Inversion of General Cyclic Heptadiagonal Matrices

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 21𝑛²−48𝑛−88 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 𝑛×𝑛 general cyclic heptadiagonal matrices take the form: 𝑑𝐻=¹𝑎₁𝐴₁𝐶₁00⋯0𝐵₁𝑏₁𝑏₂𝑑₂𝑎₂𝐴₂𝐶₂0⋯⋯⋯𝐵₂𝐵₃𝑏₃𝑑₃𝑎₃𝐴₃𝐶₃𝐷0⋯⋯0₄𝐵₄𝑏₄𝑑₄𝑎₄𝐴₄𝐶₄0…⋮0⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮0⋯0𝐷_𝑛−3𝐵_𝑛−3𝑏_𝑛−3𝑑_𝑛−3𝑎_𝑛−3𝐴_𝑛−3𝐶_𝑛−30⋯⋯0𝐷_𝑛−2𝐵_𝑛−2𝑏_𝑛−2𝑑_𝑛−2𝑎_𝑛−2𝐴_𝑛−2𝐴_𝑛−10⋯⋯0𝐷_𝑛−1𝐵_𝑛−1𝑏_𝑛−1𝑑_𝑛−1𝑎_𝑛−1𝑎_𝑛𝐴_𝑛0⋯⋯0𝐷_𝑛𝐵_𝑛𝑏_𝑛𝑑_𝑛, (1)where 𝑛≥7.

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: 𝐻𝑋=𝑅, (2) where 𝑋=(𝑥₁,𝑥₂,...,𝑥_𝑛)^𝑇 and 𝑅=(𝑟₁,𝑟₂,...,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 𝐻⁻¹ in suitable forms via the Doolittle LU factorization [10] along with parallel computation [7]. Then the elements of the remaining (𝑛−5) columns of 𝐻⁻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 [10–12]. 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 𝐿𝑈 factorization of the matrix 𝐻. It is as in the following: 𝐻=𝐿𝑈, (3) where𝑓𝐿=10⋯⋯⋯⋯⋯⋯⋯0₂𝑒10⋯⋯⋯⋯⋯⋯0₃𝑓₃𝐷10⋯⋯⋯⋯⋯0₄𝛼₁𝑒₄𝑓₄𝐷10⋯⋯⋯⋯0⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮0⋯0^𝑛−3𝛼_𝑛−6𝑒_𝑛−3𝑓_𝑛−3𝐷1⋱⋱00⋯⋯0^𝑛−2𝛼_𝑛−5𝑒_𝑛−2𝑓_𝑛−2𝑘1⋱0₁𝑘₂𝑘₃⋯⋯𝑘_𝑛−2𝑘_𝑛−1𝑘_𝑛−2ℎ10₁ℎ₂ℎ₃⋯⋯⋯ℎ_𝑛−3ℎ_𝑛−2ℎ_𝑛−11𝛼,𝑈=¹𝑔₁𝑍₁𝐶₁00⋯0𝑤₁𝑣₁0𝛼₂𝑔₂𝑍₂𝐶₂0⋯0𝑤₂𝑣₂00𝛼₃𝑔₃𝑍₃𝐶₃⋯0𝑤₃𝑣₃000𝛼₄𝑔₄𝑍₄⋱0𝑤₄𝑣₄⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮0⋯⋯⋯⋯0𝛼_𝑛−3𝑔_𝑛−3𝑤_𝑛−3𝑣_𝑛−30⋯⋯⋯⋯⋯0𝛼_𝑛−2𝑤_𝑛−2𝑣_𝑛−200⋯⋯⋯⋯⋯0𝛼_𝑛−1𝑣_𝑛−1000⋯⋯⋯⋯⋯0𝛼_𝑛. (4)(5)

