Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 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

Correspondence should be addressed to A. A. Karawia; [email protected]

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

AcademicEditor:JoaoB.R.DoVal

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.

We describe a reliable symbolic computational algorithm for inverting general cyclic heptadiagonal matrices by using parallel 2 computing along with recursion. The computational cost of it is 21𝑛 − 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 heptadiagonal matrices of the form (1) and for solving linear systems of the form: The 𝑛×𝑛general cyclic heptadiagonal matrices take the form: 𝐻𝑋=𝑅, (2)

𝑇 𝑇 𝑑1 𝑎1 𝐴1 𝐶1 0 0 ⋅⋅⋅ 0 𝐵1 𝑏1 where 𝑋=(𝑥1,𝑥2, ..., 𝑥𝑛) and 𝑅=(𝑟1,𝑟2, ..., 𝑟𝑛) . To the best of our knowledge, the inversion of a general 𝑏 𝑑 𝑎 𝐴 𝐶 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 𝐵 ( 2 2 2 2 2 2 ) ( ) cyclic heptadiagonal matrix of the form (1)hasnotbeen ( ) ( 𝐵 𝑏 𝑑 𝑎 𝐴 𝐶 0 ⋅⋅⋅ ⋅⋅⋅ 0 ) considered. Very recently in [5], the inversion of a general ( 3 3 3 3 3 3 ) ( ) ( ) cyclic pentadiagonal matrix using recursion is studied with- ( ) ( . ) out imposing any restrictive conditions on the elements of ( 𝐷 𝐵 𝑏 𝑑 𝑎 𝐴 𝐶 0 ... . ) ( 4 4 4 4 4 4 4 . ) ( . ) thematrix.Also,inthispaperwearegoingtocompute ( . ) ( 0 dddddddd . ) the inverse of a general cyclic heptadiagonal matrix of the 𝐻=( ) , ( . . ) ( . . ) form (1) without imposing any restrictive conditions on the ( . dddddddd . ) ( 0 ⋅⋅⋅ 0 𝐷 𝐵 𝑏 𝑑 𝑎 𝐴 𝐶 ) elements of the matrix 𝐻 in (1). Our approach is mainly ( 𝑛−3 𝑛−3 𝑛−3 𝑛−3 𝑛−3 𝑛−3 𝑛−3 ) ( ) ( ) basedongettingtheelementsofthelastfivecolumnsof ( 0 ⋅⋅⋅ ⋅⋅⋅ 0 𝐷 𝐵 𝑏 𝑑 𝑎 𝐴 ) −1 ( 𝑛−2 𝑛−2 𝑛−2 𝑛−2 𝑛−2 𝑛−2) 𝐻 ( ) in suitable forms via the Doolittle LU factorization [10] ( ) (𝐴 0 ⋅⋅⋅ ⋅⋅⋅ 0 𝐷 𝐵 𝑏 𝑑 𝑎 ) along with parallel computation [7]. Then the elements of 𝑛−1 𝑛−1 𝑛−1 𝑛−1 𝑛−1 𝑛−1 −1 the remaining (𝑛 − 5) columns of 𝐻 may be obtained 𝑎 𝐴 0 ⋅⋅⋅ ⋅⋅⋅ 0 𝐷 𝐵 𝑏 𝑑 𝑛 𝑛 𝑛 𝑛 𝑛 𝑛 using relevant recursive relations. The inversion algorithm ( ) (1) 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 where 𝑛≥7. systems need to pivoting, for example Gaussian elimination The inverses of cyclic heptadiagonal matrices are usually algorithm [10–12]. Overall, pivoting adds more operations required in science and engineering applications, for more to the computational cost of an algorithm. These additional details, see special cases, [1–9]. The motivation of the current operations are sometimes necessary for the algorithm to work paper is to establish efficient algorithms for inverting cyclic at all. 2 Mathematical Problems in Engineering

