# Inversion of General Cyclic Heptadiagonal Matrices

Authors: A. A. Karawia
Journal: Mathematical Problems in Engineering
 (2013)

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## 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 is21n2−48n−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.


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## Body

## 1. Introduction

Then×n general cyclic heptadiagonal matrices take the form: (1)H=(d1a1A1C100⋯0B1b1b2d2a2A2C20⋯⋯⋯B2B3b3d3a3A3C30⋯⋯0D4B4b4d4a4A4C40…⋮0⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮0⋯0Dn-3Bn-3bn-3dn-3an-3An-3Cn-30⋯⋯0Dn-2Bn-2bn-2dn-2an-2An-2An-10⋯⋯0Dn-1Bn-1bn-1dn-1an-1anAn0⋯⋯0DnBnbndn),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:
(2)HX=R,
where X=(x1,x2,...,xn)T and R=(r1,r2,...,rn)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 Section2, 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:
(3)H=LU,
where(4)L=(10⋯⋯⋯⋯⋯⋯⋯0f210⋯⋯⋯⋯⋯⋯0e3f310⋯⋯⋯⋯⋯0D4α1e4f410⋯⋯⋯⋯0⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮0⋯0Dn-3αn-6en-3fn-31⋱⋱00⋯⋯0Dn-2αn-5en-2fn-21⋱0k1k2k3⋯⋯kn-2kn-1kn-210h1h2h3⋯⋯⋯hn-3hn-2hn-11),(5)U=(α1g1Z1C100⋯0w1v10α2g2Z2C20⋯0w2v200α3g3Z3C3⋯0w3v3000α4g4Z4⋱0w4v4⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮⋮⋱⋱⋱⋱⋱⋱⋱⋱⋮0⋯⋯⋯⋯0αn-3gn-3wn-3vn-30⋯⋯⋯⋯⋯0αn-2wn-2vn-200⋯⋯⋯⋯⋯0αn-1vn-1000⋯⋯⋯⋯⋯0αn).The elements in the matricesL and U in (4) and (5) satisfy(6)αi={d1,ifi=1,d2-f2g1,ifi=2,d3-f3g2-e3Z1,ifi=3,di-figi-1-eiZi-2-D-iαi-3Ci-3,ifi=4(1)n-2,dn-1-∑j=1n-2‍wjkj,ifi=n-1,dn-∑j=1n-1‍vjhj,ifi=n,ki={An-1α1,ifi=1,k1g1α2,ifi=2,k2g2+k1Z1α3,ifi=3,ki-1gi-1+ki-2Zi-2+ki-3Ci-3αi,ifi=4(1)n-5,Dn-1-kn-5gn-5-kn-6Zn-6-kn-7Cn-7αn-4,ifi=n-4,Bn-1-kn-4gn-4-kn-5Zn-5-kn-6Cn-6αn-3,ifi=n-3,bn-1-kn-3gn-3-kn-4Zn-4-kn-5Cn-5αn-2,ifi=n-2,hi={anα1,ifi=1,-(h1g1-An)α2,ifi=2,-(h2g2+h1Z1)α3,ifi=3,-(hi-1gi-1+hi-2Zi-2+hi-3Ci-3)αi,ifi=4(1)n-4,Dn-hn-4gn-4-hn-5Zn-5-hn-6Cn-6αn-3,ifi=n-3,Bn-hn-3gn-3-hn-4Zn-4-hn-5Cn-5αn-2,ifi=n-2,bn-∑j=1n-2hjwjαn-1,ifi=n-1,vi={b1,ifi=1,-f2v1+B2,ifi=2,-f3v2-e3v1,ifi=3,-fivi-1-eivi-2-Diαi-3vi-3,ifi=4(1)n-4,-fn-3vn-4-en-3vn-5-Dn-3αn-6vn-6+Cn-3,ifi=n-3,-fn-2vn-3-en-2vn-4-Dn-2αn-5vn-5+An-2,ifi=n-2,an-1-∑j=1n-2‍kjvj,ifi=n-1,wi={B1,ifi=1,-f2w1,ifi=2,-f3w2-e3w1,ifi=3,-fiwi-1-eiwi-2-Diαi-3wi-3,ifi=4(1)n-5,-fn-4wn-5-en-4wn-6-Dn-4αn-7wn-7+Cn-4,ifi=n-4,-fn-3wn-4-en-3wn-5-Dn-3αn-6wn-6+An-3,ifi=n-3,-fn-2wn-3-en-2wn-4-Dn-2αn-5wn-5+an-2,ifi=n-2,fi={b2α1,ifi=2,(b3-e3g1)α2,ifi=3,(bi-eigi-2-(Di/αi-3)Zi-3)αi-1,ifi=4(1)n-2,ei={B3α1,ifi=3,(Bi-(Di/αi-3)gi-3)αi-2,ifi=4(1)n-2,gi={a1,ifi=1,a2-f2Z1,ifi=2,ai-fiZi-1-eiCi-2,ifi=3(1)n-3,Zi={A1,ifi=1,Ai-fiCi-1,ifi=2(1)n-4.We also have:(7)Det(H)=∏i=1n‍αi.Remark 1.
