Advances in Computer Science Research, volume 58 Modeling, Simulation and Optimization Technologies and Applications (MSOTA 2016)
A Computational Algorithm for the Inverse of a Sevendiagonal Matrix
Yanling Lin and Xiaolin Lin* School of Arts and Sciences, Shaanxi University of Science and Technology, Xi’an 710021, P.R. China *Corresponding author
Abstract—In this article, we investigate the inverse of the Without loss of generality, a sevendiagonal matrix with sevendiagonal matrix. Under certain conditions, the efficient order n can be given as algorithm which is suitable for implementation using computer algebra systems, is proposed. Also, the inverse of an anti- sevendiagonal matrix is obtained based on the special relation dcba 0 0 between the sevendiagonal matrix and the anti-sevendiagonal 1111 dcba matrix. A numerical example is provided to further illustrate the 22222 dcba validity of the corresponding algorithm. 333333 a4444 0 T Keywords-a sevendiagonal matrix; inverse matrix; determinant; 0 dn3 computer algebra system cn2 bn1 0 0 a I. INTRODUCTION nnnn (1) A large number of banded matrices are appeared in the where dcba ,,,,, and ni ),,2,1( are finite process of numerical analysis and numerical calculation, such iiiiii i as tridiagonal, pentadiagonal and sevendiagonal matrices. sequences of numbers such that i nid )3,,2,1(0 , They play an important role in theory, and they are applied in ddd 1, 0 , 0 , spline approximation, parallel computing, partial differential 12 nnn 211 321 equations(PDE) and boundary value problems(BVP). For a and nn 1 ccb n 0 . long time, the inverses of tridiagonal and pentadiagonal matrices have been researched[1-11]. Recently, the inverses of Suppose that the sevendiagonal matrix T is non-singular Toeplitz tridiagonal and pentadiagonal matrices are also and the inverse is investigated[12]. And the inverses of sevendiagonal and generalized sevendiagonal matrices have been studied[13-15]. 1 , Many of them gave algorithms to obtain the inverse matrices T 21 CCC n ),,,( by LU factorization[3-5,9,10,13-15]. A few by dividing blocks of diagonal matrices[6,12]. And Ran Rui-sheng[7] used the where C denote the jth column of T1 , ,,2,1 nj . Here Doolittle factorization to get the inverse matrix of a general j also can be written as tridiagonal matrix. C j In this paper, based on the numerical algorithms given in [1,13], we present a fast algorithm for computing the inverse j 21 ),,,( ECCCC jn of the sevendiagonal matrix. Moreover, the inverse of anti- , sevendiagonal matrix is also investigated. where E ),,,( t , ,,2,1 nj , and is the The rest of this paper is organized as follows. In Section 2, 21 jjj nj ij we present an efficient algorithm for the inverse of an n-by-n Kronecker symbol. sevendiagonal matrix, and the inverse of the anti- Through the relation 1TT I ( here is the nn sevendiagonal matrix is also given. One numerical example is n I n provided to illustrate the proposed algorithm in Section 3. identity matrix), we get the relations as Finally, some conclusions of the work are given.
