When integration sparsification fails: Banded Galerkin discretizations for Hermite functions, rational Chebyshev functions and sinh-mapped Fourier functions on an infinite domain, and Chebyshev methods for solutions with C ^∞ endpoint singularities

Chebyshev polynomial spectral methods are very accurate, but are plagued by the cost and ill-conditioning of dense discretization matrices. Modified schemes, collectively known as “integration sparsification” have mollified these problems by discretizing the highest derivative as a diagonal matrix. Here, we examine five case studies where the highest derivative diagonalization fails. Nevertheless, we show that Galerkin discretizations do yield banded matrices that retain most of the advantages of “integration sparsification”. Symbolic computer algebra greatly extends the reach of spectral methods. When spectral methods are implemented using exact rational arithmetic, as is possible for small truncation N in Maple, Mathematica and their ilk, roundoff error is irrelevant, and sparsification failure is not worrisome. When the discretization contains a parameter L, symbolic algebra spectral methods return, as answer to an eigenproblem, not discrete numbers but rather a plane algebraic curve defined as the zero set of a bivariate polynomial P(λ,L); the optimal approximations to the eigenvalues λ _j are in the middle of the straight portions of the zero contours of P(λ;L) where the isolines are parallel to the L axis.