The elements in the matrices 𝐿 and 𝑈 in (4) and (5) satisfy𝛼_𝑖=𝑑₁𝑑,if𝑖=1,₂−𝑓₂𝑔₁𝑑,if𝑖=2,₃−𝑓₃𝑔₂−𝑒₃𝑍₁𝑑,if𝑖=3,_𝑖−𝑓_𝑖𝑔_𝑖−1−𝑒_𝑖𝑍_𝑖−2−𝐷−𝑖𝛼_𝑖−3𝐶_𝑖−3(1)𝑑,if𝑖=4𝑛−2,_𝑛−1−∑^𝑛−2_𝑗=1𝑤_𝑗𝑘_𝑗𝑑,if𝑖=𝑛−1,_𝑛−∑^𝑛−1_𝑗=1𝑣_𝑗ℎ_𝑗𝑘,if𝑖=𝑛,_𝑖=𝐴_𝑛−1𝛼₁𝑘,if𝑖=1,₁𝑔₁𝛼₂𝑘,if𝑖=2,₂𝑔₂+𝑘₁𝑍₁𝛼₃𝑘,if𝑖=3,_𝑖−1𝑔_𝑖−1+𝑘_𝑖−2𝑍_𝑖−2+𝑘_𝑖−3𝐶_𝑖−3𝛼_𝑖(1)𝐷,if𝑖=4𝑛−5,_𝑛−1−𝑘_𝑛−5𝑔_𝑛−5−𝑘_𝑛−6𝑍_𝑛−6−𝑘_𝑛−7𝐶_𝑛−7𝛼_𝑛−4𝐵,if𝑖=𝑛−4,_𝑛−1−𝑘_𝑛−4𝑔_𝑛−4−𝑘_𝑛−5𝑍_𝑛−5−𝑘_𝑛−6𝐶_𝑛−6𝛼_𝑛−3𝑏,if𝑖=𝑛−3,_𝑛−1−𝑘_𝑛−3𝑔_𝑛−3−𝑘_𝑛−4𝑍_𝑛−4−𝑘_𝑛−5𝐶_𝑛−5𝛼_𝑛−2ℎ,if𝑖=𝑛−2,_𝑖=𝑎_𝑛𝛼₁−ℎ,if𝑖=1,₁𝑔₁−𝐴_𝑛𝛼₂−ℎ,if𝑖=2,₂𝑔₂+ℎ₁𝑍₁𝛼₃−ℎ,if𝑖=3,_𝑖−1𝑔_𝑖−1+ℎ_𝑖−2𝑍_𝑖−2+ℎ_𝑖−3𝐶_𝑖−3𝛼_𝑖(1)𝐷,if𝑖=4𝑛−4,_𝑛−ℎ_𝑛−4𝑔_𝑛−4−ℎ_𝑛−5𝑍_𝑛−5−ℎ_𝑛−6𝐶_𝑛−6𝛼_𝑛−3𝐵,if𝑖=𝑛−3,_𝑛−ℎ_𝑛−3𝑔_𝑛−3−ℎ_𝑛−4𝑍_𝑛−4−ℎ_𝑛−5𝐶_𝑛−5𝛼_𝑛−2𝑏,if𝑖=𝑛−2,_𝑛−∑^𝑛−2_𝑗=1ℎ_𝑗𝑤_𝑗𝛼_𝑛−1𝑣,if𝑖=𝑛−1,_𝑖=𝑏₁,if𝑖=1,−𝑓₂𝑣₁+𝐵₂,if𝑖=2,−𝑓₃𝑣₂−𝑒₃𝑣₁,if𝑖=3,−𝑓_𝑖𝑣_𝑖−1−𝑒_𝑖𝑣_𝑖−2−𝐷_𝑖𝛼_𝑖−3𝑣_𝑖−3,if𝑖=4(1)𝑛−4,−𝑓_𝑛−3𝑣_𝑛−4−𝑒_𝑛−3𝑣_𝑛−5−𝐷_𝑛−3𝛼_𝑛−6𝑣_𝑛−6+𝐶_𝑛−3,if𝑖=𝑛−3,−𝑓_𝑛−2𝑣_𝑛−3−𝑒_𝑛−2𝑣_𝑛−4−𝐷_𝑛−2𝛼_𝑛−5𝑣_𝑛−5+𝐴_𝑛−2𝑎,if𝑖=𝑛−2,_𝑛−1−∑^𝑛−2_𝑗=1𝑘_𝑗𝑣_𝑗𝑤,if𝑖=𝑛−1,_𝑖=𝐵₁,if𝑖=1,−𝑓₂𝑤₁,if𝑖=2,−𝑓₃𝑤₂−𝑒₃𝑤₁,if𝑖=3,−𝑓_𝑖𝑤_𝑖−1−𝑒_𝑖𝑤_𝑖−2−𝐷_𝑖𝛼_𝑖−3𝑤_𝑖−3,if𝑖=4(1)𝑛−5,−𝑓_𝑛−4𝑤_𝑛−5−𝑒_𝑛−4𝑤_𝑛−6−𝐷_𝑛−4𝛼_𝑛−7𝑤_𝑛−7+𝐶_𝑛−4,if𝑖=𝑛−4,−𝑓_𝑛−3𝑤_𝑛−4−𝑒_𝑛−3𝑤_𝑛−5−𝐷_𝑛−3𝛼_𝑛−6𝑤_𝑛−6+𝐴_𝑛−3,if𝑖=𝑛−3,−𝑓_𝑛−2𝑤_𝑛−3−𝑒_𝑛−2𝑤_𝑛−4−𝐷_𝑛−2𝛼_𝑛−5𝑤_𝑛−5+𝑎_𝑛−2𝑓,if𝑖=𝑛−2,_𝑖=𝑏₂𝛼₁𝑏,if𝑖=2,₃−𝑒₃𝑔₁𝛼₂𝑏,if𝑖=3,_𝑖−𝑒_𝑖𝑔_𝑖−2−𝐷_𝑖/𝛼_𝑖−3𝑍_𝑖−3𝛼_𝑖−1(1)𝑒,if𝑖=4𝑛−2,_𝑖=𝐵₃𝛼₁𝐵,if𝑖=3,_𝑖−𝐷_𝑖/𝛼_𝑖−3𝑔_𝑖−3𝛼_𝑖−2(1)𝑔,if𝑖=4𝑛−2,_𝑖=𝑎₁𝑎,if𝑖=1,₂−𝑓₂𝑍₁𝑎,if𝑖=2,_𝑖−𝑓_𝑖𝑍_𝑖−1−𝑒_𝑖𝐶_𝑖−2(1)𝑍,if𝑖=3𝑛−3,_𝑖=𝐴₁𝐴,if𝑖=1,_𝑖−𝑓_𝑖𝐶_𝑖−1,if𝑖=2(1)𝑛−4. (6)

We also have: Det(𝐻)=^𝑛_𝑖=1𝛼_𝑖. (7)

Remark 1. It is not difficult to prove that the 𝐿𝑈 decomposition (3) exists only if 𝛼_𝑖≠0, 𝑖=1(1)𝑛−1 (pivoting elements). Moreover the cyclic heptadiagonal matrix 𝐻 of the form (1) has an inverse if, in addition, 𝛼_𝑛≠0. Pivoting can be omitted by introducing auxiliary parameter 𝑡 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 𝐻 of the form (1) and can be considered as natural generalization of the symbolic algorithm DETCPENTA in [5].

