On well-conditioned spectral collocation and spectral methods by the integral reformulation

Well-conditioned spectral collocation and spectral methods have recently been proposed to solve differential equations. In this paper, we revisit the well-conditioned spectral collocation methods proposed in [T.~A. Driscoll, {\it J. Comput. Phys.}, 229 (2010), pp.~5980-5998] and [L.-L. Wang, M.~D. Samson, and X.~Zhao, {\it SIAM J. Sci. Comput.}, 36 (2014), pp.~A907--A929], and the ultraspherical spectral method proposed in [S.~Olver and A.~Townsend, {\it SIAM Rev.}, 55 (2013), pp.~462--489] for an $m$th-order ordinary differential equation from the viewpoint of the integral reformulation. Moreover, we propose a Chebyshev spectral method for the integral reformulation. The well-conditioning of these methods is obvious by noting that the resulting linear operator is a compact perturbation of the identity. The adaptive QR approach for the ultraspherical spectral method still applies to the almost-banded infinite-dimensional system arising in the Chebyshev spectral method for the integral reformulation. Numerical examples are given to confirm the well-conditioning of the Chebyshev spectral method.