A fast and well-conditioned spectral method

Abstract

A novel spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients. The method leads to matrices which are almost banded, and a numerical solver is presented that takes $O(m^{2}n)$ operations, where $m$ is the number of Chebyshev points needed to resolve the coefficients of the differential operator and $n$ is the number of Chebyshev points needed to resolve the solution to the differential equation. We prove stability of the method by relating it to a diagonally preconditioned system which has a bounded condition number, in a suitable norm. For Dirichlet boundary conditions, this reduces to stability in the standard 2-norm