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

Step 1. Set 𝛼₁=𝑑₁ If 𝛼₁=0 then 𝛼₁=𝑡 (𝑡 is just a symbolic name) end if  Set 𝑔₁=𝑎₁, 𝑍₁=𝐴₁ Set 𝑘₁=𝐴_𝑛−1/𝛼₁ Set 𝑣₁=𝑏₁ Set 𝑤₁=𝐵₁ Set ℎ₁=𝑎_𝑛/𝛼₁ Set 𝑤₁=𝐵₁ Set 𝑓₂=𝑏₂/𝛼₁ Set 𝑒₃=𝐵₃/𝛼₁ Set 𝛼₂=𝑑₂−𝑓₂∗𝑔₁ If 𝛼₂=0 then 𝛼₂=𝑡 end if   Set   𝑘₂=−𝑘₁∗𝑔₁/𝛼₂ Set 𝑣₂=𝐵₂−𝑓₂∗𝑣₁ Set 𝑤₂=−𝑓₂∗𝑤₁ Set ℎ₂=(𝐴_𝑛−ℎ₁∗𝑔₁)/𝛼₂ Set 𝑔₂=𝑎₂−𝑓₂∗𝑍₁ Set 𝑓₃=(𝑏₃−𝑒₃∗𝑔₁)/𝛼₂ Set 𝛼₃=𝑑₃−𝑒₃∗𝑍₁−𝑓₃∗𝑔₂ If 𝛼₃=0 then 𝛼₃=𝑡 end if Set 𝑘₃=−(𝑘₁∗𝑍₁+𝑘₂∗𝑔₂)/𝛼₃ Set ℎ₃=−(ℎ₁∗𝑍₁+ℎ₂∗𝑔₂)/𝛼₃ Set 𝑣₃=−𝑒₃∗𝑣₁−𝑓₃∗𝑣₂ Set 𝑤₃=−𝑓₃∗𝑤₂−𝑒₃∗𝑤₁

Step 2. Compute and simplify. For 𝑖 from 4 to 𝑛−2 do  𝑒_𝑖=(𝐵_𝑖−𝐷_𝑖∗𝑔_𝑖−3/𝛼_𝑖−3)/𝛼_𝑖−2  𝑓_𝑖=(𝑏_𝑖−𝐷_𝑖∗𝑍_𝑖−3/𝛼_𝑖−3−𝑒_𝑖∗𝑔_𝑖−2)/𝛼_𝑖−1  𝑍_𝑖−2=𝐴_𝑖−2−𝑓_𝑖−2∗𝐶_𝑖−3  𝑔_𝑖−1=𝑎_𝑖−1−𝑓_𝑖−1∗𝑍_𝑖−2−𝑒_𝑖−1∗𝐶_𝑖−3  𝛼_𝑖=𝑑_𝑖−𝐷_𝑖∗𝐶_𝑖−3/𝛼_𝑖−3−𝑒_𝑖∗𝑔_𝑖−2−𝑓_𝑖∗𝑔_𝑖−1  If 𝛼_𝑖=0 then 𝛼_𝑖=𝑡 end if End do

Step 3. Compute and simplify. For 𝑖 from 4 to 𝑛−5 do  𝑘_𝑖=−(𝑘_𝑖−3∗𝐶_𝑖−3+𝑘_𝑖−2∗𝑍_𝑖−2+𝑘_𝑖−1∗𝑔_𝑖−1)/𝛼_𝑖  𝑤_𝑖=−(𝐷_𝑖∗𝑤_𝑖−3/𝛼_𝑖−3+𝑒_𝑖∗𝑤_𝑖−2+𝑓_𝑖∗𝑤_𝑖−1) End do

Step 4. Compute and simplify. For 𝑖 from 4 to 𝑛−4 do  ℎ_𝑖=−(ℎ_𝑖−3∗𝐶_𝑖−3+ℎ_𝑖−2∗𝑍_𝑖−2+ℎ_𝑖−1∗𝑔_𝑖−1)/𝛼_𝑖  𝑣_𝑖=−(𝐷_𝑖∗𝑣_𝑖−3/𝛼_𝑖−3+𝑒_𝑖∗𝑣_𝑖−2+𝑓_𝑖∗𝑣_𝑖−1) End do