3 ( 1122 /) dCaCbCcEC nnnnnnnnn 3 II. INVERSE OF A SEVENDIAGONAL MATRIX ( 11223314 /) dCCaCbCcEC nnnnnnnnnnn 4 In this section, we mainly focus on giving some recurrence formulas for the inverse of an n-by-n sevendiagonal matrix in ( 1122334425 /) dCCCaCbCcEC nnnnnnnnnnnnn 5 ( /) dCCCCaCbCcEC terms of their columns. 3 1122 jjjjjjjjjjjjjjj 3332211 3-nj4 (2)
Copyright © 2016, the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). 298 Advances in Computer Science Research, volume 58
t DdDcDbDa 31211101 0 where E 21 jjj nj ),,,,( , ,,2,1 nj , and ij is the DdDcDbDaD 0 Kronecker symbol. 4232221202 DdDcDbDaDD 534333231303 0 It follows from (2) that if we get the last three columns 11121314151 DdDcDbDaDDD iiiiiiiiiiiiii 11 0 ,CC andC , then we can recursively determine all other nn 1 n2 ni 2,,6,5 columns ,, CC . Now we shall give the recurrence D D DDcDbDaD n3 1 nnnnnnnnnnnn 122232425262 n nnnnnnnnnn 1121314151 DDbDaDDD n1 formulas for computing ,CC nn 1 andCn2 . DDaDDD 4 3 2 nnnnnnnn 1 n2 (8) Consider the three finite sequences of numbers , BA ii From the recurrence equations (6), (7) and (8), we can give and i niD )2,,1,0( with the initial conditions of a matrix form as DDBBAA 212010 0 and DBA 012 1 , which satisfy TA EAEAEA 2112 nnnnnn , (9) A n1 TT nnn 3 0n1 A13 TB EBEBEB , (3) 2112 nnnnnn , (10)
B TD EDEDED TT n1 0 nn 2 11 2 nnnn . (11) nnn 3 B n1 13 , (4) Multiply (9),(10),(11) by ,, QQQ 131211 respectively to get the following equations D n1 TT nnn 3 0n1 D13 TA EQAEQAEQAQ , (5) 11 nnn nn 1121111211 n , (12)
t t where n I0T 333)3( , A 10 AAA n1],,,[ , TB EQBEQBEQBQ t t t 12 nnn nn 1221121212 n , (13) B 10 BBB n1],,,[ , D 10 DDD n1],,,[ , A ,, AAA nnn 21 , t t B ,, BBB nnn 21 , D ,, DDD nnn 21 . Here I 33 is the 33 identify matrix. TD EQDEQDEQDQ 13 nn 213 n 1131 nn 132 n , (14) On the basis of (3), (4) and (5) ,we can get the Eqs
multiply (9),(10),(11) by ,, QQQ 232221 respectively to get the
AdAcAbAa 31211101 0 following equations AdAcAbAaA 0 4232221202
AdAcAbAaAA 534333231303 0 TA 21 nnn nn 2121211221 EQAEQAEQAQ n 11121314151 AdAcAbAaAAA iiiiiiiiiiiiii 11 0 , (15) ni 2,,6,5 AAcAbAaAAA nn 62 52 nnnnnnnnnn 12223242 n AAbAaAAA TB EQBEQBEQBQ nnnn nnnnnn 1121314151 n1 22 nnn nn 2221221222 n , (16) AAaAAA nn 4 3 2 nnnnnn 1 n2 (6)
TD EQDEQDEQDQ 23 nn 223 n 1231 nn 232 n , (17) BdBcBbBa 31211101 0 BdBcBbBaB 0 4232221202 multiply (9),(10),(11) by ,, QQQ 333231 respectively to get the BdBcBbBaBB 0 534333231303 following equations 11121314151 BdBcBbBaBBB iiiiiiiiiiiiii 11 0 ni 2,,6,5 BBcBbBaBBB TA EQAEQAEQAQ nnnnnnnnnnnn 122232425262 n 31 nnn nn 3121311231 n , (18) nnnnnnnnnn 1121314151 BBbBaBBB n1 BBaBBB 4 3 2 nnnnnnnn 1 n2 (7) TB EQBEQBEQBQ 32 nnn nn 3221321232 n , (19)
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TD EQDEQDEQDQ 33 nn 233 n 1331 nn 332 n , (20) If n1 BA n 0,0 and An 0 , then Z is non-null and TZ 0 . where If n BA n2 0,0 and An2 0 , then Y is non-null nnnn , nnnn , BABAQDADAQDBDBQ nnnn 211213122112211211 , and TY 0 . , , BABAQDADAQDBDBQ 221 nnnn 222 22 nnnn 223 nnnn 2 , If n2 1 AAA nn ,0 then A is non-null and TA 0 . , , BABAQDADAQDBDBQ The proof is completed. 131 nnnn 321 nnnn 13311 nnnn 1 . Theorem 2.2. Suppose that , then T is invertible Let (12)+(13)+(14), (15)+(16)+(17)and(18)+(19)+(20), X n 0 thus and
