On the Inverting of a General Heptadiagonal Matrix

British Journal of Applied Science & Technology 18(5): 1-12, 2016; Article no.BJAST.31213 ISSN: 2231-0843, NLM ID: 101664541

SCIENCEDOMAIN international www.sciencedomain.org

∗ A. A. Karawia1,2

1Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 2Computer Science Unit, Deanship of Educational Services, Qassim University, P.O.Box 6595, Buraidah 51452, Saudi Arabia.

Author’s contribution

The sole author designed, analyzed and interpreted and prepared the manuscript.

Article Information DOI: 10.9734/BJAST/2016/31213 Editor(s): (1) Orlando Manuel da Costa Gomes, Professor of Economics, Lisbon Accounting and Business School (ISCAL), Lisbon Polytechnic Institute, Portugal. Reviewers: (1) Jia Liu, University of West Florida, USA. (2) Hammad Khalil, University of Education (Attock Campus), Lahore, Pakistan. (3) Gerald Bourgeois, University of French Polynesia and University of Aix-Marseille, France. Complete Peer review History: http://www.sciencedomain.org/review-history/17657

Received: 26th December 2016 Accepted: 21st January 2017 Original Research Article Published: 28th January 2017

ABSTRACT

In this paper, the author presents a new algorithm computing the inverse of any nonsingular heptadiagonal matrix. The computational cost of our algorithms is O(n2) operations in C. The algorithms are suitable for implementation using computer algebra system such as MAPLE, MATLAB and MATHEMATICA. Examples are given to illustrate the efﬁciency of the algorithms.

Keywords: Heptadiagonal matrices; LU factorization; Determinants; Computer algebra systems(CAS).

2010 Mathematics Subject Classiﬁcation: 15A15, 15A23, 68W30, 11Y05, 33F10, F.2.1, G.1.0.

*Corresponding author: E-mail: [email protected]; Karawia; BJAST, 18(5), 1-12, 2016; Article no.BJAST.31213

1 INTRODUCTION

The n × n general heptadiagonal matrices take the form:

  d1 e1 f1 g1    c2 d2 e2 f2 g2     b3 c3 d3 e3 f3 g3 0     a4 b4 c4 d4 e4 f4 g4         ......  H =  ......  , n > 4. (1.1)        a − b − c − d − e − f − g −   n 3 n 3 n 3 n 3 n 3 n 3 n 3   0 a − b − c − d − e − f −   n 2 n 2 n 2 n 2 n 2 n 2  an−1 bn−1 cn−1 dn−1 en−1 an bn cn dn where {ai}4≤i≤n, {bi}3≤i≤n, {ci}2≤i≤n, pentadiagonal matrix and they assumed few {di}1≤i≤n, {ei}1≤i≤n, {fi}1≤i≤n, and {gi}1≤i≤n conditions to avoid failure in their own algorithm. are sequences of numbers such that gi ≠ The motivation of the current paper is to establish 0, gn−2 = gn−1 = gn = 1 and fn−1 = fn = efficient algorithms for inverting heptadiagonal en = 0. matrix. I generalized the algorithm[13] to find the inverse of a general invertible heptadiagonal Heptadiagonal matrices frequently arise from matrix and we presented an efficient symbolic boundary value problems. The heptadiagonal algorithm to find the inverse of such matrices. systems emanate in many numerical models, The development of a symbolic algorithm is for example the traditional discretization of the considered in order to remove all cases where implicit method to resolve partial differential the numeric algorithm fails when at least one of equations on 3-D problems with regular grids. gi at least equals to 0, i = 1, 2, ..., n − 3. If, for Also, these kinds of matrices appear in many every i such that gi = 0, then I put gi = t for a areas of science and engineering[[1]-[9]]. So small t ≠ 0. The current work will be in the field a good technique for computing the inverse of complex numbers C. of such matrices is required. To the best of our knowledge, the inversion of a general The paper is organized as follows: In Section 2, heptadiagonal matrix of the form (1.1) has not the main result is presented. New numeric and been considered. symbolic algorithms are given in Section 3. In Section 4, illustrative examples are presented. In [10], Karawia described a reliable symbolic Conclusions of the work are given in Section 5. computational algorithm for inverting general cyclic heptadiagonal matrices by using parallel computing along with recursion. An 2 MAIN RESULTS explicit formula for the determinant of a heptadiagonal symmetric matrix is given in[6]. In this section, we present recurrence formulas Many researchers studied special cases of for the columns of the inverse of a heptadiagonal heptadiagonal matrix. In [11], the authors matrix H. presented a symbolic algorithm for finding the When the matrix H is nonsingular, its inversion is inverse of any general nonsingular tridiagonal computed as follows. matrix. A new efficient computational algorithm Let to find the inverse of a general tridiagonal matrix −1 is presented in [12] based on the Doolittle LU H = [Sij ]1≤i,j≤n = [Col1, Col2, ..., Coln] factorization. In [13], the authors introduced a computationally efficient algorithm to obtain where Colk is the kth column of the inverse the inverse of a tridiagonal matrix and a matrix H−1.