Step 5. Compute simplify. 𝑘_𝑛−4=(𝐷_𝑛−1−𝑘_𝑛−5∗𝑔_𝑛−5−𝑘_𝑛−6∗𝑍_𝑛−6−𝑘_𝑛−7∗𝐶_𝑛−7)/𝛼_𝑛−4 𝑘_𝑛−3=(𝐵_𝑛−1−𝑘_𝑛−4∗𝑔_𝑛−4−𝑘_𝑛−5∗𝑍_𝑛−5−𝑘_𝑛−6∗𝐶_𝑛−6)/𝛼_𝑛−3 𝑘_𝑛−2=(𝑏_𝑛−1−𝑘_𝑛−3∗𝑔_𝑛−3−𝑘_𝑛−4∗𝑍_𝑛−4−𝑘_𝑛−5∗𝐶_𝑛−5)/𝛼_𝑛−2 𝑤_𝑛−4=𝐶_𝑛−4−𝐷_𝑛−4∗𝑤_𝑛−7/𝛼_𝑛−7−𝑒_𝑛−4∗𝑤_𝑛−6−𝑓_𝑛−4∗𝑤_𝑛−5 𝑤_𝑛−3=𝐴_𝑛−3−𝐷_𝑛−3∗𝑤_𝑛−6/𝛼_𝑛−6−𝑒_𝑛−3∗𝑤_𝑛−5−𝑓_𝑛−3∗𝑤_𝑛−4 𝑤_𝑛−2=𝑎_𝑛−2−𝐷_𝑛−2∗𝑤_𝑛−5/𝛼_𝑛−5−𝑒_𝑛−2∗𝑤_𝑛−4−𝑓_𝑛−2∗𝑤_𝑛−3 ℎ_𝑛−3=(𝐷_𝑛−ℎ_𝑛−4∗𝑔_𝑛−4−ℎ_𝑛−5∗𝑍_𝑛−5−ℎ_𝑛−6∗𝐶_𝑛−6)/𝛼_𝑛−3 ℎ_𝑛−2=(𝐵_𝑛−ℎ_𝑛−3∗𝑔_𝑛−3−ℎ_𝑛−4∗𝑍_𝑛−4−ℎ_𝑛−5∗𝐶_𝑛−5)/𝛼_𝑛−2 𝑣_𝑛−3=𝐶_𝑛−3−𝐷_𝑛−3∗𝑣_𝑛−6/𝛼_𝑛−6−𝑒_𝑛−3∗𝑣_𝑛−5−𝑓_𝑛−3∗𝑣_𝑛−4 𝑣_𝑛−2=𝐴_𝑛−2−𝐷_𝑛−2∗𝑣_𝑛−5/𝛼_𝑛−5−𝑒_𝑛−2∗𝑣_𝑛−4−𝑓_𝑛−2∗𝑣_𝑛−3 𝑣_𝑛−1=𝑎_𝑛−1−∑^𝑛−2_𝑗=1𝑘_𝑗𝑣_𝑗 𝛼_𝑛−1=𝑑_𝑛−1−∑^𝑛−2_𝑗=1𝑤_𝑗𝑘_𝑗 If 𝛼_𝑛−1=0 then 𝛼_𝑛−1=𝑡 end if ℎ_𝑛−1=(𝑏_𝑛−∑^𝑛−2_𝑗=1ℎ_𝑗𝑤_𝑗)/𝛼_𝑛−1 𝛼_𝑛=𝑑_𝑛−∑^𝑛−1_𝑗=1𝑣_𝑗ℎ_𝑗 If 𝛼_𝑛=0 then 𝛼_𝑛=𝑡 end if

Step 6. Compute Det(𝐻)=(∏^𝑛_𝑖=1𝛼_𝑖)_𝑡=0.
The symbolic Algorithm 1 will be referred to as DETCHEPTA. The computational cost of this algorithm is 52𝑛−195 operations. The new algorithm DETCHEPTA is very useful to check the nonsingularity of the matrix 𝐻 when we consider, for example, the solution of the cyclic heptadiagonal linear systems of the form (2).

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

Let 𝐻⁻¹=𝑆_{𝑖𝑗1≤𝑖,𝑗≤𝑛}=Col₁,Col₂,…,Col_𝑛, (8) where Col_𝑚 denotes 𝑚th column of 𝐻⁻¹, 𝑚=1,2,…,𝑛.

Since the Doolittle 𝐿𝑈 factorization of the matrix 𝐻 in (1) is always possible then we can use parallel computations to get the elements of the last five columns Col_𝑖=(𝑆_1,𝑖,𝑆_2,𝑖,…,𝑆_𝑛,𝑖)^𝑇, 𝑖=𝑛,𝑛−1,𝑛−2,𝑛−3, and 𝑛−4 of 𝐻⁻¹ as follows [5].