𝐴 { 𝑛−1 The paper is organized as follows. In Section 2,new { , if 𝑖=1, { 𝛼1 symbolic computational algorithm, that will not break, is { 𝑘 𝑔 { 1 1 constructed. In Section 3, two illustrative examples are given. { , if 𝑖=2, { 𝛼2 Conclusions of the work are given in Section 4. { 𝑘 𝑔 +𝑘 𝑍 { 2 2 1 1 { , if 𝑖=3, { 𝛼3 { 𝑘 𝑔 +𝑘 𝑍 +𝑘 𝐶 { 𝑖−1 𝑖−1 𝑖−2 𝑖−2 𝑖−3 𝑖−3 2. Main Results { , if 𝑖=4(1) 𝑛−5, 𝑘 = 𝛼 𝑖 { 𝑖 { In this section we will focus on the construction of new { 𝐷𝑛−1 −𝑘𝑛−5𝑔𝑛−5 −𝑘𝑛−6𝑍𝑛−6 −𝑘𝑛−7𝐶𝑛−7 { , 𝑖=𝑛−4, { 𝛼 if symbolic computational algorithms for computing the deter- { 𝑛−4 { minant and the inverse of general cyclic heptadiagonal { 𝐵 −𝑘 𝑔 −𝑘 𝑍 −𝑘 𝐶 { 𝑛−1 𝑛−4 𝑛−4 𝑛−5 𝑛−5 𝑛−6 𝑛−6 , 𝑖=𝑛−3, matrices. The solution of cyclic heptadiagonal linear systems { 𝛼 if { 𝑛−3 of the form (2) will be taken into account. Firstly we begin { { 𝑏 −𝑘 𝑔 −𝑘 𝑍 −𝑘 𝐶 with computing the 𝐿𝑈 factorization of the matrix 𝐻.Itisas { 𝑛−1 𝑛−3 𝑛−3 𝑛−4 𝑛−4 𝑛−5 𝑛−5 , 𝑖=𝑛−2, 𝛼 if in the following: { 𝑛−2 𝑎 { 𝑛 , 𝑖=1, { 𝛼 if { 1 𝐻=𝐿𝑈, (3) { (ℎ 𝑔 −𝐴 ) { 1 1 𝑛 {− , if 𝑖=2, { 𝛼2 { (ℎ 𝑔 +ℎ 𝑍 ) where { 2 2 1 1 {− , if 𝑖=3, { 𝛼 { 3 1 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 0 { (ℎ𝑖−1𝑔𝑖−1 +ℎ𝑖−2𝑍𝑖−2 +ℎ𝑖−3𝐶𝑖−3) {− , if 𝑖=4(1) 𝑛−4, 𝑓2 1 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 0 𝛼 ℎ𝑖 ={ 𝑖 𝑒3 𝑓3 1 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 0 { ( 𝐷 ) { 𝐷𝑛 −ℎ𝑛−4𝑔𝑛−4 −ℎ𝑛−5𝑍𝑛−5 −ℎ𝑛−6𝐶𝑛−6 ( 4 𝑒 𝑓 1 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 0) { , if 𝑖=𝑛−3, ( 4 4 ) { 𝛼 ( 𝛼1 ) { 𝑛−3 ( ) { ( . .) { 𝐵𝑛 −ℎ𝑛−3𝑔𝑛−3 −ℎ𝑛−4𝑍𝑛−4 −ℎ𝑛−5𝐶𝑛−5 ( . dd d d d d d d .) { , 𝑖=𝑛−2, ( . .) { 𝛼 if ( ) { 𝑛−2 𝐿=( . .) , { ( . dd d d d d d d .) { 𝑏 −∑𝑛−2 ℎ 𝑤 ( 𝐷 ) { 𝑛 𝑗=1 𝑗 𝑗 ( 0⋅⋅⋅0 𝑛−3 𝑒 𝑓 1 dd0) { , if 𝑖=𝑛−1, ( 𝑛−3 𝑛−3 ) { 𝛼 ( 𝛼𝑛−6 ) 𝑛−1 ( 𝐷 ) ( 0 ⋅⋅⋅ ⋅⋅⋅ 0 𝑛−2 𝑒 𝑓 1 d 0) 𝑏 , 𝑖=1, ( 𝑛−2 𝑛−2 ) { 1 if 𝛼𝑛−5 {−𝑓 V +𝐵 , 𝑖=2, 𝑘 𝑘 𝑘 ⋅⋅⋅ ⋅⋅⋅ 𝑘 𝑘 𝑘 10 { 2 1 2 if 1 2 3 𝑛−2 𝑛−1 𝑛−2 { ℎ ℎ ℎ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ℎ ℎ ℎ 1 {−𝑓3V2 −𝑒3V1, if 𝑖=3, 1 2 3 𝑛−3 𝑛−2 𝑛−1 { { 𝐷𝑖 ( ) {−𝑓𝑖V𝑖−1 −𝑒𝑖V𝑖−2 − V𝑖−3, if 𝑖=4(1) 𝑛−4, { 𝛼 (4) { 𝑖−3 { 𝐷𝑛−3 V = −𝑓𝑛−3V𝑛−4 −𝑒𝑛−3V𝑛−5 − V𝑛−6 +𝐶𝑛−3, if 𝑖=𝑛−3, 𝛼 𝑔 𝑍 𝐶 00⋅⋅⋅0 𝑤 V 𝑖 { 𝛼 1 1 1 1 1 1 { 𝑛−6 0𝛼2 𝑔2 𝑍2 𝐶2 0⋅⋅⋅0 𝑤2 V2 { { 𝐷𝑛−2 00𝛼3 𝑔3 𝑍3 𝐶3 ⋅⋅⋅ 0 𝑤3 V3 {−𝑓 V −𝑒 V − V +𝐴 , 𝑖=𝑛−2, ( 000𝛼𝑔 𝑍 d 0𝑤 V ) { 𝑛−2 𝑛−3 𝑛−2 𝑛−4 𝛼 𝑛−5 𝑛−2 if ( 4 4 4 4 4 ) { 𝑛−5 ( ) { ( . . ) { 𝑛−2 ( . ddddd d d d . ) { ( ) {𝑎𝑛−1 − ∑𝑘𝑗V𝑗, if 𝑖=𝑛−1, 𝑈=( . . ) . (5) ( . ddddd d d d . ) { 𝑗=1 ( . . ) ( 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 0 𝛼𝑛−3 𝑔𝑛−3 𝑤𝑛−3 V𝑛−3) 𝐵 , 𝑖=1, ( 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 0 𝛼 𝑤 V ) { 1 if 𝑛−2 𝑛−2 𝑛−2 {−𝑓 𝑤 , 𝑖=2, 0 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 0 𝛼 V { 2 1 if 𝑛−1 𝑛−1 {−𝑓 𝑤 −𝑒 𝑤 , 𝑖=3, 0 0 0 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 0 𝛼 { 3 2 3 1 if 𝑛 { 𝐷 ( ) {−𝑓𝑤 −𝑒𝑤 − 𝑖 𝑤 , 𝑖=4(1) 𝑛−5, { 𝑖 𝑖−1 𝑖 𝑖−2 𝑖−3 if { 𝛼𝑖−3 { 𝐷 𝑤 = −𝑓 𝑤 −𝑒 𝑤 − 𝑛−4 𝑤 +𝐶 , 𝑖=𝑛−4, The elements in the matrices L and U in (4)and(5)satisfy 𝑖 { 𝑛−4 𝑛−5 𝑛−4 𝑛−6 𝛼 𝑛−7 𝑛−4 if { 𝑛−7 { { 𝐷 {−𝑓 𝑤 −𝑒 𝑤 − 𝑛−3 𝑤 +𝐴 , 𝑖=𝑛−3, { 𝑛−3 𝑛−4 𝑛−3 𝑛−5 𝛼 𝑛−6 𝑛−3 if 𝑑 , 𝑖=1, { 𝑛−6 { 1 if { {𝑑 −𝑓𝑔 , 𝑖=2, { 𝐷𝑛−2 { 2 2 1 if −𝑓 𝑤 −𝑒 𝑤 − 𝑤 +𝑎 , 𝑖=𝑛−2, {𝑑 −𝑓𝑔 −𝑒 𝑍 , 𝑖=3, { 𝑛−2 𝑛−3 𝑛−2 𝑛−4 𝛼 𝑛−5 𝑛−2 if { 3 3 2 3 1 if 𝑛−5 { 𝐷−𝑖 {𝑑 −𝑓𝑔 −𝑒𝑍 − 𝐶 , 𝑖=4(1) 𝑛−2, 𝑏2 { 𝑖 𝑖 𝑖−1 𝑖 𝑖−2 𝛼 𝑖−3 if { , if 𝑖=2, 𝛼 = 𝑖−3 { 𝛼 𝑖 { 𝑛−2 { 1 { { {𝑑𝑛−1 − ∑𝑤𝑗𝑘𝑗, if 𝑖=𝑛−1, { (𝑏 −𝑒 𝑔 ) { 3 3 1 , 𝑖=3, { 𝑗=1 𝑓𝑖 = { if { 𝑛−1 { 𝛼2 { { {𝑑 − ∑V ℎ , 𝑖=𝑛, { (𝑏 −𝑒𝑔 −(𝐷/𝛼 )𝑍 ) 𝑛 𝑗 𝑗 if { 𝑖 𝑖 𝑖−2 𝑖 𝑖−3 𝑖−3 { 𝑗=1 , if 𝑖=4(1) 𝑛−2, { 𝛼𝑖−1 Mathematical Problems in Engineering 3

