A New Algorithm for Inverting General Cyclic Heptadiagonal Matrices Recursively A.A. KARAWIA1 Computer Science Unit, Deanship of Educational Services, Qassim University, Buraidah 51452, Saudi Arabia. [email protected] ABSTRACT In this paper, we describe a reliable symbolic computational algorithm for inverting general cyclic heptadiagonal matrices by using parallel computing along with recursion. The algorithm is implementable to the Computer Algebra System(CAS) such as MAPLE, MATLAB and MATHEMATICA. An example is presented for the sake of illustration. Key Words: Cyclic heptadiagonal matrices; LU factorization; Determinants; Inverse matrix; Linear systems; Computer Algebra System(CAS). 1. INTRODUCTION The n × n general periodic heptadiagonal matrices are takes the form: d1 a 1 A 1 C 100 B 1 b 1 b d a A C00 B 2 2 2 2 2 2 B3 b 3 d 3 a 3 A 3 C 3 00 D3 B 3 b 3 d 3 a 3 A 3 C 3 0 0 H (1.1) 0 0 D B b d a A C n3 n 3 n 3 n 3 n 3 n 3 n 3 00Dn2 B n 2 b n 2 d n 2 a n 2 A n 2 A00 D B b d a n1 n 1 n 1 n 1 n 1 n 1 an A n0 0 0 D n B n b n d n Where n 8. 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 (1.1) and for solving linear systems of the form: Hx r (1.2) 1 Home address: Mathematics Department, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt. E-mail:[email protected] TT where x( x1 , x 2 ,..., xnn ) , r ( r 1 , r 2 ,..., r ) . To the best of our knowledge, the inversion of a general cyclic heptadiagonal matrix of the form (1.1) has not been considered. Very recently in [9], 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 (1.1) without imposing any restrictive conditions on the elements of the matrix H in (1.1). Our approach is mainly based on getting the elements of the last four columns of H-1 in suitable forms via the Doolittle LU factorization [10] along with parallel computation [6]. Then the elements of the remaining (n - 4) 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 [9]. The development of a symbolic algorithm is considered in order to remove all cases where the numerical algorithm fails. The paper is organized as follows. In Section 2, new symbolic computational algorithms, that will not break, is constructed. In Section 3, an illustrative example is given. Conclusions of the work are given in Section 4. 2. Main results In this section we shall 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 (1.2) 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 (2.1) where 1 0 0 0 0 0 0 0 0 f 2 1 0 0 0 0 0 0 0 ef331 0 0 0 0 0 0 D4 ef441 0 0 0 0 0 1 0 L 0 0 0efnn33 1 0 D 0 0 0n 2 ef 1 0 nn22 n 5 (2.2) k k k k k k k 10 1 2 3n 5 n 4 n 3 n 2 h1 h 2 h 3 hn 5 h n 4 h n 3 h n 2 h n 1 1 and 1g 1 z 1 C 10 0 0 w 1 v 1 0 g z 0 0 w v 2 2 2 2 2 0 03g 3 0 0 w 3 v 3 (2.3) z C w v U n5 n 5 n 5 n 5 gn4 z n 4 w n 4 v n 4 g w v n3 n 3 n 3 n 3 0 n2wv n 2 n 2 00 v nn11 0 0 0 0 0 0 n The elements in the matrices L and U in (2.2) and (2.3) satisfy: di1 if 1 d f gif i 2 2 2 1 d e z f gif i 3 3 3 1 3 2 Di di C i3 e i z i 2 f i g i 1 if i 4(5) n 2 i i 3 (2.4) n 2 dn1 w j k j if i n 1 j 1 n 1 dn v j h j if i n , j 1 An 1 ifi 1 1 kg 11 ifi 2 2 ()k z k g 1 1 2 