On the inverse of a general pentadiagonal matrix

doi:10.1016/j.amc.2008.03.004

Applied Mathematics and Computation

Volume 202, Issue 2, 15 August 2008, Pages 639-646

https://doi.org/10.1016/j.amc.2008.03.004 Get rights and content

Abstract

In this paper, employing the general Doolittle factorization, an efficient algorithm is developed to find the inverse of a general pentadiagonal matrix which is suitable for implementation using computer algebra systems software such as Matlab and Maple. Examples are given to illustrate the efficiency of the algorithm.

Introduction

A matrix of the form $A = (\begin{matrix} a_{1} & b_{1} & c_{1} \\ d_{2} & a_{2} & b_{2} & c_{2} \\ e_{3} & d_{3} & a_{3} & b_{3} & c_{3} \\ ⋱ & ⋱ & ⋱ & ⋱ & ⋱ \\ ⋱ & ⋱ & ⋱ & ⋱ & c_{n - 2} \\ e_{n - 1} & d_{n - 1} & a_{n - 1} & b_{n - 1} \\ e_{n} & d_{n} & a_{n} \end{matrix})$ is called pentadiagonal matrix. Thus $A = (a_{ij})_{1 ⩽ i, j ⩽ n}$ is pentadiagonal if $a_{ij} = 0$ for $| i - j | > 2$ . These types of matrices are widely used in areas of science and engineering, for example in numerical solution of ordinary and partial differential equations (ODE and PDE), interpolation problems, boundary value problems (BVP), etc. In many of these areas inversions of pentadiagonal matrices are necessary. So an efficient computational approach to find the inverse of the pentadiagonal matrix A in (1) is demanding. Generally, Gauss–Jordan method with partial pivoting [1] is often used. However, the method may destroy the special structure and sparsity (especially for a large scale matrix) of pentadiagonal matrix. So employing the special structure of pentadiagonal matrices, the inversions of pentadiagonal matrices could be more efficient. Shiau [2], Diele and Lopez [3], respectively, employing the special structure of pentadiagonal Toeplitz matrix, presented algorithms to invert pentadiagonal Toeplitz matrix. Recently, a new algorithm to find the inverse of a general tridiagonal matrix is presented by El-Mikkawy [4], which is extended in [5], [6] to periodic tridiagonal linear systems and comrade linear systems, respectively. This motivates the present study. Based on El-Mikkawy [4], employing general Doolittle factorization and the special structure of a pentadiagonal matrix, this paper is to develop a new algorithm to find the inverse of a general pentadiagonal matrix, which is suitable for implementation using computer algebra systems software such as Matlab and Maple. Similarly, the algorithm presented can also be extended to periodic pentadiagonal linear systems.

This paper is organized as follows: In Section 2, main results and relevant algorithms are presented. In Section 3, numerical and symbolic examples are given.

Section snippets

Main results

To give main results concerning pentadiagonal matrices of the form (1), we define the quantities relevant of A as follows: $x_{i} = \{\begin{matrix} a_{1}, & i = 1, \\ a_{2} - y_{1} z_{2}, & i = 2, \\ a_{i} - y_{i - 1} z_{i} - \frac{e_{i}}{x_{i - 2}} c_{i - 2}, & i = 3, \dots, n, \end{matrix}$ $y_{i} = \{\begin{matrix} b_{1}, & i = 1, \\ b_{i} - z_{i} c_{i - 1}, & i = 2, \dots, n - 1, \end{matrix}$ $z_{i} = \{\begin{matrix} \frac{d_{2}}{x_{1}}, & i = 2, \\ \frac{d_{i} - \frac{e_{i}}{x_{i - 2}} y_{i - 2}}{x_{i - 1}}, & i = 3, \dots, n . \end{matrix}$

