Functions of Difference Matrices Are Toeplitz Plus Hankel

被引：34

作者：

Strang, Gilbert ^{[1
]}

MacNamara, Shev ^{[1
]}

机构：

[1] MIT, Dept Math, Cambridge, MA 02139 USA

来源：

SIAM REVIEW | 2014年 / 56卷 / 03期

基金：

美国国家科学基金会;

关键词：

Toeplitz; Hankel; Laplacian; exponential; Bessel function;

D O I：

10.1137/120897572

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

When the heat equation and wave equation are approximated by u(t) = -Ku and u(tt) = -Ku (discrete in space), the solution operators involve e(-Kt), root K, cos(root Kt), and sinc(root Kt). We compute these four matrices and find accurate approximations with a variety of boundary conditions. The second difference matrix K is Toeplitz (shift-invariant) for Dirichlet boundary conditions, but we show why e(-Kt) also has a Hankel (anti-shift-invariant) part. Any symmetric choice of the four corner entries of K leads to Toeplitz plus Hankel in all functions f(K). Overall, this article is based on diagonalizing symmetric matrices, replacing sums by integrals, and computing Fourier coefficients.

引用

页码：525 / 546

页数：22

共 27 条

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