Journal of Computational and Applied Mathematics 278 (2015) 12–18
Contents lists available at ScienceDirect
Journal of Computational and Applied
Mathematics
journal homepage: www.elsevier.com/locate/cam
An explicit formula for the inverse of a pentadiagonal
Toeplitz matrix
Chaojie Wang, Hongyi Li, Di Zhao
∗
LMIB, School of Mathematics and System Science, Beihang University, Beijing, 100191, PR China
a r t i c l e i n f o
Article history:
Received 29 December 2013
Received in revised form 26 May 2014
Keywords:
Inverse
Pentadiagonal matrix
Toeplitz matrix
Factorization
Sherman–Morrison–Woodbury inversion
a b s t r a c t
In this paper, we mainly consider finding an explicit formula for the inverse of a pen-
tadiagonal Toeplitz matrix. For that purpose, we first factorize the modified form of a
pentadiagonal Toeplitz matrix by two tridiagonal Toeplitz matrices, and then use the
Sherman–Morrison–Woodbury inversion formula. As a result, an explicit inverse of a pen-
tadiagonal Toeplitz matrix is obtained under certain assumptions. And numerical experi-
ments are given to show the effectiveness of our results.
© 2014 Elsevier B.V. All rights reserved.
1. Introduction
The n-by-n pentadiagonal Toeplitz matrix takes the form
A =
x y v
z x y
.
.
.
w z x
.
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.
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v
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.
.
y
w z x
. (1.1)
Pentadiagonal Toeplitz matrices often occur when solving partial differential equations numerically using finite
difference method, finite element method and spectral method, and could be applied to the mathematical representation
of high dimensional, nonlinear electromagnetic interference signals [1–7]. Pentadiagonal matrices are a certain class of
special matrices, and other common types of special matrices are Jordan, Frobenius, generalized Vandermonde, Hermite,
centrosymmetric, and arrowhead matrices [8–12]. Usually, after the original partial differential equations are processed
with these numerical methods, we need to solve the pentadiagonal Toeplitz systems of linear equations for obtaining the
numerical solutions of the original partial differential equations. Thus, the inversion of pentadiagonal Toeplitz matrices is
necessary to solve the partial differential equations in these cases.
Recently, J. Jia, T. Sogabe and M. El-Mikkawy have proposed an explicit inverse of the tridiagonal Toeplitz matrix which is
strictly row diagonally dominant [13]. However, this result cannot be generalized to the case of the pentadiagonal Toeplitz
matrix directly. This motivates us to investigate whether an explicit inverse of the pentadiagonal Toeplitz matrix also exists.
∗
Corresponding author.
E-mail addresses: zdbuaa@163.com, hylbuaa@163.com (D. Zhao).
http://dx.doi.org/10.1016/j.cam.2014.08.010
0377-0427/© 2014 Elsevier B.V. All rights reserved.
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