2 Karawia; BJAST, 18(5), 1-12, 2016; Article no.BJAST.31213

−1 By using the fact HH = In, where In is the identity matrix, the ﬁrst (n−3) columns can be obtained by relations

where Ek is the kth unit vector.

From (2.1), we note that if we know the last three columns Coln, Coln−1, and Coln−2 then we can recursively compute the remaining (n − 3) columns Coln−3, Coln−4, ..., Col1.

At this point it is convenient to give recurrence formulas for computing Coln, Coln−1, and Coln−2.

Consider the sequence of numbers {Ai}1≤i≤n+3, {Bi}1≤i≤n+3,and {Ci}1≤i≤n+3 characterized by a term recurrence relations  A1 = 0,   A = 0,  2   A3 = 1, 

d1A1 + e1A2 + f1A3 + g1A4 = 0, (2.1)   c2A1 + d2A2 + e2A3 + f2A4 + g2A5 = 0,   b3A1 + c3A2 + d3A3 + e3A4 + f3A5 + g3A6 = 0,   aiAi−3 + biAi−2 + ciAi−1 + diAi + eiAi+1 + fiAi+2 + giAi+3 = 0, i ≥ 4,  B1 = 0,   B = 1,  2   B3 = 0, 

d1B1 + e1B2 + f1B3 + g1B4 = 0, (2.2)   c2B1 + d2B2 + e2B3 + f2B4 + g2B5 = 0,   b3B1 + c3B2 + d3B3 + e3B4 + f3B5 + g3B6 = 0,   aiBi−3 + biBi−2 + ciBi−1 + diBi + eiBi+1 + fiBi+2 + giBi+3 = 0, i ≥ 4, and  C1 = 1,   C = 0,  2   C3 = 0, 

d1C1 + e1C2 + f1C3 + g1C4 = 0, (2.3)   c2C1 + d2C2 + e2C3 + f2C4 + g2C5 = 0,   b3C1 + c3C2 + d3C3 + e3C4 + f3C5 + g3C6 = 0,   aiCi−3 + biCi−2 + ciCi−1 + diCi + eiCi+1 + fiCi+2 + giCi+3 = 0, i ≥ 4.

3 Karawia; BJAST, 18(5), 1-12, 2016; Article no.BJAST.31213

Now, we can give matrix forms for term recurrences (2.2), (2.3) and (2.4)

HA = −An+1En−2 − An+2En−1 − An+3En, (2.4)

HB = −Bn+1En−2 − Bn+2En−1 − Bn+3En, (2.5)

HC = −Cn+1En−2 − Cn+2En−1 − Cn+3En, (2.6) t t t where A = [A1,A2, ..., An] , B = [B1,B2, ..., Bn] , and C = [C1,C2, ..., Cn] .

Let’s deﬁne the following determinants:

Ai An+2 An+3

Xi = Bi Bn+2 Bn+3 , i = 1, 2, ..., n + 1, (2.7)

Ci Cn+2 Cn+3

Ai An+1 An+3

Yi = Bi Bn+1 Bn+3 , i = 1, 2, ..., n + 2, (2.8)

Ci Cn+1 Cn+3

Ai An+1 An+2

Zi = Bi Bn+1 Bn+2 , i = 1, 2, ..., n + 3. (2.9)

Ci Cn+1 Cn+2 By simple calculations, we have HX = −Xn+1En−2, (2.10)

HY = −Yn+2En−1, (2.11)

HZ = −Zn+3En, (2.12) t t t where X = [X1,X2, ..., Xn] , Y = [Y1,Y2, ..., Yn] , and Z = [Z1,Z2, ..., Zn] .