It is not difficult to prove that theLU decomposition (3) exists only if αi≠0, i=1(1)n-1 (pivoting elements). Moreover the cyclic heptadiagonal matrix H of the form (1) has an inverse if, in addition, αn≠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 matrixH of the form (1) and can be considered as natural generalization of the symbolic algorithm DETCPENTA in [5].Algorithm 1.
To computeDet(H) for the cyclic heptadiagonal matrix H in (1), we may proceed as follows.Step 1.
Setα1=d1Ifα1=0 then α1=t (t is just a symbolic name) end ifSetg1=a1, Z1=A1Setk1=An-1/α1Setv1=b1Setw1=B1Seth1=an/α1Setw1=B1Setf2=b2/α1Sete3=B3/α1Setα2=d2-f2*g1Ifα2=0 then α2=t end ifSetk2=-k1*g1/α2Setv2=B2-f2*v1Setw2=-f2*w1Seth2=(An-h1*g1)/α2Setg2=a2-f2*Z1Setf3=(b3-e3*g1)/α2Setα3=d3-e3*Z1-f3*g2Ifα3=0 then α3=t end ifSetk3=-(k1*Z1+k2*g2)/α3Seth3=-(h1*Z1+h2*g2)/α3Setv3=-e3*v1-f3*v2Setw3=-f3*w2-e3*w1Step 2.
Compute and simplify.Fori from 4 to n-2 do
ei=(Bi-Di*gi-3/αi-3)/αi-2fi=(bi-Di*Zi-3/αi-3-ei*gi-2)/αi-1Zi-2=Ai-2-fi-2*Ci-3gi-1=ai-1-fi-1*Zi-2-ei-1*Ci-3αi=di-Di*Ci-3/αi-3-ei*gi-2-fi*gi-1Ifαi=0 then αi=t end ifEnd doStep 3.
Compute and simplify.Fori from 4 to n-5 do
ki=-(ki-3*Ci-3+ki-2*Zi-2+ki-1*gi-1)/αiwi=-(Di*wi-3/αi-3+ei*wi-2+fi*wi-1)End doStep 4.
Compute and simplify.Fori from 4 to n-4 do
hi=-(hi-3*Ci-3+hi-2*Zi-2+hi-1*gi-1)/αivi=-(Di*vi-3/αi-3+ei*vi-2+fi*vi-1)End doStep 5.
Compute simplify.kn-4=(Dn-1-kn-5*gn-5-kn-6*Zn-6-kn-7*Cn-7)/αn-4kn-3=(Bn-1-kn-4*gn-4-kn-5*Zn-5-kn-6*Cn-6)/αn-3kn-2=(bn-1-kn-3*gn-3-kn-4*Zn-4-kn-5*Cn-5)/αn-2wn-4=Cn-4-Dn-4*wn-7/αn-7-en-4*wn-6-fn-4*wn-5wn-3=An-3-Dn-3*wn-6/αn-6-en-3*wn-5-fn-3*wn-4wn-2=an-2-Dn-2*wn-5/αn-5-en-2*wn-4-fn-2*wn-3hn-3=(Dn-hn-4*gn-4-hn-5*Zn-5-hn-6*Cn-6)/αn-3hn-2=(Bn-hn-3*gn-3-hn-4*Zn-4-hn-5*Cn-5)/αn-2vn-3=Cn-3-Dn-3*vn-6/αn-6-en-3*vn-5-fn-3*vn-4vn-2=An-2-Dn-2*vn-5/αn-5-en-2*vn-4-fn-2*vn-3vn-1=an-1-∑j=1n-2‍kjvjαn-1=dn-1-∑j=1n-2‍wjkjIfαn-1=0 then αn-1=t end ifhn-1=(bn-∑j=1n-2‍hjwj)/αn-1αn=dn-∑j=1n-1‍vjhjIfαn=0 then αn=t end ifStep 6.
