Computable theoretical error bounds for Nyström methods for 1-D Fredholm integral equations of the second kind

New expressions for computable error bounds are derived for Nyström method approximate solutions of one-dimensional second-kind Fredholm integral equations. The bounds are computed using only the numerical solution, and so require no a priori knowledge of the exact solution. The analysis is implemented on test problems with both well-behaved and “Runge-phenomenon” solutions, and the computed predictions are shown to be in impressive quantitative agreement with the true errors obtained from known exact solutions of the test problems. For independent computational validation, both Lagrange and barycentric interpolation are employed on grids with both regularly spaced nodes and those located at the roots or extrema of orthogonal polynomials. For independent theoretical validation, asymptotic estimates are derived for the convergence rates of the observed computational errors.