We introduce a method to numerically compute equilibrium measures for problems with attractive-repulsive power law kernels of the form
Kpx ´ yq “ |x´y|
α
α ´
|x´y|
β
β
using recursively generated banded and approximately banded operators acting on expansions in ultraspherical polynomial
bases. The proposed method reduces what is na¨ıvely a difficult to approach
optimization problem over a measure space to a straightforward optimization problem over one or two variables fixing the support of the equilibrium
measure. The structure and rapid convergence properties of the obtained operators results in high computational efficiency in the individual optimization
steps. We discuss stability and convergence of the method under a Tikhonov
regularization and use an implementation to showcase comparisons with analytically known solutions as well as discrete particle simulations. Finally, we
numerically explore open questions with respect to existence and uniqueness
of equilibrium measures as well as gap forming behaviour in parameter ranges
of interest for power law kernels, where the support of the equilibrium measure
splits into two intervals.