Barycentric Rational Interpolation and Spectral Methods

For the numerical solution of differential equations spectral methods typically give excellent accuracy with relatively few points (small N), but certain numerical issues arise with larger N. This thesis focuses on spectral collocation methods, also known as pseudo-spectral methods, that rely on interpolation at collocation points. A relatively new class of interpolants will be considered, namely the Floater-Hormann family of rational interpolants. These interpolants and their properties will be studied, including their use in differentiation by means of differentiation matrices based on rational interpolants in the barycentric form. Then, consideration will be given to the solution of singularly perturbed boundary value problems through the use of boundary layer resolving coordinate transformations. Finally, coupled systems of singularly perturbed boundary value problems will be investigated, though only with the standard Chebyshev collocation method.