Theorem 2.1

If $x_{i} \neq 0$ , $i = 1, \dots, n - 1$ , the $LU$ factorization of the matrix A in (1) exists, that is $A = L_{1} U_{1}$ , where $L_{1} = (\begin{matrix} 1 & 0 & 0 & \dots & \dots & 0 \\ z_{2} & 1 & 0 & ⋱ & ⋮ \\ \frac{e_{3}}{x_{1}} & z_{3} & 1 & ⋱ & ⋱ & ⋮ \\ 0 & \frac{e_{4}}{x_{2}} & ⋱ & ⋱ & ⋱ & 0 \\ ⋮ & ⋱ & ⋱ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & \frac{e_{n}}{x_{n - 2}} & z_{n} & 1 \end{matrix}),$ $U_{1} = (\begin{matrix} x_{1} & y_{1} & c_{1} & \dots & \dots & 0 \\ 0 & x_{2} & y_{2} & ⋱ & ⋮ \\ 0 & 0 & x_{3} & ⋱ & ⋱ & ⋮ \\ 0 & 0 & ⋱ & ⋱ & ⋱ & c_{n - 2} \\ ⋮ & ⋱ & ⋱ & ⋱ & ⋱ & y_{n - 1} \\ 0 & \dots & 0 & 0 & 0 & x_{n} \end{matrix}) .$

Proof

By using the Doolittle factorization, it is easy to be

Examples

In this section, examples are given to illustrate the above results.

Example 3.1

Consider the pentadiagonal matrix $A = (\begin{matrix} 1 & 1 & 3 & 0 & 0 & 0 \\ 1 & 1 & 2 & 2 & 0 & 0 \\ 2 & 6 & 3 & 1 & 5 & 0 \\ 0 & 2 & 4 & 2 & 3 & 1 \\ 0 & 0 & 1 & 3 & 1 & 4 \\ 0 & 0 & 0 & 3 & 5 & \frac{1}{2} \end{matrix}) .$

Applying Algorithm 2.2, it gives $P (t) = - 139 t - 868$ , $\det (A) = - 868$ , $A^{- 1} = (\begin{matrix} 0.7880 & 0.0058 & 0.1031 & - 0.7062 & 0.1400 & 0.2926 \\ - 0.4931 & 0.1601 & 0.1665 & 0.1671 & - 0.0086 & - 0.2650 \\ 0.2350 & - 0.0553 & - 0.0899 & 0.1797 & - 0.0438 & - 0.0092 \\ - 0.3825 & 0.4724 & - 0.0449 & 0.0899 & - 0.0219 & - 0.0046 \\ 0.2120 & - 0.2558 & 0.0219 & - 0.0438 & - 0.0150 & 0.2074 \\ 0.1751 & - 0.2765 & 0.0507 & - 0.1014 & 0.2811 & - 0.0461 \end{matrix}) .$

Example 3.2

Consider $B = (\begin{matrix} 2 & 1 & 1 & 0 & 0 \\ 2 & 2 & 2 & - 1 & 0 \\ 1 & 2 & 1 & 1 & α \\ 0 & 1 & 0 & 2 & α \\ 0 & 0 & - 1 & β + 2 & 3 + α \end{matrix}) .$

Applying

Acknowledgement

The authors would like to express their great gratitude to the referees and editor Dr. Melvin Scott for their constructive comments and suggestions that lead to enhancement of this paper.

References (9)

J.J.H. Shiau
A new algorithm for 5-band Toeplitz matrix inversion with application to GCV smoothing spline computation
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F. Diele et al.
The use of the factorization of five-diagonal matrices by tridiagonal Toeplitz matrices
Appl. Math. Lett.
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M.E.A. El-Mikkawy
On the inverse of a general tridiagonal matrix
Appl. Math. Comput.
(2004)
M.E.A. El-Mikkawy
A new computational algorithm for solving periodic tridiagonal linear systems
Appl. Math. Comput.
(2005)

There are more references available in the full text version of this article.

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^☆: This research was supported by 973 Programs (2008CB317110), NSFC (10771030), the Scientific and Technological Key Project of the Chinese Ministry of Education (107098), the PhD. Programs Fund of Chinese Universities (20070614001), Sichuan Province Project for Applied Basic Research (2008JY0052) and the Project for Academic Leader and Group of UESTC.

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On the inverse of a general pentadiagonal matrix☆