ComputeDet(H)=(∏i=1n‍αi)t=0.
The symbolic Algorithm1 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 matrixH is nonsingular, its inversion is computed as follows.Let(8)H-1=[Sij]1≤i,j≤n=[Col1,Col2,…,Coln],
where Colm denotes mth column of H-1, m=1,2,…,n.Since the DoolittleLU 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 Coli=(S1,i,S2,i,…,Sn,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 matrixL is given by (4)
(9)L(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)⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Qn-4(n)Qn-4(n-1)Qn-4(n-2)Qn-4(n-3)Qn-4(n-4)Qn-3(n)Qn-3(n-1)Qn-3(n-2)Qn-3(n-3)Qn-3(n-4)Qn-2(n)Qn-2(n-1)Qn-2(n-2)Qn-2(n-3)Qn-2(n-4)Qn-1(n)Qn-1(n-1)Qn-1(n-2)Qn-1(n-3)Qn-1(n-4)Qn(n)Qn(n-1)Qn(n-2)Qn(n-3)Qn(n-4))=(0000000000⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮0000100010001000100010000),
we get
(10)(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)⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Qn-4(n)Qn-4(n-1)Qn-4(n-2)Qn-4(n-3)Qn-4(n-4)Qn-3(n)Qn-3(n-1)Qn-3(n-2)Qn-3(n-3)Qn-3(n-4)Qn-2(n)Qn-2(n-1)Qn-2(n-2)Qn-2(n-3)Qn-2(n-4)Qn-1(n)Qn-1(n-1)Qn-1(n-2)Qn-1(n-3)Qn-1(n-4)Qn(n)Qn(n-1)Qn(n-2)Qn(n-3)Qn(n-4))=(0000000000⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮000010001-fn-3001-fn-2M601-kn-2M4M51-hn-1M1M2M3),
where
(11)M1=-hn-2+hn-1kn-2,M2=-hn-3+hn-2fn-2+hn-1kn-3-hn-1kn-2fn-2,M3=-hn-4+hn-3fn-3+hn-2en-2-hn-2fn-2fn-3+hn-1kn-4-hn-1kn-3fn-3-hn-1kn-2en-2+hn-1kn-2fn-2fn-3,M4=-kn-3+kn-2fn-2,M5=-kn-4+kn-3fn-3+kn-2en-2-kn-2fn-2fn-3,M6=-en-2+fn-2fn-3.
