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 operations, where is the number of Chebyshev points needed to resolve the coefficients of the differential operator and 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.