrod rules to compute embedded error estimates (otherwise refines the panel). The singularity should be included in the integrand, not separated into a integration measure. The quadrature rules attempt to integrate functions of the form smooth plus singular times smooth to attain faster convergence for a wide class of functions.

Currently, the following function spaces are available:

• GaussKronrod(): integrates smooth functions with embedded error estimates. (Default function space)
• GaussLegendre(): integrates smooth functions with refined error estimates.
• LogAtMin(): integrates logarithmic singularities at left endpoints. Refined error estimates
• LogAtMax(): integrates logarithmic singularities at right endpoints. Refined error estimates

TODO: add a FunSpace for changes of variables, such as FunSpace(Transformed(GaussKronrod, x -> sqrt(x), y -> y^2, y -> 2y), 0, 1)

TODO: add power laws with GeneralizedChebyshevQuadrature.jl

TODO: add Jacobi weight functions with QuadJGK.jl