Improved projector choices for collocation in Hammerstein-equation solvers

Identify and design improved choices of the projector P in the Kumar–Sloan collocation framework for solving the Hammerstein integral equations governing sparse random matrix spectra, in order to achieve stable and accurate performance at higher polynomial orders J and lower average degrees c, overcoming Runge-type instabilities and excessive precision demands of naive Taylor-based projectors.

Background

The numerical approach in the paper approximates the nonlinear term via expansions (Laguerre or half-range Hermite) and enforces a collocation projector to obtain a finite-dimensional nonlinear system. While effective at modest orders, performance degrades at low c or when increasing J, due in part to limitations of the interpolating projector and potential Runge phenomena.

The authors note that alternative projector choices could improve stability and accuracy but do not specify them; they explicitly leave this task for future investigation, suggesting that better projectors may avoid the need for very high precision required by naive Taylor expansions.