Hence, solving the following standard linear systems whose coefficient matrix U is given by (5)
(12)U(S1,nS1,n-1S1,n-2S1,n-3S1,n-4S2,nS2,n-1S2,n-2S2,n-3S2,n-4⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Sn-4,nSn-4,n-1Sn-4,n-2Sn-4,n-3Sn-4,n-4Sn-3,nSn-3,n-1Sn-3,n-2Sn-3,n-3Sn-3,n-4Sn-2,nSn-2,n-1Sn-2,n-2Sn-2,n-3Sn-2,n-4Sn-1,nSn-1,n-1Sn-1,n-2Sn-1,n-3Sn-1,n-4Sn,nSn,n-1Sn,n-2Sn,n-3Sn,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)⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮Qn-4(n)Qn-4(n-1)Qn-4(n-2)Qn-4(n-3)Qn-4(n-4)Qn-3(n)Qn-3(n-1)Qn-3(n-2)Qn-3(n-3)Qn-3(n-4)Qn-2(n)Qn-2(n-1)Qn-2(n-2)Qn-2(n-3)Qn-2(n-4)Qn-1(n)Qn-1(n-1)Qn-1(n-2)Qn-1(n-3)Qn-1(n-4)Qn(n)Qn(n-1)Qn(n-2)Qn(n-3)Qn(n-4))
gives the five columns Coli, i=n,n-1,n-2,n-3, and n-4 in the forms:
(13)Sn,n=1αn,Sn-1,n=-vn-1Sn,nαn-1,Si,n=-(giSi+1,n+ZiSi+2,n+CiSi+3,n+wiSi-1,n+viSn,n)αi,i=n-5(-1)1,Sn,n-1=-hn-1αn,Sn-1,n-1=(1-vn-1Sn,n-1)αn-1,Si,n-1=-(giSi+1,n-1+ZiSi+2,n-1+CiSi+3,n-1+wiSi-1,n-1+viSn,n-1)(αi)-1,i=n-5(-1)1,Sn,n-2=(-hn-2+hn-1kn-2)αn,Sn-1,n-2=-(kn-2+vn-1Sn,n-2)αn-1,Sn-2,n-2=(1-wn-2Sn-1,n-2-vn-2Sn,n-2)αn-2,Si,n-2=-(giSi+1,n-2+ZiSi+2,n-2+CiSi+3,n-2+wiSi-1,n-2+viSn,n-2)(αi)-1,i=n-5(-1)1,Sn,n-3=(-hn-3+hn-2fn-2-hn-1(kn-2fn-2-kn-3))αn,Sn-1,n-3=(kn-2fn-2-kn-3-vn-1Sn,n-3)αn-1,Sn-2,n-3=-(fn-2+wn-2Sn-1,n-3+vn-2Sn,n-3)αn-2,Sn-3,n-3=(1-gn-3Sn-2,n-3-wn-3Sn-1,n-3-vn-3Sn,n-3)αn-3,Sn,n-4=(-hn-4+hn-3fn-3-hn-2(en-2-fn-2fn-3)αn+hn-1(kn-4-kn-3fn-3-kn-2(en-2-fn-2fn-3))αn),Sn-1,n-4=(-kn-4+kn-3fn-3-kn-2(en-2-fn-2fn-3)-vn-1Sn,n-4)αn-1,Sn-2,n-4=-((en-2-fn-2fn-3)+wn-2Sn-1,n-4+vn-2Sn,n-4)αn-2,Sn-3,n-4=-(fn-3+gn-3Sn-2,n-4+wn-3Sn-1,n-4+vn-3Sn,n-4)αn-3,Sn-4,n-4=(1-gn-4Sn-3,n-4-Zn-4Sn-2,n-4-wn-4Sn-1,n-3-vn-4Sn,n-4)(αn-4)-1,Si,n-4=-(giSi+1,n-4+ZiSi+2,n-4+CiSi+3,n-4+wiSn-1,n-4+viSn,n-4)(αn-4)-1,i=n-5(-1)1.The remaining(n-5) columns are obtained by using the fact HH-1=In where In is the identity matrix. They are as in the following:
(14)Coln-5=(En-2-An-4Coln-4-an-3Coln-3-dn-2Coln-2-bn-1Coln-1-BnColn)(Cn-5)-1,(15)Coli=(Ei+3-Ai+1Coli+1-ai+2Coli+2-di+3Coli+3-bi+4Coli+4-Bi+5Coli+5-Di+6Coli+6)(Ci)-1,i=n-6(-1)1,
where Ei is the ith unit vector.Remark 2.
Equations (14) and (15) suggest an additional assumption ∏i=1n-5‍Ci≠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 then×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,Di,Bi,bi,di,ai,Ai,Ci,i=1,2,…,n,whereD1=D2=D3=Cn-2=Cn-1=Cn=0.OUTPUT. Inverse matrix.Step 1.
IfCi=0 for any i=1,2,…,n-5 set Ci=t (t is just a symbolic name).Step 2.
IfBi=0 for any i=6,7,…,n,Bi=t.Step 3.
Use theDETCHEPTA 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.
Fori=1,2,…,n, compute and simplify the components Si,n,Si,n-1,Si,n-2, Si,n-3, and Si,n-4 of the columns Cj,j=n,n-1,n-2,n-3, and n-4, respectively, by using (13).Step 5.
Fori=1,2,…,n, compute and simplify the components Si,n-5 by using (14).Step 6.