Remark 2.1. Xn+1 = −Yn+2 = Zn+3.

Theorem 2.1.(generalization version of theorem 3.1 in [13]) H is invertible iff Xn+1 ≠ 0 and , for ̸ every i, gi = 0. Moreover [ ] t −Z1 −Z2 −Zn Coln = , , ..., , (2.13) Xn+1 Xn+1 Xn+1 [ ] t Y1 Y2 Yn Coln−1 = , , ..., ] , (2.14) Xn+1 Xn+1 Xn+1 [ ] t −X1 −X2 −Xn Coln−2 = , , ..., . (2.15) Xn+1 Xn+1 Xn+1 (∏ ) − n−3 ̸ ̸ Proof. Since det(H) = i=1 gi Xn+1. So if H is invertible then Xn+1 = 0 and if Xn+1 = 0 and gi ≠ 0 for every i then det(H) ≠ 0 and H is invertible.From (2.11), (2.12) and (2.13) we obtain Coln, Coln−1, and Coln−2. The proof is completed. 2

̸ Remark 2.2. If, for every i such that gi = 0, we put gi = t for a small t = 0, and if H|t=0 is invertible, then Xn+1(t) ≠ 0.then H is invertible.

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3 NEW NUMERIC AND SYMBOLIC ALGORITHMS FOR THE INVERSE OF HEPTADIAGONAL MATRIX

In this section, we formulate the result in the previous section . It is a numerical algorithm to compute the inverse of a general heptadiagonal matrix of the form (1.1) when it exists.

Algorithm 3.1. To ﬁnd the inverse of heptadiagonal matrix (1.1). let fn−1 = fn = en = 0 and gn−2 = gn−1 = gn = 1.

INPUT: Order of the matrix n and the components ai, bj , ck, dl, el, fl, and gl for i = 4, 5, ..., n, j = 3, 4, ..., n, k = 2, 3, ..., n, and l = 1, 2, ..., n, OUTPUT: The inverse of heptadiagonal matrix H−1.

Step 1: Compute the sequence of numbers Ai,Bi, and Ci for i = 1, 2, ..., n + 3 using (2.2), (2.3) and (2.4) respectively. Step 2: Compute Xi, i = 1, 2, ..., n + 1 using (2.8), Yi, i = 1, 2, ..., n + 2 using (2.9) and Zi, i = 1, 2, ..., n + 3 using (2.10). Step 3: Compute the last three columns Coln, Coln−1, and Coln−2 using (2.14), (2.15), and (2.16) respectively. Step 4: Compute the remaining (n-3)-columns Colj , j = n − 3, n − 4, ..., 1 using (2.1). −1 Step 5: Set H = [Col1, Col2, ..., Coln].

The numeric algorithm 3.1 will be referred to as NINVHEPTA algorithm in the sequel. The computational cost of NINVHEPTA algorithm is 11n2 − 75n + 21 operations.

As can be easily seen, it breaks down unless the conditions gi ≠ 0 are satisﬁed for all i = 1, 2, ..., n−3. So the following symbolic algorithm is developed in order to remove the cases where the numeric algorithm fails.

Algorithm 3.2. To ﬁnd the inverse of heptadiagonal matrix (1.1). let fn−1 = fn = en = 0 and gn−2 = gn−1 = gn.

Step 1: If gi = 0 for any i = 1, 2, ..., n − 3 set gi = t(t is just a symbolic name). Step 2: Compute the sequence of numbers Ai,Bi, and Ci for i = 1, 2, ..., n + 3 using (2.2), (2.3) and (2.4)respectively. Step 3: Compute Xi, i = 1, 2, ..., n + 1 using (2.8), Yi, i = 1, 2, ..., n + 2 using (2.9) and Zi, i = 1, 2, ..., n + 3 using (2.10). Step 4: Compute the last three columns Coln, Coln−1, and Coln−2 using (2.14), (2.15), and (2.16) respectively. Step 5: Compute the remaining (n-3)-columns Colj , j = n − 3, n − 4, ..., 1 using (2.1). Step 6: Substitute the actual value t = 0 in all expressions to obtain the elements of columns Colj , j = 1, 2, ..., n. −1 Step 7: Set H = [Col1, Col2, ..., Coln].

