Accurate Evaluation of Ellipsoidal Harmonics Using Tanh-Sinh Quadrature

Abstract

Ellipsoidal coordinates are an orthogonal coordinate system under which the Laplace equation can be solved by separation of variables. While this has many benefits over spherical coordinates for a variety of potential problems, computation in ellipsoidal coordinates is difficult. Most notably, high-order harmonics can lack closed-form solutions and the associated normalization constants require approximating a singular integral. We provide a method for computing normalization constants to machine precision using tanh-sinh quadrature which exhibits exponential convergence for a large class of functions with singular endpoints. Combined with previous efforts to make ellipsoidal harmonics more accessible, the result is a library which makes computation in ellipsoidal coordinates as accessible as computation in spherical coordinates. Finally, we apply our implementation to the mixed-dielectric solvation problem and provide work-precision analysis for the results.