Inversion of General Cyclic Heptadiagonal Matrices

whereX = (x 1 , x 2 , ..., x n ) T and R = (r 1 , r 2 , ..., r n ) T. 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 H in (1). Our approach is mainly based on getting the elements of the last five columns of


Introduction
The × general cyclic heptadiagonal matrices take the form: where  ≥ 7.
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  −1 in suitable forms via the Doolittle LU factorization [10] along with parallel computation [7].Then the elements of the remaining ( − 5) columns of  −1 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][11][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.

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: ( ) .
The elements in the matrices L and U in (4) and (5) satisfy We also have: Remark 1.It is not difficult to prove that the LU decomposition (3) exists only if   ̸ = 0,  = 1(1) − 1 (pivoting elements).Moreover the cyclic heptadiagonal matrix  of the form (1) has an inverse if, in addition,   ̸ = 0. 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 Det() for the cyclic heptadiagonal matrix  in (1), we may proceed as follows.
For  from 4 to  − 2 do Step 3. Compute and simplify.
For  from 4 to  − 5 do

End do
Step 4. Compute and simplify.
For  from 4 to  − 4 do

End do
Step 5. Compute simplify.

𝑘
Step 6. Compute Det() = (∏  =1   ) =0 .The symbolic Algorithm 1 will be referred to as DETCHEPTA.The computational cost of this algorithm is 52 − 195 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 Col  denotes th column of  −1 ,  = 1, 2, . . ., .
Remark 2. Equations ( 14) and ( 15) suggest an additional assumption ∏ −5 =1   ̸ = 0, 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).
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.
The symbolic Algorithm 2 will be referred to as CHINV algorithm.The computational cost of CHINV algorithm is 21 By using (17) and after simple calculations we can obtain the solution of cyclic heptadiagonal linear system (16): )=0