AnexplicitformulafortheinverseofapentadiagonalToeplitzmatrix资源-CSDN文库

69 浏览量 2021-02-22 05:45:30 上传评论收藏 360KB PDF 举报

资源推荐

资源详情

资源评论

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.

1. Introduction

The n-by-n pentadiagonal Toeplitz matrix takes the form

A =







x y v

z x 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

C. Wang et al. / Journal of Computational and Applied Mathematics 278 (2015) 12–18 13

Based on the results presented in [13], we derive an explicit formula for the inverse of the pentadiagonal Toeplitz ma-

trix under some assumptions by using the Sherman–Morrison–Woodbury inversion formula and the factorization of the

modified pentadiagonal Toeplitz matrices [14].

The remainder of this paper is organized as follows: in Section 2, we perform a comparison with other methods. In

Section 3, a method for obtaining the inverse of a pentadiagonal Toeplitz matrix is given. In Section 4, a numerical example

is provided to show the effectiveness of our results. Finally, we present some concluding remarks in Section 5.

2. Comparison with other methods

In recent years, many researchers have studied the inversion of the pentadiagonal Toeplitz matrix or related matrices:

• In [15,16], the authors proposed two explicit formulas for the inverse of general Toeplitz matrices, respectively. These

two methods could be directly applied to inverting the pentadiagonal Toeplitz matrix, but they could not make full use

of the sparse structure of the pentadiagonal Toeplitz matrix.

• [17] gave a method to obtain the inverse of a general pentadiagonal matrix which was not specific to the pentadiagonal

Toeplitz matrix and required general Doolittle factorization.

• [18] presented a numerical method for computing the inverse of the pentadiagonal Toeplitz matrix that involves solving

a system of linear equations.

In summary, all the above methods are not very efficient for the inversion of the pentadiagonal Toeplitz matrix. This

motivates us to find a better method for computing the inverse of the pentadiagonal Toeplitz matrix.

3. Main results

In this section, we derive an explicit formula for computing the inverse of a pentadiagonal Toeplitz matrix.

3.1. The factorization of the modified pentadiagonal Toeplitz matrix by tridiagonal Toeplitz matrices

It is easy to prove that the product of any two tridiagonal Toeplitz matrices is not a pentadiagonal Toeplitz matrix because

the first and last elements in the principle diagonal are different from the other ones [19,20]. Thus, it is necessary to consider

the factorization of a pentadiagonal Toeplitz matrix modified in the first and the last element of the principle diagonal.

Let the modified form of A be an n-by-n pentadiagonal matrix as the following







x − α y v

z x y

w z x

w z x − β







. (3.1)

The real tridiagonal Toeplitz matrix of size n is given by

T =







b c

a b







. (3.2)

Denote it by

T = trid

[a, b, c].

Factorize A

by two tridiagonal Toeplitz matrices as

= T

(3.3)

where

= trid

, b

, c

], T

= trid

, b

, c

]. (3.4)

剩余6页未读，继续阅读

评论收藏

内容反馈

weixin_38629873

粉丝: 2
资源: 967

An explicit formula for the inverse of a pentadiagonal Toeplitz ...

最新资源

An explicit formula for the inverse of a pentadiagonal Toeplitz ...

A hybrid expansion method for FRFs of non-proportionally damped systems

GEE错误集-Cannot add an object of type ＜Element＞ to the map. Might be fixable with an explicit .pdf

Study on the approximative formula for the far-field of a Gaussian beam under circular aperture diffraction and its divergence

a project model for the FreeBSD Project.7z

MySQL 5.6 中的 TIMESTAMP 和 explicit_defaults_for_timestamp 参数

A Mathematical Introduction to Robotic Manipulation.pdf

The Craft of System Security

Programming Environments Manual for 32-Bit Implementations of the PowerPC™ Architecture

Matrix Diﬀerential Calculus with Applications in Statistics and Econometrics

Denoising based new Noise model

Extension of The Implicit Curve-Fitting Method for Fast Calculation of Thermodynamic Properties to Subcooled Refrigerant

Make3D: Learning 3D Scene Structure from a Single Still Image

Explicit estimate for convergence rates of continuous time markov chains (I)

Methods of Information Geometry

The Reasoned Schemer, 2nd Edition

Hollo-Bolt_3D_IGS_1074.zip_GTN VUMAT_donepj1_ds18b20_needleman_o

euler-explicit.f.zip_code_euler explicit_explicit_explicit fortr

Model Predictive Control：THEORY AND PRACTICE

An End-to-End Network for Generating Social Relationship Graphs.pdf

Programming Guide for Linux USB Device.pdf

在 Fortran ( Abaqus Explicit VUMAT) 中实现的复合材料 3D 连续介质损伤力学模型

Timing Optimization Through Clock Skew Scheduling

Scala for the Impatient.pdf

Universal-USB-Installer

WizFlow网页编辑

最新资源