𝐵3 { , 𝑖=3, Set 𝑘3 =−(𝑘1 ∗𝑍1 +𝑘2 ∗𝑔2)/𝛼3 { 𝛼 if 𝑒 = 1 𝑖 { (𝐵 −(𝐷/𝛼 )𝑔 ) ℎ =−(ℎ ∗𝑍 +ℎ ∗𝑔 )/𝛼 { 𝑖 𝑖 𝑖−3 𝑖−3 Set 3 1 1 2 2 3 , if 𝑖=4(1) 𝑛−2, { 𝛼𝑖−2 Set V3 =−𝑒3 ∗ V1 −𝑓3 ∗ V2

Set 𝑤3 =−𝑓3 ∗𝑤2 −𝑒3 ∗𝑤1 {𝑎1, if 𝑖=1, 𝑔𝑖 = {𝑎2 −𝑓2𝑍1, if 𝑖=2, {𝑎𝑖 −𝑓𝑖𝑍𝑖−1 −𝑒𝑖𝐶𝑖−2, if 𝑖=3(1) 𝑛−3, Step 2. Compute and simplify. 𝑖 4 𝑛−2 𝐴1, if 𝑖=1, For from to do 𝑍𝑖 ={ 𝐴𝑖 −𝑓𝑖𝐶𝑖−1, if 𝑖=2(1) 𝑛−4. 𝑒 =(𝐵 −𝐷 ∗𝑔 /𝛼 )/𝛼 (6) 𝑖 𝑖 𝑖 𝑖−3 𝑖−3 𝑖−2 𝑓𝑖 =(𝑏𝑖 −𝐷𝑖 ∗𝑍𝑖−3/𝛼𝑖−3 −𝑒𝑖 ∗𝑔𝑖−2)/𝛼𝑖−1 We also have: 𝑍𝑖 −2=𝐴𝑖−2 −𝑓𝑖−2 ∗𝐶𝑖−3 𝑛 𝑔𝑖−1 =𝑎𝑖−1 −𝑓𝑖−1 ∗𝑍𝑖−2 −𝑒𝑖−1 ∗𝐶𝑖−3 Det (𝐻) = ∏𝛼𝑖. (7) 𝑖=1 𝛼𝑖 =𝑑𝑖 −𝐷𝑖 ∗𝐶𝑖−3/𝛼𝑖−3 −𝑒𝑖 ∗𝑔𝑖−2 −𝑓𝑖 ∗𝑔𝑖−1

Remark 1. It is not difficult to prove that the LU decomposi- If 𝛼𝑖 =0then 𝛼𝑖 =𝑡end if tion (3) exists only if 𝛼𝑖 ≠ 0, 𝑖 = 1(1)𝑛 −1 (pivoting elements). Moreover the cyclic heptadiagonal matrix 𝐻 of the form (1) End do has an inverse if, in addition, 𝛼𝑛 =0̸ .Pivotingcanbeomitted by introducing auxiliary parameter 𝑡 in Algorithm 1 given Step 3. Compute and simplify. later. So no pivoting is included in our algorithm. For 𝑖 from 4 to 𝑛−5do

At this point it is convenient to formulate our first result. 𝑘 =−(𝑘 ∗𝐶 +𝑘 ∗𝑍 +𝑘 ∗𝑔 )/𝛼 It is a symbolic algorithm for computing the determinant 𝑖 𝑖−3 𝑖−3 𝑖−2 𝑖−2 𝑖−1 𝑖−1 𝑖 of a cyclic heptadiagonal matrix 𝐻 of the form (1)and 𝑤𝑖 =−(𝐷𝑖 ∗𝑤𝑖−3/𝛼𝑖−3 +𝑒𝑖 ∗𝑤𝑖−2 +𝑓𝑖 ∗𝑤𝑖−1) can be considered as natural generalization of the symbolic algorithm DETCPENTA in [5]. End do

Algorithm 1. To compute Det(𝐻) for the cyclic heptadiagonal 𝐻 matrix in (1), we may proceed as follows. Step 4. Compute and simplify.

Step 1. Set 𝛼1 =𝑑1 For 𝑖 from 4 to 𝑛−4do If 𝛼1 =0then 𝛼1 =𝑡(𝑡is just a symbolic name) end if ℎ𝑖 =−(ℎ𝑖−3 ∗𝐶𝑖−3 +ℎ𝑖−2 ∗𝑍𝑖−2 +ℎ𝑖−1 ∗𝑔𝑖−1)/𝛼𝑖 Set 𝑔1 =𝑎1, 𝑍1 =𝐴1 V𝑖 =−(𝐷𝑖 ∗ V𝑖−3/𝛼𝑖−3 +𝑒𝑖 ∗ V𝑖−2 +𝑓𝑖 ∗ V𝑖−1) Set 𝑘1 =𝐴𝑛−1/𝛼1 Set V1 =𝑏1 End do Set 𝑤1 =𝐵1 Step 5. Compute simplify. Set ℎ1 =𝑎𝑛/𝛼1