Forj=n-6,n-7,…,1, doFori=1,2,…,n, doCompute and simplify the componentsSi,j by using (15).End doEnd doStep 7.
Substitute the actual valuet=0 in all expressions to obtain the elements, Si,j,i,j=1,2,…,n.The symbolic Algorithm2 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 the10×10 cyclic heptadiagonal linear system(16)(1-11-200002-11111-10000121-112300002-2315-60000011111120000-1-1-1-1-1-110000222231-30000-2-21135300003134-12400002341)(x1x2x3x4x5x6x7x8x9x10)=(21533043-2447707894).By using the coefficient matrix of the system (16) and applying the CHINV algorithm, we get the inverse of the coefficient matrix (H-1): (17)H-1=(-1266432715-292132715-189810905295732715-2339932715-2441932715-20696543-667632715137143271563163271526863271541693271590210905-111832715800632715880132715416543-236632715-6391327154571327155417109054693109053443635-24110905584210905329210905280218175810905-279210905-165810905293109055347109055913635146109051393109055698109058532181157710905-183810905-198210905634432715-49643271529831090515983271512259327159484327154336543262132715-637432715-167632715938363535736357883635-3393635102336355133635567272973635-4133635-4773635117810905-90810905-2543635-60410905-1323210905-1401210905385218149710905-165810905371810905-635610905-496910905-1382363577810905353910905-130610905-9222181-1904109055891109051941090516382327151073832715324410905-1216327151409232715278323271527256543807832715-1128232715-515332715-80832715-3617327151174109054432715594732715-186832715-938654346583271519332715-164332715).By using (17) and after simple calculations we can obtain the solution of cyclic heptadiagonal linear system (16):
(18)X=(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10)T=(1,2,3,4,5,6,7,8,9,10)T.Example 2.
Consider the8×8 cyclic heptadiagonal matrix
(19)H=(0-11-2002-11111-100121-1123002-2315-6000111111200-1-1-1-1-1-1300222232400-2-211).
By applying the CHINV algorithm, it yields(20)H-1=(438438t+29831061438t+2983402438t+2983133438t+2983-135438t+29831361438t+2983388438t+2983-156438t+2983-72438t+2983-9(2t+33)438t+29832(42t+253)438t+29836t+19438t+29834(24t+169)438t+29832(6t-71)438t+2983-(78t+595)438t+298312(6t+43)438t+2983693438t+29833(3t+580)438t+2983-14(3t-25)438t+2983-(3t-190)438t+29833(57t+317)438t+2983-9(25t-69)438t+2983-36(5t+17)438t+2983-12(3t+41)438t+2983-1125438t+2983321t-539438t+298327(4t-11)438t+298339t-76438t+2983-(471t+2861)438t+2983-(141t+4456)438t+298325(6t+1)438t+29835(6t+121)438t+29836438t+2983-(108t+721)438t+298366t+455438t+298336t+247438t+29832(69t+469)438t+298372t+509438t+2983-(30t+199)438t+2983-(6t+43)438t+2983334438t+2983-2(13t-316)438t+29832(12t+235)438t+2983-(40t+171)438t+298330(3t+17)438t+2983-(80t-493)438t+2983-4(16t+35)438t+2983-3(14t+135)438t+298320438t+2983-(214t+1409)438t+2983-8(9t+59)438t+2983-(26t+171)438t+2983-45(6t+41)438t+2983-5(98t+655)438t+298346t+331438t+2983126t+851438t+298372438t+29839(2t+33)438t+2983-2(42t+253)438t+2983-(6t+19)438t+29833(114t+769)438t+2983426t+3125438t+298378t+595438t+2983-12(6t+43)438t+2983)t=0=(43829831061298340229837157-1352983136129833882983-1562983-722983-2972983506298311576762983-1422983-5952983516298369329831740298335029831015795129836212983-6122983-4922983-11252983-5392983-2972983-4157-28612983-44562983252983605298362983-721298345529831315793829835092983-1992983-432983334298363229834702983-915751029834932983-1402983-4052983202983-14092983-4722983-9157-18452983-32752983331298385129837229832972983-5062983-115723072983312529835952983-5162983).


## 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.



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*Source: 321032-2013-04-07.xml*