2 ifi 3 3 ()ki3 C i 3 k i 2 z i 2 k i 1 g i 1 ki if i 4(5) n 5 (2.5) i ()D k C k z k g n1 n 7 n 7 n 6 n 6 n 5 n 5 ifin 4 n 4 ()B k C k z k g n1 n 6 n 6 n 5 n 5 n 4 n 4 ifin 3 n 3 (b k C k z k g ) n1 n 5 n 5 n 4 n 4 nn33 ifin 2, n 2 a n ifi 1 1 A h g n 11 ifi 2 2 ()h z h g 1 1 2 2 ifi 3 3 ()hi3 C i 3 h i 2 z i 2 h i 1 g i 1 hi if i 4(5) n 4 i ()D h C h z h g n n6 n 6 n 5 n 5 n 4 n 4 ifin 3 n 3 ()B h C h z h g n n5 n 5 n 4 n 4 n 3 n 3 ifin 2 n 2 n 2 ()bn h j w j j 1 ifin 1, (2.6) n 1 bi1 if 1 B f vif i 2 2 2 1 e3 v 1 f 3 v 2 if i 3 (2.7) Di vi v i3 e i v i 2 f i v i 1 if i 4(5) n 4 i 3 Dn 3 Cn3 v n 6 e n 3 v n 5 f n 3 v n 4 if i n 3 n 6 Dn 2 An2 v n 5 e n 2 v n 4 f n 2 v n 3 if i n 2 n 5 n 2 an1 v j k j if i n 1, j 1 Biif 1 1 f wif i 2 21 f3 w 2 e 3 w 1 if i 3 (2.8) Di wi w i3 e i w i 2 f i w i 1 if i 4(5) n 5 i 3 Dn 4 Cn4 W n 7 e n 4 w n 6 f n 4 w n 5 if i n 4 n 7 Dn 3 An3 w n 6 e n 3 w n 5 f n 3 w n 4 if i n 3 n 6 Dn 2 an2 w n 5 e n 2 w n 4 f n 2 w n 3 if i n 2, n 5 b 2 if i 2 1 b3 e 3 g 1 (2.9) fii if 3 2 D bi z e g i i32 i i i 3 if in4(5) 2, i 1 B 3 if i 3 1 (2.10) ei D Bg i ii 3 i 3 if in4(5) 2, i 2 ai1 if 1 (2.11) gi a2 f 2 z 1 if i 2 ai f i z i12 e i C i if i 3(4) n 3, Ai1 if 1 (2.12) z i Ai f i C i 1 if i 2(3) n 4. We also have: n detH i . (2.13) i 1 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 (1.1) and can be considered as natural generalization of the symbolic algorithm DETCPENTA in [9]. Algorithm 2.1. To compute det H for the cyclic heptadiagonal matrix H in (1.1), we may proceed as follows: Step 1: Set α1=d1. If α1=0 then α1= t end if. Set g1=a1, z1=A1, k1=An-1/ α1, v1=b1, w1=B1, h1=an/ α1, w1=B1 , f2=b2/ α1, e3=B3/ α1, α2=d2-f2 *g1. If α2=0 then α2= t end if. Set K2=-k1*g1/ α2, v2=B2- f2*v1, w2=-f2*w1, h2=(An-h1*g1)/ α2, g2=a2-f2*z1, f3=(b3-e3*g1)/α2, α3=d3-e3*z1-f3*g2. If α3=0 then α3= t end if. Set k3=-(k1*z1+k2*g2)/α3, h3=-(h1z1+h2g2)/α3, v3=-e3*v1-f3*v2, w3=-f3w2-e3w1. Step 2: Compute and simplify: For i from 4 to n-2 do ei=(Bi-Di* gi-3/ αi-3)/ αi-2 fi=(bi- Di* zi-3/ αi-3-ei*gi-2)/ αi-1 zi-2=Ai-2-fi-2*Ci-3 gi-1=ai-1-fi-1*zi-2-ei-1*Ci-3 αi=(di- Di* Ci-3/ αi-3-ei*zi-2-fi*gi-1) If αi=0 then αi= t end if End do Step 3: Compute and simplify: For i from 4 to n-5 do Ki=-(ki-3*Ci-3+ki-2*zi-2+ki-1*gi-1)/ αi wi=-(Di*wi-3/ αi-3+ei*wi-2+fi*wi-1) End do Step 4: Compute and simplify: For i from 4 to n-4 do hi=-(hi-3*Ci-3+hi-2*zi-2+hi-1*gi-1)/ αi vi=-(Di*vi-3/ αi-3+ei*vi-2+fi*vi-1) End do Step 5: Compute simplify: kn-4=(Dn-1-kn-5*gn-5-kn-6*zn-6-kn-7*Cn-7)/ αn-4 kn-3=(Bn-1-kn-4*gn-4-kn-5*zn-5-kn-6*Cn-6)/ αn-3 kn-2=(bn-1-kn-3*gn-3-kn-4*zn-4-kn-5*Cn-5)/ αn-2 wn-4=Cn-4-Dn-4*wn-7/αn-7-en-4*wn-6-fn-4wn-5 wn-3=An-3-Dn-3*wn-6/αn-6-en-3*wn-5-fn-3wn-4 wn-2=an-2-Dn-2*wn-5/αn-5-en-2*wn-4-fn-2wn-3 hn-3=(Dn-hn-4*gn-4-hn-5*zn-5-hn-6*Cn-6)/ αn-3 hn-2=(Bn-hn-3*gn-3-hn-4*zn-4-hn-5*Cn-5)/ αn-2 vn-3=Cn-3-Dn-3*vn-6/αn-6-en-3*vn-5-fn-3*vn-4 vn-2=An-2-Dn-2*vn-5/αn-5-en-2*vn-4-fn-2*vn-3 n 2 vn-1=an-1- kvjj j 1 n 2 αn-1=dn-1- kwjj j 1 If αn-1=0 then αn-1= t end if n 2 hn-1=(bn- hwjj)/ αn-1 j 1 n 1 αn=dn- hvjj j 1 If αn=0 then αn= t end if n Step 6: Compute det H =i .