Set 𝑤1 =𝐵1 𝑘𝑛−4 =(𝐷𝑛−1−𝑘𝑛−5∗𝑔𝑛−5−𝑘𝑛−6∗𝑍𝑛−6−𝑘𝑛−7∗𝐶𝑛−7)/𝛼𝑛−4 Set 𝑓2 =𝑏2/𝛼1 𝑘𝑛−3 =(𝐵𝑛−1−𝑘𝑛−4∗𝑔𝑛−4−𝑘𝑛−5∗𝑍𝑛−5−𝑘𝑛−6∗𝐶𝑛−6)/𝛼𝑛−3 Set 𝑒3 =𝐵3/𝛼1 𝑘𝑛−2 =(𝑏𝑛−1−𝑘𝑛−3∗𝑔𝑛−3−𝑘𝑛−4∗𝑍𝑛−4−𝑘𝑛−5∗𝐶𝑛−5)/𝛼𝑛−2 Set 𝛼2 =𝑑2 −𝑓2 ∗𝑔1 𝑤𝑛−4 =𝐶𝑛−4−𝐷𝑛−4∗𝑤𝑛−7/𝛼𝑛−7−𝑒𝑛−4∗𝑤𝑛−6−𝑓𝑛−4∗𝑤𝑛−5 If 𝛼2 =0then 𝛼2 =𝑡end if 𝑤𝑛−3 =𝐴𝑛−3−𝐷𝑛−3∗𝑤𝑛−6/𝛼𝑛−6−𝑒𝑛−3∗𝑤𝑛−5−𝑓𝑛−3∗𝑤𝑛−4 Set 𝑘2 =−𝑘1 ∗𝑔1/𝛼2 𝑤𝑛−2 =𝑎𝑛−2−𝐷𝑛−2∗𝑤𝑛−5/𝛼𝑛−5−𝑒𝑛−2∗𝑤𝑛−4−𝑓𝑛−2∗𝑤𝑛−3 Set V2 =𝐵2 −𝑓2 ∗ V1 ℎ𝑛−3 =(𝐷𝑛−ℎ𝑛−4∗𝑔𝑛−4−ℎ𝑛−5∗𝑍𝑛−5−ℎ𝑛−6∗𝐶𝑛−6)/𝛼𝑛−3 Set 𝑤2 =−𝑓2 ∗𝑤1 ℎ𝑛−2 =(𝐵𝑛−ℎ𝑛−3∗𝑔𝑛−3−ℎ𝑛−4∗𝑍𝑛−4−ℎ𝑛−5∗𝐶𝑛−5)/𝛼𝑛−2 Set ℎ2 =(𝐴𝑛 −ℎ1 ∗𝑔1)/𝛼2 V𝑛−3 =𝐶𝑛−3 −𝐷𝑛−3 ∗V𝑛−6/𝛼𝑛−6 −𝑒𝑛−3 ∗V𝑛−5 −𝑓𝑛−3 ∗V𝑛−4 Set 𝑔2 =𝑎2 −𝑓2 ∗𝑍1 V =𝐴 −𝐷 ∗V /𝛼 −𝑒 ∗V −𝑓 ∗V 𝑓 =(𝑏 −𝑒 ∗𝑔 )/𝛼 𝑛−2 𝑛−2 𝑛−2 𝑛−5 𝑛−5 𝑛−2 𝑛−4 𝑛−2 𝑛−3 Set 3 3 3 1 2 𝑛−2 V𝑛−1 =𝑎𝑛−1 −∑𝑗=1 𝑘𝑗V𝑗 Set 𝛼3 =𝑑3 −𝑒3 ∗𝑍1 −𝑓3 ∗𝑔2 𝑛−2 If 𝛼3 =0then 𝛼3 =𝑡end if 𝛼𝑛−1 =𝑑𝑛−1 −∑𝑗=1 𝑤𝑗𝑘𝑗 4 Mathematical Problems in Engineering

00000 If 𝛼𝑛−1 =0then 𝛼𝑛−1 =𝑡end if 00000 ℎ =(𝑏 − ∑𝑛−2 ℎ 𝑤 )/𝛼 (. . . . .) 𝑛−1 𝑛 𝑗=1 𝑗 𝑗 𝑛−1 (. . . . .) ( ) (. . . . .) (. . . . .) 𝛼 =𝑑 − ∑𝑛−1 V ℎ ( ) 𝑛 𝑛 𝑗=1 𝑗 𝑗 = (. . . . .) , (. . . . .) (. . . . .) (00001) If 𝛼𝑛 =0then 𝛼𝑛 =𝑡end if ( ) (00010) 𝑛 00100 Step 6. Compute Det(𝐻) =∏ ( 𝛼𝑖)𝑡=0. 𝑖=1 01000 The symbolic Algorithm 1 will be referred to as 10000 DETCHEPTA. The computational cost of this algorithm ( ) is 52𝑛 − 195 operations. The new algorithm DETCHEPTA (9) is very useful to check the nonsingularity of the matrix 𝐻 we get when we consider, for example, the solution of the cyclic (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4) heptadiagonal linear systems of the form (2). 𝑄1 𝑄1 𝑄1 𝑄1 𝑄1 Now, when the matrix 𝐻 is nonsingular, its inversion is (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4) ( 𝑄2 𝑄2 𝑄2 𝑄2 𝑄2 ) ( ) computed as follows. ( ) ( ) Let ( . . . . . ) ( . . . . . ) ( ) ( . . . . . ) 𝐻−1 =[𝑆 ] =[ , ,..., ], ( . . . . . ) 𝑖𝑗 1≤𝑖,𝑗≤𝑛 Col1 Col2 Col𝑛 (8) ( ) ( . . . . . ) ( . . . . . ) ( ) (𝑄(𝑛) 𝑄(𝑛−1) 𝑄(𝑛−2) 𝑄(𝑛−3) 𝑄(𝑛−4)) 𝑚 𝐻−1 𝑚=1,2,...,𝑛 ( 𝑛−4 𝑛−4 𝑛−4 𝑛−4 𝑛−4 ) where Col𝑚 denotes th column of , . ( ) 𝐿𝑈 𝐻 ( (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4)) Since the Doolittle factorization of the matrix in (𝑄 𝑄 𝑄 𝑄 𝑄 ) ( 𝑛−3 𝑛−3 𝑛−3 𝑛−3 𝑛−3 ) (1)isalwayspossiblethenwecanuseparallelcomputa- ( ) = ( (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4)) tionstogettheelementsofthelastfivecolumnsCol𝑖 (𝑄 𝑄 𝑄 𝑄 𝑄 ) (𝑆 ,𝑆 ,...,𝑆 )𝑇 𝑖=𝑛,𝑛−1,𝑛−2,𝑛−3 𝑛−4 𝐻−1 ( 𝑛−2 𝑛−2 𝑛−2 𝑛−2 𝑛−2 ) 1,𝑖 2,𝑖 𝑛,𝑖 , ,and of ( ) as follows [5]. (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4) 𝑄𝑛−1 𝑄𝑛−1 𝑄𝑛−1 𝑄𝑛−1 𝑄𝑛−1 (10) Solving in parallel the standard linear systems whose coefficient matrix 𝐿 is given by (4) (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4) ( 𝑄𝑛 𝑄𝑛 𝑄𝑛 𝑄𝑛 𝑄𝑛 )

