Functions of Difference Matrices Are Toeplitz Plus Hankel
Author(s)
MacNamara, Shevarl; Strang, Gilbert
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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.
Date issued
2014-08Department
Massachusetts Institute of Technology. Department of MathematicsJournal
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
ISSN
0036-1445
1095-7200