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The symbolic algorithm 3.2 will be referred to as SINVHEPTA algorithm in the sequel. The SINVHEPTA algorithm use O(n2) elementary operations. Based on SINVHEPTA algorithm, a MAPLE procedure [14] for inverting a general nonsingular heptadiagonal matrix H is listed as an Appendix.

4 ILLUSTRATIVE EXAMPLES

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

Example 4.1. (Case I: gi ≠ 0 for all i) Find the inverse of following 10 × 10 heptadiagonal matrix   2 1 4 −1 0 0 0 0 0 0      5 1 1 2 2 0 0 0 0 0       1 2 −3 2 7 2 0 0 0 0       6 1 3 2 3 −1 3 0 0 0       0 1 −1 2 2 −3 4 1 0 0    H1 =   (4.1)    0 0 4 4 4 1 2 1 1 0     − − −   0 0 0 1 2 1 3 3 2 1       0 0 0 0 3 1 2 1 11 3       0 0 0 0 0 4 −3 2 1 1  0 0 0 0 0 0 −7 1 1 2 Solution: By applying the NINVHEPTA algorithm, it yields • − 9 53 21 83 − 93 663 − 67 − − Step 1: A = [0, 0, 1, 4, 2 , 4 , 4 , 4 , 2 , 4 , 4 , 198, 269], − 3 13 19 41 − 47 341 − 53 − 107 − 124 B = [0, 1, 0, 1, 2 , 4 , 12 , 12 , 6 , 12 , 12 , 3 , 3 ], and − 9 53 67 269 − 221 1781 − 881 − 578 − 730 C = [1, 0, 0, 2, 2 , 4 , 12 , 12 , 6 , 12 , 12 , 3 , 3 ]. • 4231 − 10942 − 2146 − 4687 − 31741 − 19873 51721 19773 − 154735 − 905413 Step 2: X = [ 3 , 3 , 3 , 3688, 2 , 12 , 12 , 12 , 2 , 12 , 12 ], 1983 − 62693 11759 − 82109 49839 − 220819 − 141107 − 5312 160577 − 563539 905413 Y = [ 4 , 4 , 6 , 12 , 4 , 12 , 12 , 3 , 12 , 12 , 0, 12 ], and

3325 − 33928 21211 − 22109 − 19327 − 84955 50981 − 15235 76363 − 905413 Z = [ 12 , 3 , 12 , 6 , 7763, 2 , 12 , 12 , 4 , 6 , 0, 0, 12 ]. • Step 3: 3325 − 135712 21211 − 44218 93156 − 115962 − 84955 50981 − 45705 152726 t Col10 = [ 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 ] , − 5949 188079 − 23518 82109 − 149517 220819 141107 21248 − 160577 563539 t Col9 = [ 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 ] ,and 16924 − 43768 − 8584 − 44256 28122 − 31741 − 19873 51721 118638 − 154735 t Col8 = [ 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 ] .

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• Step 4: − 51473 214649 6848 139095 − 121161 106328 88422 − 278373 − 103927 500627 t Col7 = [ 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , ] , − 80594 − 217 83035 170735 − 10659 31638 − 14393 − 84414 14531 − 15434 t Col6 = [ 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 ] , − 82176 447486 − 28082 170806 − 175068 − 6589 102273 9510 − 78091 392246 t Col5 = [ 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 ] , 205297 − 910556 6935 − 472222 410790 − 147233 42741 427692 − 147953 9724 t Col4 = [ 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 ] , − 2619 − 30890 − 25421 − 137812 172515 − 19216 − 46090 62775 11493 − 198449 t Col3 = [ 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 ] ,

− 29328 877900 − 53389 605688 − 491917 172702 − 34896 − 501141 156703 50083 t Col2 = [ 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , ] , and

− 88555 552363 125378 − 28648 − 88835 19552 − 17938 − 61611 46355 − 55155 t Col1 = [ 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 , 905413 ] . • Step 5:

Example 4.2.(Case II: gi = 0 for at least one of i) Find the inverse of following 5 × 5 heptadiagonal matrix   2 3 4 1 0      −1 1 −2 3 0      H2 =  3 5 1 −1 2  (4.2)      4 −1 3 2 6  0 2 1 4 −3

Solution: i- By applying the NINVHEPTA algorithm, it breaks down since g2 = 0. ii- By applying the SINVHEPTA algorithm, it yields • − −1 − 5 x+28 5 x−84 5 x+14 Step 1: A = [0, 0, 1, 4, 14 x , x , x , 3 x ], − −1 − x+2 7 x−48 5 x+12 B = [0, 1, 0, 3, 8 x , 8 x , x , 2 x ], and − −1 − 5 x+14 − −1 8 x+21 C = [1, 0, 0, 2, 7 x , x , 42 x , x ].