00 0 0 0 𝑄(𝑛) 𝑄(𝑛−1) 𝑄(𝑛−2) 𝑄(𝑛−3) 𝑄(𝑛−4) 1 1 1 1 1 00 0 0 0 (. . . . . ) ( 𝑄(𝑛) 𝑄(𝑛−1) 𝑄(𝑛−2) 𝑄(𝑛−3) 𝑄(𝑛−4)) (. . . . . ) ( 2 2 2 2 2 ) ( ) ( ) (. . . . . ) ( ) (. . . . . ) ( ) ( ) ( . . . . . ) = . . . . . , ( . . . . . ) (. . . . . ) ( . . . . . ) (. . . . . ) ( ) (00 0 0 1) ( ) ( ) ( ) (00 0 1−𝑓) ( . . . . . ) ( 𝑛−3) ( . . . . . ) 00 1−𝑓 𝑀 ( . . . . . ) 𝑛−2 6 ( ) 01−𝑘 𝑀 𝑀 ( ) 𝑛−2 4 5 ( ) (1−ℎ 𝑀 𝑀 𝑀 ) ( . . . . . ) 𝑛−1 1 2 3 ( . . . . . ) 𝐿 ( ) where ( ) ( ) ( (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4)) 𝑀1 =−ℎ𝑛−2 +ℎ𝑛−1𝑘𝑛−2, (𝑄𝑛−4 𝑄𝑛−4 𝑄𝑛−4 𝑄𝑛−4 𝑄𝑛−4 ) ( ) ( ) 𝑀 =−ℎ +ℎ 𝑓 ( ) 2 𝑛−3 𝑛−2 𝑛−2 (𝑄(𝑛) 𝑄(𝑛−1) 𝑄(𝑛−2) 𝑄(𝑛−3) 𝑄(𝑛−4)) ( 𝑛−3 𝑛−3 𝑛−3 𝑛−3 𝑛−3 ) +ℎ 𝑘 −ℎ 𝑘 𝑓 , ( ) 𝑛−1 𝑛−3 𝑛−1 𝑛−2 𝑛−2 ( ) ( (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4)) (𝑄 𝑄 𝑄 𝑄 𝑄 ) 𝑀3 =−ℎ𝑛−4 +ℎ𝑛−3𝑓𝑛−3 +ℎ𝑛−2𝑒𝑛−2 ( 𝑛−2 𝑛−2 𝑛−2 𝑛−2 𝑛−2 ) ( ) ( ) −ℎ𝑛−2𝑓𝑛−2𝑓𝑛−3 +ℎ𝑛−1𝑘𝑛−4 ( (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4)) 𝑄𝑛−1 𝑄𝑛−1 𝑄𝑛−1 𝑄𝑛−1 𝑄𝑛−1 −ℎ𝑛−1𝑘𝑛−3𝑓𝑛−3 −ℎ𝑛−1𝑘𝑛−2𝑒𝑛−2 (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4) ( 𝑄𝑛 𝑄𝑛 𝑄𝑛 𝑄𝑛 𝑄𝑛 ) +ℎ𝑛−1𝑘𝑛−2𝑓𝑛−2𝑓𝑛−3, Mathematical Problems in Engineering 5

(𝑔 𝑆 +𝑍𝑆 +𝐶𝑆 +𝑤𝑆 + V 𝑆 ) 𝑀4 =−𝑘𝑛−3 +𝑘𝑛−2𝑓𝑛−2, 𝑖 𝑖+1,𝑛 𝑖 𝑖+2,𝑛 𝑖 𝑖+3,𝑛 𝑖 𝑖−1,𝑛 𝑖 𝑛,𝑛 𝑆𝑖,𝑛 =− , 𝛼𝑖 𝑀5 =−𝑘𝑛−4 +𝑘𝑛−3𝑓𝑛−3 𝑖=𝑛−5(−1) 1, +𝑘𝑛−2𝑒𝑛−2 −𝑘𝑛−2𝑓𝑛−2𝑓𝑛−3, ℎ 𝑆 =− 𝑛−1 , 𝑀6 =−𝑒𝑛−2 +𝑓𝑛−2𝑓𝑛−3. 𝑛,𝑛−1 𝛼𝑛 (11) (1 − V𝑛−1𝑆𝑛,𝑛−1) 𝑆𝑛−1,𝑛−1 = , 𝛼𝑛−1 𝑆 =−(𝑔𝑆 +𝑍𝑆 +𝐶𝑆 Hence, solving the following standard linear systems whose 𝑖,𝑛−1 𝑖 𝑖+1,𝑛−1 𝑖 𝑖+2,𝑛−1 𝑖 𝑖+3,𝑛−1 𝑈 −1 coefficient matrix is given by (5) +𝑤𝑖𝑆𝑖−1,𝑛−1 + V𝑖𝑆𝑛,𝑛−1)(𝛼𝑖) , 𝑖=𝑛−5(−1) 1,