t TX EX n n n2120 ZZZZZZC nn 21 ]/,,/,/[ nn 2 , (21) t n1 n n1110 YYYYYYC nn 11 ]/,,/,/[ XXXXXXC ]/,,/,/[ t n2 0 n 1 n 1 nn (27) TY EY nn 11 , (22) Proof. According to the determinant of tridiagonal and pentadiagonal matrix (see [1]), we can get n TZ 2 EZ nn n , so T is invertible. From the (21), , (23) T k Xd n 0)()1()det( k1 t t (22), (23) and (27), we have T EC nn , T n EC n11 and where X 10 ,,, XXX n1 , Y 10 ,,, YYY n1 , t T n2 EC n2 . The proof is completed. Z 10 ,,, ZZZ n1 , and for ni 20 Algorithm 2.3. To find the nn inverse matrix of the AAA sevendiagonal matrix in (1) by using the relations (2)-(27). n2 1 in X det BBB i n2 1 in INPUT: Order of the matrix n and the components a , b , i i n2 1 DDD in , (24) ci , di , i , i , i , ,,2,1 ni , where
n 12 ddd nn 1 , . AAA 1 bcc nnn 321211 0 n2 in 1 Yi det n2 BBB in OUTPUT: inverse matrix T . n2 DDD in , (25) Step 1: If dk 0 for any nk 3,,2,1 , then set
k td (t is just a symbolic name). AAA Step 2: Set AA 0 , A 1, then for ni 2,,3 , n1 in 10 2 compute the components by using (6). Zi det n1 BBB in Ai n1 DDD in . (26) Step 3: Set BB 20 0 , B1 1, then for ni 2,,3 ,
compute the components Bi by using (7). We can easily get nn 1 ZYX n2 by (24), (25) and (26).
Step 4: Set D0 1, DD 21 0 , then for ni 2,,3 , Lemma 2.1. If X n 0 , then the matrix T is singular. compute the components Di by using (8). Proof. We have Step 5: For ni 20 , compute the components
,YX ii and Zi by using (24), (25) and (26). nnnn , 2021120 nnnn , 102 BABAZBABAYBABAX nnnn 1 Step 6: Substitute the actual value t 0 , and compute If BA 0,0 and A 0 , then X is non-null n n2 n1 n1 T n )()1()det( Xd . If the matrix T is singular , then and TX 0 . k n k 1 OUTPUT (‘Singular Matrix’); Stop.
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Step 7: Compute the components ,CC nn 12 andCn by using Example 3.1. Consider the 66 matrices T and H (27). given by
Step 8: Compute the components ,CC nn 43 and Cn5 by using the first three equations in (2). 1 21 1 0 0 Step 9: For nj 4:1:3 , compute the components 1 111 1 0 C by using the last equation in (2). 2 0 111 1 j3 T 11 1 2 1 1 Remark 2.4. Here the algorithm complexity of algorithm 2.3 will be computed. 10 1 2 1 4 The algorithm complexity in step 1 is n 3 , in each step 0 0 1 11 1 2-4 is n 3 , in step 5 is n 93 , in the step 6-8 are 3,3, nnn 2 respectively, and in the last step is 6nn . So the algorithm 0 0 21 1 1 complexity of algorithm 2.3 is 2 nn 158 . 0 111 1 1 Here we will discuss the inverse of nn anti- 111 1 0 2 H sevendiagonal matrix, the anti-sevendiagonal matrix of T is 1 1 2 11 1 denoted by H , take the form 4 1 2 11 0 11 1 1 0 0 0 0 abcd 1111 abcd 22222 By applying the algorithm 2.3, we get abcd 333333 0 a 4444 H t t d 0 n3 A ]4,3,2,1,0,0[ and A ]4,24,12[ , (step c n2 2) bn1 a 0 0 nnnn (28) B ]3,2,1,0,1,0[ t B ]2,17,8[ t To do this we take into account the fact that for the nn and , (step 3) matrix R rij )( given by
t t 0 10 D ]5,2,1,0,0,1[ D ]4,24,8[ and , (step 01 4) R Rt 0 t X ,20,20,20,0,20[ 6 ,80and]20 7 ,0 XXX 8 0 01 0 (29) t Y 6 7 ,80,0and]24,0,8,16,16,8[ YYY 8 0 t From [1] we can get Z and]4,0,28,56,96,12[ 6 ,0 7 ZZZ 8 80,0 ,
H TR . (30) (step 5) Moreover, if the matrix T is invertible and its inverse is 1 T , then we obtain 6 6 T k Xd 6 080)()1()det( k 1 111 1 , (step H R RTT . (31) 6)