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• 55 x+294 63+40 x − 183+56 x − 2 x+15 − −1 − 315 x+901 Step 2: X = [ x , x , x , 3 x , 79 x , x ], − 35 x+88 5 x+13 7 x+18 − 21 x+65 − −1 315 x+901 Y = [ 2 x , 7 x , 2 x , x , 14 x , 0, x ], and − 5 x−548 5 x−28 − 35 x+194 3 x+4 − −1 − 315 x+901 Z = [ x , 5 x , x , 25 x , 325 x , 0, 0, x ]. • − 5 x−548 5 x−28 − 35 x+194 3 x+4 −325 t Step 3: Col5 = [ 315 x+901 , 5 315 x+901 , 315 x+901 , 25 315 x+901 , 315 x+901 ] , 35 x+88 − 5 x+13 − 7 x+18 21 x+65 14 t Col4 = [2 315 x+901 , 7 315 x+901 , 2 315 x+901 , 315 x+901 , 315 x+901 ] , and 55 x+294 63+40 x − 183+56 x − 2 x+15 −79 t Col3 = [ 315 x+901 , 315 x+901 , 315 x+901 , 3 315 x+901 , 315 x+901 ] . • −545 205 91 111 315 t Step 4: Col2 = [ 315 x+901 , 315 x+901 , 315 x+901 , 315 x+901 , 315 x+901 ] , and

13 x+123 x+19 2 x+7 33 x+7 248 t Col1 = [−5 , 10 , 56 , − , ] .  315 x+901 315 x+901 315 x+901 315 x+901 315 x+901  −5 13 x+123 −545 55 x+294 2 35 x+88 − 5 x−548  315 x+901 315 x+901 315 x+901 315 x+901 315 x+901   −   10 x+19 205 63+40 x −7 5 x+13 5 5 x 28   315 x+901 315 x+901 315 x+901 315 x+901 315 x+901  − • 1  2 x+7 91 183+56 x 7 x+18 35 x+194  Step 5: H2 =  56 − −2 −   315 x+901 315 x+901 315 x+901 315 x+901 315 x+901     − 33 x+7 111 − 2 x+15 21 x+65 3 x+4  315 x+901 315 x+901 3 315 x+901 315 x+901 25 315 x+901 248 315 −79 14 −325 315 x+901 315 x+901 315 x+901 315 x+901 315 x+901   x=0 − 615 − 545 294 176 548  901 901 901 901 901   190 205 63 91 140   − −   901 901 901 901 901  −1  392 91 183 36 194  • Step 6: H =  − − − . 2  901 901 901 901 901   − 7 111 − 45 65 100   901 901 901 901 901  248 315 − 79 14 − 325 901 901 901 901 901 Example 4.3. We consider the following n × n heptadiagonal matrix in order to demonstrate the efﬁciency of SINVHEPTA algorithm.   −2 −1 2 1  − −   3 2 1 2 1   − −   1 3 2 1 2 1 0   − −   2 1 3 2 1 2 1         ......  H =  ......  . (4.3)        2 1 3 −2 −1 2 1     0 2 1 3 −2 −1 2    2 1 3 −2 −1 2 1 3 −2

In Table 1. we give a comparison of the NINVHEPTA is recorded in table 2, which gives mean time between SINVHEPTA, CHEPTA[10] us the mean values of time elapsed(in seconds) −1 (symbolic algorithm to ﬁnd the inverse of and relative error ||H · H − In||F /||In||F , over Cyclic Heptadiagonal matrix) algorithms and 100 trials, for the computation of the inverse of MatrixInverse function in Maple 13.0 for different a heptadiagonal matrix H with random integers orders, over 100 trials. It was tested in an Intel(R) entries hij for |i−j| ≤ 3 and with nonzero entries Core(TM) i7-4700MQ [email protected] 2.40 GHz. on its third superdiagonal, and hij = 0 otherwise. Example 4.4. The performance of the algorithm

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Table 1. The mean values of the time elapsed(in seconds) over 100 trials, of the proposed algorithm, CHEPTA[10] algorithm and MatrixInverse function in Maple 13.0.