(−ℎ𝑛−2 +ℎ𝑛−1𝑘𝑛−2) 𝑆1,𝑛 𝑆1,𝑛−1 𝑆1,𝑛−2 𝑆1,𝑛−3 𝑆1,𝑛−4 𝑆𝑛,𝑛−2 = , 𝛼𝑛 𝑆2,𝑛 𝑆2,𝑛−1 𝑆2,𝑛−2 𝑆2,𝑛−3 𝑆2,𝑛−4 (𝑘 + V 𝑆 ) ( . . . . . ) 𝑛−2 𝑛−1 𝑛,𝑛−2 ( . . . . . ) 𝑆𝑛−1,𝑛−2 =− , ( ) 𝛼 ( . . . . . ) 𝑛−1 ( . . . . . ) ( ) (1 − 𝑤 𝑆 − V 𝑆 ) ( . . . . . ) 𝑆 = 𝑛−2 𝑛−1,𝑛−2 𝑛−2 𝑛,𝑛−2 , 𝑈 ( . . . . . ) 𝑛−2,𝑛−2 𝛼 ( ) 𝑛−2 (𝑆𝑛−4,𝑛 𝑆𝑛−4,𝑛−1 𝑆𝑛−4,𝑛−2 𝑆𝑛−4,𝑛−3 𝑆𝑛−4,𝑛−4) ( ) 𝑆𝑖,𝑛−2 =−(𝑔𝑖𝑆𝑖+1,𝑛−2 +𝑍𝑖𝑆𝑖+2,𝑛−2 +𝐶𝑖𝑆𝑖+3,𝑛−2 (𝑆𝑛−3,𝑛 𝑆𝑛−3,𝑛−1 𝑆𝑛−3,𝑛−2 𝑆𝑛−3,𝑛−3 𝑆𝑛−3,𝑛−4) ( ) −1 (𝑆𝑛−2,𝑛 𝑆𝑛−2,𝑛−1 𝑆𝑛−2,𝑛−2 𝑆𝑛−2,𝑛−3 𝑆𝑛−2,𝑛−4) +𝑤𝑖𝑆𝑖−1,𝑛−2 + V𝑖𝑆𝑛,𝑛−2)(𝛼𝑖) , 𝑆𝑛−1,𝑛 𝑆𝑛−1,𝑛−1 𝑆𝑛−1,𝑛−2 𝑆𝑛−1,𝑛−3 𝑆𝑛−1,𝑛−4 𝑖=𝑛−5(−1) 1, 𝑆𝑛,𝑛 𝑆𝑛,𝑛−1 𝑆𝑛,𝑛−2 𝑆𝑛,𝑛−3 𝑆𝑛,𝑛−4 ( ) (−ℎ𝑛−3 +ℎ𝑛−2𝑓𝑛−2 −ℎ𝑛−1 (𝑘𝑛−2𝑓𝑛−2 −𝑘𝑛−3)) 𝑆𝑛,𝑛−3 = , (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4) 𝛼𝑛 𝑄1 𝑄1 𝑄1 𝑄1 𝑄1 (𝑘 𝑓 −𝑘 − V 𝑆 ) (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4) (12) 𝑛−2 𝑛−2 𝑛−3 𝑛−1 𝑛,𝑛−3 ( 𝑄2 𝑄2 𝑄2 𝑄2 𝑄2 ) 𝑆𝑛−1,𝑛−3 = , ( ) 𝛼 ( ) 𝑛−1 ( . . . . . ) ( . . . . . ) (𝑓𝑛−2 +𝑤𝑛−2𝑆𝑛−1,𝑛−3 + V𝑛−2𝑆𝑛,𝑛−3) ( . . . . . ) 𝑆 =− , ( ) 𝑛−2,𝑛−3 𝛼 ( . . . . . ) 𝑛−2 ( . . . . . ) ( ) (1 − 𝑔 𝑆 −𝑤 𝑆 − V 𝑆 ) ( . . . . . ) 𝑆 = 𝑛−3 𝑛−2,𝑛−3 𝑛−3 𝑛−1,𝑛−3 𝑛−3 𝑛,𝑛−3 , ( . . . . . ) 𝑛−3,𝑛−3 = ( ) 𝛼𝑛−3 (𝑄(𝑛) 𝑄(𝑛−1) 𝑄(𝑛−2) 𝑄(𝑛−3) 𝑄(𝑛−4)) ( 𝑛−4 𝑛−4 𝑛−4 𝑛−4 𝑛−4 ) ( ) 𝑆𝑛,𝑛−4 ( (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4)) (𝑄 𝑄 𝑄 Q 𝑄 ) ( 𝑛−3 𝑛−3 𝑛−3 𝑛−3 𝑛−3 ) −ℎ𝑛−4 +ℎ𝑛−3𝑓𝑛−3 −ℎ𝑛−2 (𝑒𝑛−2 −𝑓𝑛−2𝑓𝑛−3) ( ) = ( ( (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4)) 𝛼 (𝑄 𝑄 𝑄 𝑄 𝑄 ) 𝑛 ( 𝑛−2 𝑛−2 𝑛−2 𝑛−2 𝑛−2 ) ( ) ℎ (𝑘 −𝑘 𝑓 −𝑘 (𝑒 −𝑓 𝑓 )) (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4) + 𝑛−1 𝑛−4 𝑛−3 𝑛−3 𝑛−2 𝑛−2 𝑛−2 𝑛−3 ), 𝑄𝑛−1 𝑄𝑛−1 𝑄𝑛−1 𝑄𝑛−1 𝑄𝑛−1 𝛼𝑛 (𝑛) (𝑛−1) (𝑛−2) (𝑛−3) (𝑛−4) ( 𝑄𝑛 𝑄𝑛 𝑄𝑛 𝑄𝑛 𝑄𝑛 ) 𝑆𝑛−1,𝑛−4 (−𝑘 +𝑘 𝑓 −𝑘 (𝑒 −𝑓 𝑓 )−V 𝑆 ) = 𝑛−4 𝑛−3 𝑛−3 𝑛−2 𝑛−2 𝑛−2 𝑛−3 𝑛−1 𝑛,𝑛−4 , 𝛼𝑛−1 gives the five columns Col𝑖, 𝑖=𝑛,𝑛−1,𝑛−2,𝑛−3,and𝑛−4 in the forms: ((𝑒𝑛−2 −𝑓𝑛−2𝑓𝑛−3)+𝑤𝑛−2𝑆𝑛−1,𝑛−4 + V𝑛−2𝑆𝑛,𝑛−4) 𝑆𝑛−2,𝑛−4 =− , 𝛼𝑛−2 (𝑓 +𝑔 𝑆 +𝑤 𝑆 + V 𝑆 ) 𝑆 =− 𝑛−3 𝑛−3 𝑛−2,𝑛−4 𝑛−3 𝑛−1,𝑛−4 𝑛−3 𝑛,𝑛−4 , 1 𝑛−3,𝑛−4 𝛼 𝑆𝑛,𝑛 = , 𝑛−3 𝛼𝑛 𝑆𝑛−4,𝑛−4 =(1−𝑔𝑛−4𝑆𝑛−3,𝑛−4 −𝑍𝑛−4𝑆𝑛−2,𝑛−4 V𝑛−1𝑆𝑛,𝑛 𝑆𝑛−1,𝑛 =− , −1 𝛼𝑛−1 −𝑤𝑛−4𝑆𝑛−1,𝑛−3 − V𝑛−4𝑆𝑛,𝑛−4)(𝛼𝑛−4) , 6 Mathematical Problems in Engineering