Solving in parallel the standard linear systems whose coefficient matrix 𝐿 is given by (4) 𝐿𝑄₁^(𝑛)𝑄₁^(𝑛−1)𝑄₁^(𝑛−2)𝑄₁^(𝑛−3)𝑄₁^(𝑛−4)𝑄₂^(𝑛)𝑄₂^(𝑛−1)𝑄₂^(𝑛−2)𝑄₂^(𝑛−3)𝑄₂^(𝑛−4)𝑄⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮_{(𝑛)𝑛−4}𝑄^(𝑛−1)_𝑛−4𝑄^(𝑛−2)_𝑛−4𝑄^(𝑛−3)_𝑛−4𝑄^(𝑛−4)_𝑛−4𝑄^(𝑛)_𝑛−3𝑄^(𝑛−1)_𝑛−3𝑄^(𝑛−2)_𝑛−3𝑄^(𝑛−3)_𝑛−3𝑄^(𝑛−4)_𝑛−3𝑄^(𝑛)_𝑛−2𝑄^(𝑛−1)_𝑛−2𝑄^(𝑛−2)_𝑛−2𝑄^(𝑛−3)_𝑛−2𝑄^(𝑛−4)_𝑛−2𝑄^(𝑛)_𝑛−1𝑄^(𝑛−1)_𝑛−1𝑄^(𝑛−2)_𝑛−1𝑄^(𝑛−3)_𝑛−1𝑄^(𝑛−4)_𝑛−1𝑄_𝑛^(𝑛)𝑄_𝑛^(𝑛−1)𝑄_𝑛^(𝑛−2)𝑄_𝑛^(𝑛−3)𝑄_𝑛^(𝑛−4)=,0000000000⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮0000100010001000100010000 (9) we get 𝑄₁^(𝑛)𝑄₁^(𝑛−1)𝑄₁^(𝑛−2)𝑄₁^(𝑛−3)𝑄₁^(𝑛−4)𝑄₂^(𝑛)𝑄₂^(𝑛−1)𝑄₂^(𝑛−2)𝑄₂^(𝑛−3)𝑄₂^(𝑛−4)𝑄⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮_{(𝑛)𝑛−4}𝑄^(𝑛−1)_𝑛−4𝑄^(𝑛−2)_𝑛−4𝑄^(𝑛−3)_𝑛−4𝑄^(𝑛−4)_𝑛−4𝑄^(𝑛)_𝑛−3𝑄^(𝑛−1)_𝑛−3𝑄^(𝑛−2)_𝑛−3𝑄^(𝑛−3)_𝑛−3𝑄^(𝑛−4)_𝑛−3𝑄^(𝑛)_𝑛−2𝑄^(𝑛−1)_𝑛−2𝑄^(𝑛−2)_𝑛−2𝑄^(𝑛−3)_𝑛−2𝑄^(𝑛−4)_𝑛−2𝑄^(𝑛)_𝑛−1𝑄^(𝑛−1)_𝑛−1𝑄^(𝑛−2)_𝑛−1𝑄^(𝑛−3)_𝑛−1𝑄^(𝑛−4)_𝑛−1𝑄_𝑛^(𝑛)𝑄_𝑛^(𝑛−1)𝑄_𝑛^(𝑛−2)𝑄_𝑛^(𝑛−3)𝑄_𝑛^(𝑛−4)=0000000000⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮000010001−𝑓_𝑛−3001−𝑓_𝑛−2𝑀₆01−𝑘_𝑛−2𝑀₄𝑀₅1−ℎ_𝑛−1𝑀₁𝑀₂𝑀₃, (10) where 𝑀₁=−ℎ_𝑛−2+ℎ_𝑛−1𝑘_𝑛−2,𝑀₂=−ℎ_𝑛−3+ℎ_𝑛−2𝑓_𝑛−2+ℎ_𝑛−1𝑘_𝑛−3−ℎ_𝑛−1𝑘_𝑛−2𝑓_𝑛−2,𝑀₃=−ℎ_𝑛−4+ℎ_𝑛−3𝑓_𝑛−3+ℎ_𝑛−2𝑒_𝑛−2−ℎ_𝑛−2𝑓_𝑛−2𝑓_𝑛−3+ℎ_𝑛−1𝑘_𝑛−4−ℎ_𝑛−1𝑘_𝑛−3𝑓_𝑛−3−ℎ_𝑛−1𝑘_𝑛−2𝑒_𝑛−2+ℎ_𝑛−1𝑘_𝑛−2𝑓_𝑛−2𝑓_𝑛−3,𝑀₄=−𝑘_𝑛−3+𝑘_𝑛−2𝑓_𝑛−2,𝑀₅=−𝑘_𝑛−4+𝑘_𝑛−3𝑓_𝑛−3+𝑘_𝑛−2𝑒_𝑛−2−𝑘_𝑛−2𝑓_𝑛−2𝑓_𝑛−3,𝑀₆=−𝑒_𝑛−2+𝑓_𝑛−2𝑓_𝑛−3. (11) Hence, solving the following standard linear systems whose coefficient matrix 𝑈 is given by (5) 𝑈𝑆_1,𝑛𝑆_1,𝑛−1𝑆_1,𝑛−2𝑆_1,𝑛−3𝑆_1,𝑛−4𝑆_2,𝑛𝑆_2,𝑛−1𝑆_2,𝑛−2𝑆_2,𝑛−3𝑆_2,𝑛−4𝑆⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮_{𝑛−4,𝑛}𝑆_{𝑛−4,𝑛−1}𝑆_{𝑛−4,𝑛−2}𝑆_{𝑛−4,𝑛−3}𝑆_{𝑛−4,𝑛−4}𝑆_{𝑛−3,𝑛}𝑆_{𝑛−3,𝑛−1}𝑆_{𝑛−3,𝑛−2}𝑆_{𝑛−3,𝑛−3}𝑆_{𝑛−3,𝑛−4}𝑆_{𝑛−2,𝑛}𝑆_{𝑛−2,𝑛−1}𝑆_{𝑛−2,𝑛−2}𝑆_{𝑛−2,𝑛−3}𝑆_{𝑛−2,𝑛−4}𝑆_{𝑛−1,𝑛}𝑆_{𝑛