Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 321032, 9 pages

http://dx.doi.org/10.1155/2013/321032

Research Article

Inversion of General Cyclic Heptadiagonal Matrices

A. A. Karawia

Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 23 December 2012; Revised 26 February 2013; Accepted 27 February 2013

Academic Editor: Joao B. R. Do Val

Copyright © 2013 A. A. Karawia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We describe a reliable symbolic computational algorithm for inverting general cyclic heptadiagonal matrices by using parallel computing along with recursion. The computational cost of it is operations. The algorithm is implementable to the Computer Algebra System (CAS) such as MAPLE, MATLAB, and MATHEMATICA. Two examples are presented for the sake of illustration.

1. Introduction

The general cyclic heptadiagonal matrices take the form:

where .

The inverses of cyclic heptadiagonal matrices are usually required in science and engineering applications, for more details, see special cases, [1–9]. The motivation of the current paper is to establish efficient algorithms for inverting cyclic heptadiagonal matrices of the form (1) and for solving linear systems of the form:

where and .

To the best of our knowledge, the inversion of a general cyclic heptadiagonal matrix of the form (1) has not been considered. Very recently in [5], the inversion of a general cyclic pentadiagonal matrix using recursion is studied without imposing any restrictive conditions on the elements of the matrix. Also, in this paper we are going to compute the inverse of a general cyclic heptadiagonal matrix of the form (1) without imposing any restrictive conditions on the elements of the matrix in (1). Our approach is mainly based on getting the elements of the last five columns of in suitable forms via the Doolittle LU factorization [10] along with parallel computation [7]. Then the elements of the remaining columns of may be obtained using relevant recursive relations. The inversion algorithm of this paper is a natural generalization of the algorithm presented in [5]. The development of a symbolic algorithm is considered in order to remove all cases where the numerical algorithm fails. Many algorithms for solving banded linear systems need to pivoting, for example Gaussian elimination algorithm [10–12]. Overall, pivoting adds more operations to the computational cost of an algorithm. These additional operations are sometimes necessary for the algorithm to work at all.

The paper is organized as follows. In Section 2, new symbolic computational algorithm, that will not break, is constructed. In Section 3, two illustrative examples are given. Conclusions of the work are given in Section 4.

2. Main Results

In this section we will focus on the construction of new symbolic computational algorithms for computing the determinant and the inverse of general cyclic heptadiagonal matrices. The solution of cyclic heptadiagonal linear systems of the form (2) will be taken into account. Firstly we begin with computing the factorization of the matrix . It is as in the following:

where

The elements in the matrices and in (4) and (5) satisfy

We also have:

Remark 1. It is not difficult to prove that the decomposition (3) exists only if , (pivoting elements). Moreover the cyclic heptadiagonal matrix of the form (1) has an inverse if, in addition, . Pivoting can be omitted by introducing auxiliary parameter in Algorithm 1 given later. So no pivoting is included in our algorithm.

At this point it is convenient to formulate our first result. It is a symbolic algorithm for computing the determinant of a cyclic heptadiagonal matrix of the form (1) and can be considered as natural generalization of the symbolic algorithm DETCPENTA in [5].

Algorithm 1. To compute for the cyclic heptadiagonal matrix in (1), we may proceed as follows.

Step 1. Set

 If then   is just a symbolic name) end if  Set ,  Set  Set  Set  Set  Set  Set  Set  Set  If then end if   Set    Set  Set  Set  Set  Set  Set  If then end if Set  Set  Set  Set

Step 2. Compute and simplify.

 For from to do

            If then end if

 End do

Step 3. Compute and simplify.

 For from to do

 End do

Step 4. Compute and simplify.

 For from to do

 End do

Step 5. Compute simplify.

             If then end if   If then end if

Step 6. Compute .

The symbolic Algorithm 1 will be referred to as DETCHEPTA. The computational cost of this algorithm is operations. The new algorithm DETCHEPTA is very useful to check the nonsingularity of the matrix when we consider, for example, the solution of the cyclic heptadiagonal linear systems of the form (2).

Now, when the matrix is nonsingular, its inversion is computed as follows.

Let

where denotes column of , .

Since the Doolittle factorization of the matrix in (1) is always possible then we can use parallel computations to get the elements of the last five columns , , and of as follows [5].

Solving in parallel the standard linear systems whose coefficient matrix is given by (4)

we get

where

Hence, solving the following standard linear systems whose coefficient matrix is given by (5)

gives the five columns , , and in the forms:

The remaining columns are obtained by using the fact where is the identity matrix. They are as in the following:

where is the th unit vector.