Algorithms n 10 20 50 100 150 200 500 SINVHEPTA Mean time(s) 0.00563 0.01326 0.08171 0.44915 1.29597 2.95462 48.74440 CHEPTA[10] Mean time(s) 0.00233 0.01189 0.06819 0.43924 1.27376 2.95963 49.35749 MatrixInverse Mean time(s) 0.00375 0.01512 0.17214 1.31768 4.41329 10.82438 198.68109

Table 2. The mean values of the time elapsed(in seconds)and the mean of the relative −1 error(MRE)||H · H − In||F /||In||F , over 100 trials, of the proposed algorithm, NINVHEPTA algorithm to obtain more accurate information about the full inverse of a random heptadiagonal matrix H.

n 10 20 50 100 150 200 500 Mean time(s) 5.60e − 3 1.36e − 2 7.68e − 2 3.37e − 1 9.24e − 1 2.05 5.74 MRE 4.12e − 14 7.42e − 13 1.62e − 12 8.09e − 12 5.38e − 11 9.66e − 11 3.54e − 10

5 CONCLUSIONS [4] Yamamoto T, Ikebe Y. Inversion of band matrices. Linear Algebra Appl. 1979;24:105- In this work new numeric and symbolic algorithms 111. have been developed for finding the inverse [5] Trench WF. An algorithm for the inversion of any nonsingular heptadiagonal matrix. The of finite Toeplitz matrices. J. SIAM. symbolic algorithm removes the cases where 1964;12:515-522. the numeric algorithms fail when at least one of gi = 0, i = 1, 2, ..., n−3. It has the smallest mean [6] Elouafi M. A note for an explicit formula time the methods in literature for large sizes. for the determinant of pentadiagonal and heptadiagonal symmetric Toeplitz matrices. ACKNOWLEDGEMENT Appl. Math. Comput. 2013;219:4789-4791. [7] Solary M. Finding eigenvalues for The author would like to thank anonymous heptadiagonal symmetric Toeplitz matrices. referees for pointing out several comments and J. Math. Anal. Appl. 2013;402:719-730. suggestions. [8] Ting D, Gu M, Chi X, Cao J. Numerical COMPETING INTERESTS Acceleration of Three-Dimensional Quantum Transport Method Using a Seven- Author has declared that no competing interests Diagonal Pre-Conditioner. J. Comput. exist. Electron. 2002;1:93-97. [9] Gu S, Peng J, Cui R. A Polynomial Time References Solvable Algorithm to Binary Quadratic Programming Problems with Q Being a [1] Bo¨ttcher A, Grudsky S. Spectral Properties Seven-Diagonal Matrix and Its Neural of Banded Toeplitz Matrices. SIAM, Network Implementation. Advances in Philadelphia; 2005. Neural Networks ISNN 2014, Lecture [2] Golub GH, Van Loan CF. Matrix Notes in Computer Science. 2014:338-346. Computations. third ed., The Johns Hopkins [10] Karawia AA. Inversion of General Cyclic University Press, Baltimore and London; Heptadiagonal Matrices. Math. Probl. Eng. 1996. 2013;9. Article ID 321032. [3] Burden RL, Faires JD. Numerical Analysis. [11] El-Mikkawy M, Karawia AA. Inversion of seventh ed., Books & Cole Publishing, general tridiagonal matrices. Appl. Math. Pacific Grove, CA; 2001. Lett. 2006;19:712-720.

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[12] El-Mikkawy M. On the inverse of a general pentadiagonal matrix. Appl. Math. Comput. tridiagonal matrix. Appl. Math. Comput. 2008;202:441-445. 2004;150:669-679. [14] Maple software. [13] Hadj A, Elouaﬁ M. A fast numerical Available: http://www.maplesoft.com/ algorithm for the inverse of a tridiagonal and

10 Karawia; BJAST, 18(5), 1-12, 2016; Article no.BJAST.31213

11 Karawia; BJAST, 18(5), 1-12, 2016; Article no.BJAST.31213

—————————————————————————————————————————————————- ⃝c 2016 Karawia; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Peer-review history: The peer review history for this paper can be accessed here: http://sciencedomain.org/review-history/17657