Solving the signed equilibrium problem on several real intervals

Given the union $I$ of real disjoint intervals and a smooth external field $Q$ on $I$, the signed equilibrium problem consist of finding a signed measure $\mu$ of mass $1$ supported on $I$ such that $U^\mu+Q$ is equal to a constant on $I$. Taking derivatives, we may reformulate this problem as an integral equation with a Cauchy kernel, and can solve the problem via a block polynomial spectral method via Chebyshev polynomials. However, small gaps between consecutive subintervals deteriorate the rate of convergence. Instead, we suggest a new spectral method based on Chebyshev rational orthogonal functions, leading to much smaller systems of linear equations. We present several numerical examples, and combine this technique with the iterated balayage algorithm of Dragnev, in order to solve also positive equilibrium problems.