Remark 2. Equations (14) and (15) suggest an additional assumption , which is only formal and can be omitted by introducing auxiliary parameter in Algorithm 2 given later.

Now we formulate a second result. It is a symbolic computational algorithm to compute the inverse of a general cyclic heptadiagonal matrix of the form (1) when it exists.

Algorithm 2. To find the inverse matrix of the general cyclic heptadiagonal matrix in (1) by using the relations (13)–(15).

 INPUT. Order of the matrix and the components,                   ,                  where .  OUTPUT. Inverse matrix.

Step 1. If for any set ( is just a symbolic name).

Step 2. If for any .

Step 3. Use the DETCHEPTA algorithm to check the nonsingularity of the matrix . If the matrix is singular then OUTPUT (“The matrix is singular”); stop.

Step 4. For , compute and simplify the components , , and of the columns , and , respectively, by using (13).

Step 5. For , compute and simplify the components by using (14).

Step 6. For , do

 For , do     Compute and simplify the components by using (15). End do End do

Step 7. Substitute the actual value in all expressions to obtain the elements, .

The symbolic Algorithm 2 will be referred to as CHINV algorithm. The computational cost of CHINV algorithm is operations. The Algorithms 2.2, 2.3, and 2.2 in [5, 13, 14], respectively, are now special cases of the CHINV algorithm.

3. Two Illustrative Examples

In this section we give two examples for the sake of illustration.

Example 1. Consider the cyclic heptadiagonal linear system

By using the coefficient matrix of the system (16) and applying the CHINV algorithm, we get the inverse of the coefficient matrix ():

By using (17) and after simple calculations we can obtain the solution of cyclic heptadiagonal linear system (16):

Example 2. Consider the cyclic heptadiagonal matrix

By applying the CHINV algorithm, it yields

4. Conclusions

In this work new recursive computational algorithms have been developed for computing the determinant and inverse of general cyclic heptadiagonal matrices and solving linear systems of cyclic heptadiagonal type. The algorithms are reliable, are computationally efficient, and will not fail. The algorithms are natural generalizations of some algorithms in current use.

References

A. A. Karawia, “A computational algorithm for solving periodic penta-diagonal linear systems,” Applied Mathematics and Computation, vol. 174, no. 1, pp. 613–618, 2006.

A. D. A. Hadj and M. Elouafi, “A fast numerical algorithm for the inverse of a tridiagonal and pentadiagonal matrix,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 441–445, 2008.

I. M. Navon, “A periodic pentadiagonal systems solver,” Communications in Applied Numerical Methods, vol. 3, pp. 63–69, 1987.

M. Batista, “method for solving cyclic block penta-diagonal systems of linear equations,” Mathematical Software, 2008, http://arxiv.org/abs/0803.0874

M. El-Mikkawy and E.-D. Rahmo, “Symbolic algorithm for inverting cyclic pentadiagonal matrices recursively: derivation and implementation,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1386–1396, 2010.

S. N. Neossi Nguetchue and S. Abelman, “A computational algorithm for solving nearly penta-diagonal linear systems,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 629–634, 2008.

T. Sogabe, “New algorithms for solving periodic tridiagonal and periodic pentadiagonal linear systems,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 850–856, 2008.

X.-G. Lv and J. Le, “A note on solving nearly penta-diagonal linear systems,” Applied Mathematics and Computation, vol. 204, no. 2, pp. 702–712, 2008.

X.-L. Zhao and T.-Z. Huang, “On the inverse of a general pentadiagonal matrix,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 639–646, 2008.

J. W. Demmel, Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1997.

G. Engeln-Müllges and F. Uhlig, Numerical Algorithms with C, Springer, Berlin, Germany, 1996.

M. B. Allen III and E. L. Isaacson, Numerical Analysis For Applied Science, John Wiley and Sons, 1997.

M. El-Mikkawy and E.-D. Rahmo, “A new recursive algorithm for inverting general tridiagonal and anti-tridiagonal matrices,” Applied Mathematics and Computation, vol. 204, no. 1, pp. 368–372, 2008.

M. El-Mikkawy and E.-D. Rahmo, “A new recursive algorithm for inverting general periodic pentadiagonal and anti-pentadiagonal matrices,” Applied Mathematics and Computation, vol. 207, no. 1, pp. 164–170, 2009.