𝑆 =−(𝑔𝑆 +𝑍𝑆 +𝐶𝑆 𝑖,𝑛−4 𝑖 𝑖+1,𝑛−4 𝑖 𝑖+2,𝑛−4 𝑖 𝑖+3,𝑛−4 Step 1. If 𝐶𝑖 =0for any 𝑖 = 1,2,...,𝑛−5set 𝐶𝑖 =𝑡(𝑡 is just −1 a symbolic name). +𝑤𝑖𝑆𝑛−1,𝑛−4 + V𝑖𝑆𝑛,𝑛−4)(𝛼𝑛−4) , 𝐵 =0 𝑖=6,7,...,𝑛,𝐵 =𝑡 𝑖=𝑛−5(−1) 1. Step 2. If 𝑖 for any 𝑖 . (13) Step 3. Use the DETCHEPTA algorithm to check the non- singularity of the matrix 𝐻.Ifthematrix𝐻 is singular then The remaining (𝑛 − 5) columns are obtained by using the OUTPUT (“The matrix 𝐻 is singular”); stop. −1 fact 𝐻𝐻 =𝐼𝑛 where 𝐼𝑛 is the identity matrix. They are as in the following: Step 4. For 𝑖 = 1,2,...,𝑛, compute and simplify the 𝑆 ,𝑆 ,𝑆 𝑆 𝑆 components 𝑖,𝑛 𝑖,𝑛−1 𝑖,n−2, 𝑖,𝑛−3,and 𝑖,𝑛−4 of the columns Col𝑛−5 =(𝐸𝑛−2 −𝐴𝑛−4Col𝑛−4 𝐶𝑗,𝑗 = 𝑛,𝑛−1,𝑛−2,𝑛−3,and𝑛−4,respectively,byusing (13). −𝑎 −𝑑 𝑛−3Col𝑛−3 𝑛−2Col𝑛−2 (14) Step 5. For 𝑖 = 1,2,...,𝑛, compute and simplify the −1 components 𝑆𝑖,𝑛−5 by using (14). −𝑏𝑛−1Col𝑛−1 −𝐵𝑛Col𝑛)(𝐶𝑛−5) , Step 6. For 𝑗=𝑛−6,𝑛−7,...,1,do Col𝑖 =(𝐸𝑖+3 −𝐴𝑖+1Col𝑖+1 −𝑎𝑖+2Col𝑖+2

−𝑑𝑖+3Col𝑖+3 −𝑏𝑖+4Col𝑖+4 For 𝑖=1,2,...,𝑛,do −1 (15) −𝐵𝑖+5Col𝑖+5 −𝐷𝑖+6Col𝑖+6)(𝐶𝑖) , Compute and simplify the components 𝑆𝑖,𝑗 by using 𝑖=𝑛−6(−1) 1, (15). where 𝐸𝑖 is the ith unit vector. End do Remark 2. Equations (14)and(15) suggest an additional ∏𝑛−5𝐶 =0̸ assumption 𝑖=1 𝑖 , which is only formal and can be End do omitted by introducing auxiliary parameter 𝑡 in Algorithm 2 given later. Step 7. Substitute the actual value 𝑡=0in all expressions to obtain the elements, 𝑆𝑖,𝑗,𝑖,𝑗=1,2,...,𝑛. Now we formulate a second result. It is a symbolic computational algorithm to compute the inverse of a general The symbolic Algorithm 2 will be referred to as CHINV cyclic heptadiagonal matrix of the form (1) when it exists. algorithm. The computational cost of CHINV algorithm is 2 21𝑛 − 48𝑛 − 88 operations. The Algorithms 2.2, 2.3, and 2.2 𝑛×𝑛 Algorithm 2. To find the inverse matrix of the general in [5, 13, 14], respectively, are now special cases of the CHINV 𝐻 cyclic heptadiagonal matrix in (1) by using the relations algorithm. (13)–(15). INPUT. Order of the matrix 𝑛 and the components,

𝐷𝑖,𝐵𝑖,𝑏𝑖,𝑑𝑖,𝑎𝑖,𝐴𝑖,𝐶𝑖,𝑖=1,2,...,𝑛, 3. Two Illustrative Examples 𝐷 =𝐷 =𝐷 =𝐶 =𝐶 = where 1 2 3 𝑛−2 𝑛−1 Inthissectionwegivetwoexamplesforthesakeofillustra- 𝐶 =0 𝑛 . tion.

OUTPUT. Inverse matrix. Example 1. Consider the 10 × 10 cyclic heptadiagonal linear system

1−11−200 0 02−1 𝑥1 2 1111−100001 𝑥2 15 21−11230000 𝑥 33 ( ) ( 3 ) ( ) (2−2315−60000) ( 𝑥 ) ( 0 ) ( ) ( 4 ) ( ) (0111111200) ( 𝑥 ) ( 43 ) ( ) ( 5 ) = ( ) . (0 0 −1 −1 −1 −1 −1 −1 1 0 ) ( 𝑥 ) (−24) (16) ( ) ( 6 ) ( ) (000222231−3) ( 𝑥7 ) ( 47 ) 0000−2−21135 𝑥8 70 300003134−1 𝑥9 78 (2400002341) (𝑥10) ( 94 ) Mathematical Problems in Engineering 7

By using the coefficient matrix of the system (16) and applying the CHINV algorithm, we get the inverse of the coefficient −1 matrix (𝐻 ):

12664 2921 1898 2957 23399 24419 2069 6676 13714 6316 − − − − − − − 32715 32715 10905 32715 32715 32715 6543 32715 32715 32715 ( 2686 4169 902 1118 8006 8801 41 2366 6391 4571 ) ( ) ( − − − ) ( 32715 32715 10905 32715 32715 32715 6543 32715 32715 32715 ) ( ) ( ) ( 5417 4693 344 241 5842 3292 280 758 2792 1658 ) ( − − − ) ( 10905 10905 3635 10905 10905 10905 2181 10905 10905 10905) ( ) ( ) ( 293 5347 591 146 1393 5698 853 1577 1838 1982 ) ( − − ) ( ) ( 10905 10905 3635 10905 10905 10905 2181 10905 10905 10905) ( ) ( ) ( 6344 4964 2983 1598 12259 9484 433 2621 6374 1676 ) ( − − − ) ( 32715 32715 10905 32715 32715 32715 6543 32715 32715 32715) ( ) 𝐻−1 = ( ) . ( 938 357 788 339 1023 513 56 297 413 477 ) (17) ( − − − ) ( 3635 3635 3635 3635 3635 3635 727 3635 3635 3635 ) ( ) ( ) ( 1178 908 254 604 13232 14012 385 497 1658 3718 ) ( ) ( − − − − − − ) ( 10905 10905 3635 10905 10905 10905 2181 10905 10905 10905 ) ( ) ( ) ( 6356 4969 1382 778 3539 1306 922 1904 5891 194 ) (− − − − − − ) ( 10905 10905 3635 10905 10905 10905 2181 10905 10905 10905 ) ( ) ( ) ( 16382 10738 3244 1216 14092 27832 2725 8078 11282 5153 ) ( − − − ) ( ) ( 32715 32715 10905 32715 32715 32715 6543 32715 32715 32715) 808 3617 1174 44 5947 1868 938 4658 193 1643 − − − − − 32715 32715 10905 32715 32715 32715 6543 32715 32715 32715 ( )

By using (17) and after simple calculations we can obtain the Example 2. Consider the 8×8cyclic heptadiagonal matrix solution of cyclic heptadiagonal linear system (16):

