On semi-separability and differentiation matrices
Arieh Iserles
TL;DR
This paper analyzes differentiation matrices arising from Jacobi-based W-functions in univariate spectral methods with zero boundary conditions on a compact interval. It derives an explicit Jacobi differentiation matrix and proves it is semi-separable of rank 2, with the square becoming semi-separable of rank 4, enabling fast O(N) linear algebra for discretized PDEs. A key contribution is a general result showing that the product of semi-separable matrices preserves a bounded rank, which has practical implications for forming operators like the Laplacian. The work also explores structural connections to the Askey scheme and discusses broader geometric and computational consequences for spectral-method frameworks.
Abstract
The theory of spectral methods for partial differential equations leads to infinite-dimensional matrices which represent the derivative operator with respect to an underlying orthonormal basis. Favourable properties of such differentiation matrices are crucial in the design of good spectral methods. It is known that bases using Laguerre and ultraspherical polynomials lead to semi-separable differentiation matrices of rank 1. In this paper we consider orthonormal bases constructed from Jacobi polynomials and prove that the underlying differentiation matrices are semi-separable of rank 2. This requires new results on semi-separable matrices which might be interesting in a wider context.