Integration Preconditioning Matrix for Ultraspherical Pseudospectral Operators

In this paper, we give an integration rectangular preconditioner for ultraspherical pseudospectral operators. We present an explicit solution of matrix equations with the structure provided by the pseudospectral discretization of the nth-order differential operator, subject to various types of boundary conditions. This occurs in two steps. In the first step we introduce an integration rectangular preconditioner. In the second step we use a new strategy to impose the boundary conditions, on the condition that the coefficients matrix has a known inverse. Computational examples are given.