−1,𝑛−1}𝑆_{𝑛−1,𝑛−2}𝑆_{𝑛−1,𝑛−3}𝑆_{𝑛−1,𝑛−4}𝑆_𝑛,𝑛𝑆_{𝑛,𝑛−1}𝑆_{𝑛,𝑛−2}𝑆_{𝑛,𝑛−3}𝑆_{𝑛,𝑛−4}=𝑄₁^(𝑛)𝑄₁^(𝑛−1)𝑄₁^(𝑛−2)𝑄₁^(𝑛−3)𝑄₁^(𝑛−4)𝑄₂^(𝑛)𝑄₂^(𝑛−1)𝑄₂^(𝑛−2)𝑄₂^(𝑛−3)𝑄₂^(𝑛−4)𝑄⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮_{(𝑛)𝑛−4}𝑄^(𝑛−1)_𝑛−4𝑄^(𝑛−2)_𝑛−4𝑄^(𝑛−3)_𝑛−4𝑄^(𝑛−4)_𝑛−4𝑄^(𝑛)_𝑛−3𝑄^(𝑛−1)_𝑛−3𝑄^(𝑛−2)_𝑛−3Q^(𝑛−3)_𝑛−3𝑄^(𝑛−4)_𝑛−3𝑄^(𝑛)_𝑛−2𝑄^(𝑛−1)_𝑛−2𝑄^(𝑛−2)_𝑛−2𝑄^(𝑛−3)_𝑛−2𝑄^(𝑛−4)_𝑛−2𝑄^(𝑛)_𝑛−1𝑄^(𝑛−1)_𝑛−1𝑄^(𝑛−2)_𝑛−1𝑄^(𝑛−3)_𝑛−1𝑄^(𝑛−4)_𝑛−1𝑄_𝑛^(𝑛)𝑄_𝑛^(𝑛−1)𝑄_𝑛^(𝑛−2)𝑄_𝑛^(𝑛−3)𝑄_𝑛^(𝑛−4) (12) gives the five columns Col_𝑖, 𝑖=𝑛,𝑛−1,𝑛−2,𝑛−3, and 𝑛−4 in the forms: 𝑆_𝑛,𝑛=1𝛼_𝑛,𝑆_{𝑛−1,𝑛}𝑣=−^𝑛−1𝑆_𝑛,𝑛𝛼_𝑛−1,𝑆_𝑖,𝑛𝑔=−^𝑖𝑆_𝑖+1,𝑛+𝑍_𝑖𝑆_𝑖+2,𝑛+𝐶_𝑖𝑆_𝑖+3,𝑛+𝑤_𝑖𝑆_{𝑖−1,𝑛}+𝑣_𝑖𝑆_𝑛,𝑛𝛼_𝑖,()𝑆𝑖=𝑛−5−11,_{𝑛,𝑛−1}ℎ=−^𝑛−1𝛼_𝑛,𝑆_{𝑛−1,𝑛−1}=1−𝑣_𝑛−1𝑆_{𝑛,𝑛−1}𝛼_𝑛−1,𝑆_{𝑖,𝑛−1}=−𝑔_𝑖𝑆_{𝑖+1,𝑛−1}+𝑍_𝑖𝑆_{𝑖+2,𝑛−1}+𝐶_𝑖𝑆_{𝑖+3,𝑛−1}+𝑤_𝑖𝑆_{𝑖−1,𝑛−1}+𝑣_𝑖𝑆_{𝑛,𝑛−1}𝛼_𝑖⁻¹,()𝑆𝑖=𝑛−5−11,_{𝑛,𝑛−2}=−ℎ_𝑛−2+ℎ_𝑛−1𝑘_𝑛−2𝛼_𝑛,𝑆_{𝑛−1,𝑛−2}𝑘=−^𝑛−2+𝑣_𝑛−1𝑆_{𝑛,𝑛−2}𝛼_𝑛−1,𝑆_{𝑛−2,𝑛−2}=1−𝑤_𝑛−2𝑆_{𝑛−1,𝑛−2}−𝑣_𝑛−2𝑆_{𝑛,𝑛−2}𝛼_𝑛−2,𝑆_{𝑖,𝑛−2}=−𝑔_𝑖𝑆_{𝑖+1,𝑛−2}+𝑍_𝑖𝑆_{𝑖+2,𝑛−2}+𝐶_𝑖𝑆_{𝑖+3,𝑛−2}+𝑤_𝑖𝑆_{𝑖−1,𝑛−2}+𝑣_𝑖𝑆_{𝑛,𝑛−2}𝛼_𝑖⁻¹,()𝑆𝑖=𝑛−5−11,_{𝑛,𝑛−3}=−ℎ_𝑛−3+ℎ_𝑛−2𝑓_𝑛−2−ℎ_𝑛−1𝑘_𝑛−2𝑓_𝑛−2−𝑘_𝑛−3𝛼_𝑛,𝑆_{𝑛−1,𝑛−3}=𝑘_𝑛−2𝑓_𝑛−2−𝑘_𝑛−3−𝑣_𝑛−1𝑆_{𝑛,𝑛−3}𝛼_𝑛−1,𝑆_{𝑛−2,𝑛−3}𝑓=−^𝑛−2+𝑤_𝑛−2𝑆_{𝑛−1,𝑛−3}+𝑣_𝑛−2𝑆_{𝑛,𝑛−3}𝛼_𝑛−2,𝑆_{𝑛−3,𝑛−3}=1−𝑔_𝑛−3𝑆_{𝑛−2,𝑛−3}−𝑤_𝑛−3𝑆_{𝑛−1,𝑛−3}−𝑣_𝑛−3𝑆_{𝑛,𝑛−3}𝛼_𝑛−3,𝑆_{𝑛,𝑛−4}=−ℎ_𝑛−4+ℎ_𝑛−3𝑓_𝑛−3−ℎ_𝑛−2𝑒_𝑛−2−𝑓_𝑛−2𝑓_𝑛−3𝛼_𝑛+ℎ_𝑛−1𝑘_𝑛−4−𝑘_𝑛−3𝑓_𝑛−3−𝑘_𝑛−2𝑒_𝑛−2−𝑓_𝑛−2𝑓_𝑛−3𝛼_𝑛,𝑆_{𝑛−1,𝑛−4}=−𝑘_𝑛−4+𝑘_𝑛−3𝑓_𝑛−3−𝑘_𝑛−2𝑒_𝑛−2−𝑓_𝑛−2𝑓_𝑛−3−𝑣_𝑛−1𝑆_{𝑛,𝑛−4}𝛼_𝑛−1,𝑆_{𝑛−2,𝑛−4}𝑒=−^𝑛−2−𝑓_𝑛−2𝑓_𝑛−3+𝑤_𝑛−2𝑆_{𝑛−1,𝑛−4}+𝑣_𝑛−2𝑆_{𝑛,𝑛−4}𝛼_𝑛−2,𝑆_{𝑛−3,𝑛−4}𝑓=−^𝑛−3+𝑔_𝑛−3𝑆_{𝑛−2,𝑛−4}+𝑤_𝑛−3𝑆_{𝑛−1,𝑛−4}+𝑣_𝑛−3𝑆_{𝑛,𝑛−4}𝛼_𝑛−3,𝑆_{𝑛−4,𝑛−4}=1−𝑔_𝑛−4𝑆_{𝑛−3,𝑛−4}−𝑍_𝑛−4𝑆_{𝑛−2,𝑛−4}−𝑤_𝑛−4𝑆_{𝑛−1,𝑛−3}−𝑣_𝑛−4𝑆_{𝑛,𝑛−4}𝛼_𝑛−4⁻¹,𝑆_{𝑖,𝑛−4}=−𝑔_𝑖𝑆_{𝑖+1,𝑛−4}+𝑍_𝑖𝑆_{𝑖+2,𝑛−4}+𝐶_𝑖𝑆_{𝑖+3,𝑛−4}+𝑤_𝑖𝑆_{𝑛−1,𝑛−4}+𝑣_𝑖𝑆_{𝑛,𝑛−4}𝛼_𝑛−4⁻¹,𝑖=𝑛−5(−1)1. (13)

