Spectral differentiation: Integration and inversion

Pseudospectral differentiation matrices suffer from large round-off error, and give rise to illconditioned systems used to solve differential equations numerically. This thesis presents two types of matrices designed to precondition these systems and improve robustness towards this round-off error for spectral methods on Chebyshev-Gauss-Lobatto points. The first of these is a generalization of a pseudospectral integration matrix described by Wang et al. [18]. The second uses this integration matrix to construct the matrix representing the inverse operator of the differential equation. Comparison is made between expected and calculated eigenvalues. Both preconditioners are tested on several examples. In many cases, accuracy is improved over the standard methodology by several orders of magnitude. Using these matrices on general sets of points is briefly discussed.