# Condition Numbers for the Sinc Matrices Associated with Discretizing the Second-Order Differential Operator

• Kelly M. McArthur
Part of the Progress in Systems and Control Theory book series (PSCT, volume 15)

## Abstract

The Sinc-Galerkin discretizations of the model equations
$$\begin{gathered} \mathcal{L}u(x)\bar = - u(x) = f(x),0 < x < 1 \hfill \\ u(0) = u(1) = 0 \hfill \\ \end{gathered}$$
(1.1)
and
$$\begin{gathered} \mathcal{L}u(x,y)\bar = - (\frac{{\partial ^2 }} {{\partial x^2 }} + \frac{{\partial ^2 }} {{\partial y^2 }})u(x,y) = f(x,y),(x,y) \in (0,1)^2 \hfill \\ u|\partial (0,2)^2 = 0 \hfill \\ \end{gathered}$$
(1.2)
are briefly outlined below in Sections 2 and 3, respectively. Detailed discussions of the discretization of more general, linear, second-order spatial operators are found in [9, 10, 5, 8]. The emphasis here is presenting results regarding condition numbers for the linear systems affiliated with (1.1) and (1.2). These condition numbers vary widely depending on how the linear systems are posed. Adroit changes of variables can lead to well-conditioned linear systems while poor choices can magnify existing problems. The importance of these choices is especially evident when developing a Sinc-Galerkin ADI scheme [7]. It should be pointed out that conditioning problems are not unique to this spectral scheme. Canuto [2] and Deville and Mund [3] have implemented finite element preconditioners for their spectral methods applied to (1.2). The changes of variables represented below may be thought of as preconditioners.

Dition Kelly

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