The remaining (𝑛−5) columns are obtained by using the fact 𝐻𝐻⁻¹=𝐼_𝑛 where 𝐼_𝑛 is the identity matrix. They are as in the following: Col_𝑛−5=𝐸_𝑛−2−𝐴_𝑛−4Col_𝑛−4−𝑎_𝑛−3Col_𝑛−3−𝑑_𝑛−2Col_𝑛−2−𝑏_𝑛−1Col_𝑛−1−𝐵_𝑛Col_𝑛𝐶_𝑛−5⁻¹,Col_𝑖=𝐸_𝑖+3−𝐴_𝑖+1Col_𝑖+1−𝑎_𝑖+2Col_𝑖+2−𝑑_𝑖+3Col_𝑖+3−𝑏_𝑖+4Col_𝑖+4−𝐵_𝑖+5Col_𝑖+5−𝐷_𝑖+6Col_𝑖+6𝐶_𝑖⁻¹,𝑖=𝑛−6(−1)1, (14)(15) where 𝐸_𝑖 is the 𝑖th unit vector.

Remark 2. Equations (14) and (15) suggest an additional assumption ∏^𝑛−5_𝑖=1𝐶_𝑖≠0, which is only formal and can be omitted by introducing auxiliary parameter 𝑡 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 𝑛×𝑛 inverse matrix of the general cyclic heptadiagonal matrix 𝐻 in (1) by using the relations (13)–(15). INPUT. Order of the matrix 𝑛 and the components,       𝐷_𝑖,𝐵_𝑖,𝑏_𝑖,𝑑_𝑖,𝑎_𝑖,𝐴_𝑖,𝐶_𝑖,𝑖=1,2,…,𝑛,      where 𝐷₁=𝐷₂=𝐷₃=𝐶_𝑛−2=𝐶_𝑛−1=𝐶_𝑛=0.  OUTPUT. Inverse matrix.

Step 1. If 𝐶_𝑖=0 for any 𝑖=1,2,…,𝑛−5 set 𝐶_𝑖=𝑡 (𝑡 is just a symbolic name).

Step 2. If 𝐵_𝑖=0 for any 𝑖=6,7,…,𝑛,𝐵_𝑖=𝑡.

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

Step 4. For 𝑖=1,2,…,𝑛, compute and simplify the components 𝑆_𝑖,𝑛,𝑆_{𝑖,𝑛−1},𝑆_{𝑖,𝑛−2}, 𝑆_{𝑖,𝑛−3}, and 𝑆_{𝑖,𝑛−4} of the columns 𝐶_𝑗,𝑗=𝑛,𝑛−1,𝑛−2,𝑛−3, and 𝑛−4, respectively, by using (13).

Step 5. For 𝑖=1,2,…,𝑛, compute and simplify the components 𝑆_{𝑖,𝑛−5} by using (14).

