Functions of Difference Matrices Are Toeplitz Plus Hankel
Name
Strang-2014-Functions of differe.pdf
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479.64 KB
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Author(s)
MacNamara, Shevarl
Strang, Gilbert
Date Issued
August 2014
Journal
SIAM Review
Publisher
Society for Industrial and Applied Mathematics
Citation
Strang, Gilbert, and Shev MacNamara. “Functions of Difference Matrices Are Toeplitz Plus Hankel.” SIAM Review 56, no. 3 (January 2014): 525–546. © 2014, Society for Industrial and Applied Mathematics
Version
Final published version
Abstract
When the heat equation and wave equation are approximated by $\bm{u}_t = -\bm{K} \bm{u}$ and $\bm{u}_{tt} = -\bm{K} \bm{u}$ (discrete in space), the solution operators involve $e^{-\bm{K}t}$, $\sqrt{\bm{K}}$, $\cos(\sqrt{\bm{K}}t)$, and $\mathrm{sinc}(\sqrt{\bm{K}}t)$. We compute these four matrices and find accurate approximations with a variety of boundary conditions. The second difference matrix $\bm{K}$ is Toeplitz (shift-invariant) for Dirichlet boundary conditions, but we show why $e^{\bm{-Kt}}$ also has a Hankel (anti-shift-invariant) part. Any symmetric choice of the four corner entries of $\bm{K}$ leads to Toeplitz plus Hankel in all functions $f(\bm{K})$. Overall, this article is based on diagonalizing symmetric matrices, replacing sums by integrals, and computing Fourier coefficients.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/120897572