III. ONE ILLUSTRATIVE EXAMPLE In this section we are going to give one illustrative example.
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t REFERENCES C6 ,15.0[ ,2.1 ,7.0 ,35.0 ,0 ]05.0 t [1] A. D. A. Hadj, M. Elouafi, A fast numerical algorithm for the inverse of C5 [ ,1.0 ,2.0 ,2.0 ,1.0 ,0 ]3.0 a tridiagonal and pentadiagonal matrix, Appl. Math. Comput. 202 (2008) C [ ,25.0 ,0 ,25.0 ,25.0 ,25.0 ]25.0 t 441-445. 4 , [2] M. E. Kanal, N. A. Baykara, M. Demiralp, Theory and algorithm of the inversion method for pentadiagonal matrices, J. Math. Chem. 50 (2012) 289-299. (step 7) [3] E.Kilic, Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions, Appl. Math. Comput. 197 (2008) 345-357. [4] M. El-Mikkawy, E.-D. Rahmo, A new recursive algorithm for inverting general periodic pentadiagonal and anti-pentadiagonal matrices, Appl. t Math. Comput. 207 (2009) 164-170. C3 [ ,3.0 ,4.0 ,0.15 ,0.2 ,0.25 ].10 t [5] A. A. Karawia, A computational algorithm for solving periodic penta- C2 [ ,0 ,1 ,.50 ,0 ,5.0 ]0 diagonal linear systems, Appl. Math. Comput. 174 (2006) 613-618. t [6] C.Y. Yu, Y. H. Chen, A new algorithm for computing the inverse of a C1 ,15.0[ ,0.2 ,0.45 ,.150 ,25.0 ].050 . (step 8) periodic tridiagonal matrix, Math. Prac. Theo. 40 (2010) 192-198. [7] R. S. Ran, T. Z. Huang, X.P.Liu, T. X. Gu, Algorithm for the inverse of So, a general tridiagonal matrix, Appl. Math. Mech. 30 (2009) 238-244. [8] D. Tang, Fast inversion of periodic tridiagonal matrices, J. SHH. DJ. Univ. 16 (2013) 300-304. 325603 [9] M.EI-Mikkawy, E.-D. Rahmo, Symbolic algorithm for inverting cyclic 24408204 . pentadiagonal matrices recursively-Derivation and implementation, Comput. Math. Appl. 59 (2010) 1386-1396. 1 14453109 T1 [10] R. S. Ran, T. Z. Huang, Inverses of block tridiagonal matrices, J. Univ. 20 725403 Elect. Scie. Tech. Chn. 36 (2007) 341-342. 0055105 [11] F. Chen, Q. Lu, Z. J. Yuan, Fast algorithm for inverting a block 165201 pentadiagonal matrix, J. HF. Univ. Tech. 31 (2008) 1904-1907. [12] G. Liu, T. Z. Huang, An algorithm for the inverse of tri-diagonal and Hence, using (31) gives five-diagonal Toeplitz matrices, Pure. Appl. Math. 26 (2010) 292-299. [13] X. L. Lin, P. P. Huo, J. T. Jia, A new recursive algorithm for inverting general periodic sevendiagonal and anti-sevendiagonal matrices, F.E.J. 165201 Appl. Math. 86 (2014) 41-55. [14] J. T. Jia, X.L.Lin, A new computational algorithm for inverting general 0055105 . periodic seven-diagonal matrices, Pure. Appl. Math. 26 (2010) 1040- 1 725403 1046. H1 20 14453109 [15] A. A. Karawia, Inversion of general cyclic heptadiagonal matrices, Mathe. Prob. Engin. (2013) 1-9. 24408204 325603
IV. CONCLUSION In this paper, we mainly give a fast algorithm for the inverse of a sevendiagonal matrix, and the fast algorithm is appropriate for implementation using computer algebra systems. What’s more, in the light of algorithm 2.3, then we can obtain that the algorithm complexity is 2 nn 158 . Base on the illustrative example of section 3 which confirm the validity of this algorithm. The recent algorithm presented in [1] for inverting a tridiagonal and pentadiagonal matrix is a special case of this new algorithm.
ACKNOWLEDGMENT This work was supported by Research Project Funding Academic Team of Shaanxi University of Science and Technology (2013XSD39) and Research Projects of Provincial Key Laboratory of Science and Technology Department of Shaanxi Province (2011HBSZS014).
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