0−11−20 0 2−1 1111−1001 21−112300 ( ) 𝑇 (2−2315−600) 𝑋=(𝑥1,𝑥2,𝑥3,𝑥4,𝑥5,𝑥6,𝑥7,𝑥8,𝑥9,𝑥10) 𝐻=( ) . (01111112) (19) 𝑇 0 0 −1−1−1−1−1−1 = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) . 30022223 (18) (2 4 0 0 −2 −2 1 1 ) 8 Mathematical Problems in Engineering

By applying the CHINV algorithm, it yields

438 1061 402 133 −135 1361 388 −156 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 ( ) ( −72 −9 (2𝑡 + 33) 2 (42𝑡 + 253) 6𝑡 + 19 4 (24𝑡 + 169) 2 (6𝑡 − 71) − (78𝑡 + 595) 12 (6𝑡 + 43) ) ( ) ( 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 ) ( ) ( ) ( 693 3 (3𝑡 + 580) −14 (3𝑡 − 25) − (3𝑡 − 190) 3 (57𝑡 + 317) −9 (25𝑡 − 69) −36 (5𝑡 + 17) −12 (3𝑡+41) ) ( ) ( 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 ) ( ) ( ) ( −1125 321𝑡 − 539 27 (4𝑡 − 11) 39𝑡 − 76 − (471𝑡 + 2861) − (141𝑡 + 4456) 25 (6𝑡 + 1) 5 (6𝑡 + 121) ) ( ) ( ) ( 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 ) 𝐻−1 = ( ) ( ) ( 6 − (108𝑡 + 721) 66𝑡 + 455 36𝑡 + 247 2 (69𝑡 + 469) 72𝑡 + 509 − (30𝑡 + 199) − (6𝑡 + 43) ) ( ) ( 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 ) ( ) ( ) ( 334 −2 (13𝑡 − 316) 2 (12𝑡 + 235) − (40𝑡 + 171) 30 (3𝑡 + 17) − (80𝑡 − 493) −4 (16𝑡 + 35) −3 (14𝑡 + 135) ) ( ) ( ) ( 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 ) ( ) ( ) ( 20 − (214𝑡 + 1409) −8 (9𝑡 + 59) − (26𝑡 + 171) −45 (6𝑡 + 41) −5 (98𝑡 + 655) 46𝑡 + 331 126𝑡 + 851 ) ( 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 ) 72 9 (2𝑡 + 33) −2 (42𝑡 + 253) − (6𝑡 + 19) 3 (114𝑡 + 769) 426𝑡 + 3125 78𝑡 + 595 −12 (6𝑡 + 43)

( 438 𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 438𝑡 + 2983 )𝑡=0 (20) 438 1061 402 7 135 1361 388 156 − − 2983 2983 2983 157 2983 2983 2983 2983 ( 72 297 506 1 676 142 595 516 ) ( ) (− − − − ) ( 2983 2983 2983 157 2983 2983 2983 2983 ) ( ) ( ) ( 693 1740 350 10 951 621 612 492 ) ( − − ) ( 2983 2983 2983 157 2983 2983 2983 2983 ) ( ) ( ) ( 1125 539 297 4 2861 4456 25 605 ) (− − − − − − ) ( 2983 2983 2983 157 2983 2983 2983 2983 ) ( ) = ( ) . ( 6 721 455 13 938 509 199 43 ) ( − − − ) ( ) ( 2983 2983 2983 157 2983 2983 2983 2983 ) ( ) ( ) ( 334 632 470 9 510 493 140 405 ) ( − − − ) ( 2983 2983 2983 157 2983 2983 2983 2983 ) ( ) ( ) ( 20 1409 472 9 1845 3275 331 851 ) ( − − − − − ) 2983 2983 2983 157 2983 2983 2983 2983 72 297 506 1 2307 3125 595 516 − − − ( 2983 2983 2983 157 2983 2983 2983 2983 )

4. Conclusions [4]M.Batista,“methodforsolvingcyclicblockpenta-diagonal systems of linear equations,” Mathematical Software,2008, In this work new recursive computational algorithms have http://arxiv.org/abs/0803.0874. been developed for computing the determinant and inverse [5] M. El-Mikkawy and E.-D. Rahmo, “Symbolic algorithm for of general cyclic heptadiagonal matrices and solving linear inverting cyclic pentadiagonal matrices recursively: derivation systems of cyclic heptadiagonal type. The algorithms are and implementation,” Computers & Mathematics with Applica- reliable, are computationally efficient, and will not fail. The tions,vol.59,no.4,pp.1386–1396,2010. algorithms are natural generalizations of some algorithms in [6]S.N.NeossiNguetchueandS.Abelman,“Acomputational current use. algorithm for solving nearly penta-diagonal linear systems,” Applied Mathematics and Computation,vol.203,no.2,pp.629– 634, 2008. References [7] T. Sogabe, “New algorithms for solving periodic tridiagonal and periodic pentadiagonal linear systems,” Applied Mathematics [1] A. A. Karawia, “Acomputational algorithm for solving periodic and Computation,vol.202,no.2,pp.850–856,2008. penta-diagonal linear systems,” Applied Mathematics and Com- putation,vol.174,no.1,pp.613–618,2006. [8] X.-G. Lv and J. Le, “A note on solving nearly penta-diagonal linear systems,” Applied Mathematics and Computation,vol. [2]A.D.A.HadjandM.Elouafi,“Afastnumericalalgorithmfor 204, no. 2, pp. 702–712, 2008. the inverse of a tridiagonal and pentadiagonal matrix,” Applied [9] X.-L. Zhao and T.-Z. Huang, “On the inverse of a general Mathematics and Computation,vol.202,no.2,pp.441–445, pentadiagonal matrix,” Applied Mathematics and Computation, 2008. vol. 202, no. 2, pp. 639–646, 2008. [3] I. M. Navon, “A periodic pentadiagonal systems solver,” Com- [10] J. W. Demmel, Applied Numerical Linear Algebra, Society for munications in Applied Numerical Methods,vol.3,pp.63–69, Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, 1987. USA, 1997. Mathematical Problems in Engineering 9

[11] G. Engeln-Mullges¨ and F. Uhlig, Numerical Algorithms with C, Springer, Berlin, Germany, 1996. [12] M. B. Allen III and E. L. Isaacson, Numerical Analysis For Applied Science,JohnWileyandSons,1997. [13] 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. [14] M. El-Mikkawy and E.-D. Rahmo, “A new recursive algo- rithm for inverting general periodic pentadiagonal and anti- pentadiagonal matrices,” Applied Mathematics and Computa- tion,vol.207,no.1,pp.164–170,2009. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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