Step 6. For 𝑗=𝑛−6,𝑛−7,…,1, do For 𝑖=1,2,…,𝑛, do  Compute and simplify the components 𝑆_𝑖,𝑗 by using (15). End do End do

Step 7. Substitute the actual value 𝑡=0 in all expressions to obtain the elements, 𝑆_𝑖,𝑗,𝑖,𝑗=1,2,…,𝑛.

The symbolic Algorithm 2 will be referred to as CHINV algorithm. The computational cost of CHINV algorithm is 21𝑛²−48𝑛−88 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 10×10 cyclic heptadiagonal linear system1−11−200002−1𝑥1111−10000121−112300002−2315−60000011111120000−1−1−1−1−1−110000222231−30000−2−21135300003134−12400002341¹𝑥₂𝑥₃𝑥₄𝑥₅𝑥₆𝑥₇𝑥₈𝑥₉𝑥₁₀=20153343−2447707894. (16)By using the coefficient matrix of the system (16) and applying the CHINV algorithm, we get the inverse of the coefficient matrix (𝐻⁻¹): 𝐻⁻¹=^{−12664−327152921−327151898109052957−3271523399−3271524419−327152069−65436676327151371432715631632715268632715416932715902−1090511183271580063271588013271541−65432366−32715639132715457132715541710905469310905344−3635241109055842109053292109052802181758−109052792−109051658109052931090553471090559136351461090513931090556981090585321811577−109051838−109051982109056344−32715496432715298310905159832715122593271594843271543365432621−327156374−32715167632715}_{93836353573635788−3635339363510233635513363556727297−3635413−363547736351178−10905908−10905254−3635604−1090513232−1090514012109053852181497−109051658109053718−109056356−109054969−1090513823635778109053539−109051306−10905922−218119041090558911090519410905163823271510738327153244−1090512163271514092327152783232715272565438078−3271511282−327155153−32715808−3271536173271511741090544327155947−327151868−327159386543465832715193−32715164332715}. (17)By using (17) and after simple calculations we can obtain the solution of cyclic heptadiagonal linear system (16): 𝑋=𝑥₁,𝑥₂,𝑥₃,𝑥₄,𝑥₅,𝑥₆,𝑥₇,𝑥₈,𝑥₉,𝑥₁₀^𝑇=(1,2,3,4,5,6,7,8,9,10)^𝑇. (18)

Example 2. Consider the 8×8 cyclic heptadiagonal matrix 𝐻=0−11−2002−11111−100121−1123002−2315−6000111111200−1−1−1−1−1−1300222232400−2−211. (19) By applying the CHINV algorithm, it yields𝐻⁻¹=438438𝑡+29831061438𝑡+2983402438𝑡+2983133438𝑡+2983−135438𝑡+29831361438𝑡+2983388438𝑡+2983−156438𝑡+2983−72()438𝑡+2983−92𝑡+332()438𝑡+298342𝑡+253438𝑡+29836𝑡+194()438𝑡+298324𝑡+1692()438𝑡+29836𝑡−71−()438𝑡+298378𝑡+595()438𝑡+2983126𝑡+43438𝑡+29836933()438𝑡+29833𝑡+580()438𝑡+2983−143𝑡−25−()438𝑡+29833𝑡−1903()438𝑡+298357𝑡+317()438𝑡+2983−925𝑡−69()438𝑡+2983−365𝑡+17()438𝑡+2983−123𝑡+41438𝑡+2983−1125438𝑡+2983321𝑡−539()438𝑡+2983274𝑡−11438𝑡+298339𝑡−76−()438𝑡+2983471𝑡+2861−()438𝑡+2983141𝑡+4456()438𝑡+2983256𝑡+15()438𝑡+29836𝑡+1216438𝑡+2983−()438𝑡+2983108𝑡+721438𝑡+298366𝑡+455438𝑡+298336𝑡+2472()438𝑡+298369𝑡+469438𝑡+298372𝑡+509−()438𝑡+298330𝑡+199−()438𝑡+29836𝑡+43438𝑡+2983334()438𝑡+2983−213𝑡−3162()438𝑡+298312𝑡+235−()438𝑡+298340𝑡+171()438𝑡+2983303𝑡+17−()438𝑡+298380𝑡−493()438𝑡+2983−416𝑡+35()438𝑡+2983−314𝑡+135438𝑡+298320−()438𝑡+2983214𝑡+1409()438𝑡+2983−89𝑡+59−()438𝑡+298326𝑡+171()438𝑡+2983−456𝑡+41()438𝑡+2983−598𝑡+655438𝑡+298346𝑡+331438𝑡+2983126𝑡+851438𝑡+2983729()438𝑡+29832𝑡+33()438𝑡+2983−242𝑡+253−()438𝑡+29836𝑡+193()438𝑡+2983114𝑡+769438𝑡+2983426𝑡+3125438𝑡+298378𝑡+595()438𝑡+2983−126𝑡+43438𝑡+2983_𝑡=0=43829831061298340272983−157135298313612983388−2983156−298372−2983297298350612983157676−2983142−2983595298351629836932983174029833502983101579512983621−2983612−2983492−29831125−2983539−2983297−42983−1572861−29834456298325298360562983−298372129834552983131579382983509−2983199−298343298333429836322983470−929831575102983493−2983140−2983405298320−29831409−2983472−92983−1571845−29833275298333129838512983722983297−2983506−129831572307298331252983595−2983516.2983 (20)

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.