A Fast and Well-Conditioned Spectral Method

Abstract

A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients and general boundary conditions. The method leads to matrices that are almost banded, and a numerical solver is presented that takes ${\cal O}(m^2n)$ 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 coefficients needed to resolve the solution to the differential equation. We prove stability of the method by relating it to a diagonally preconditioned system that has a bounded condition number, in a suitable norm. For Dirichlet boundary conditions, this implies stability in the standard $2$-norm. An adaptive QR factorization is developed to efficiently solve the resulting linear system and automatically choose the optimal number of Chebyshev coefficients needed to represent the solution. The resulting algorithm can efficiently and reliably solve for solutions that require as many as a million unknowns.