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A Numerical Algorithm for Computing the Inverse of a Toeplitz Pentadiagonal Matrix

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A Numerical Algorithm for Computing the Inverse of a Toeplitz Pentadiagonal Matrix Journal of Applied Mathematics and Computational Mechanics 2018, 17(3), 83-95 www.amcm.pcz.pl p-ISSN 2299-9965 DOI: 10.17512/jamcm.2018.3.08 e-ISSN 2353-0588 A NUMERICAL ALGORITHM FOR COMPUTING THE INVERSE OF A TOEPLITZ PENTADIAGONAL MATRIX 1 2 1 Boutaina Talibi , Ahmed Driss Aiat Hadj , Driss Sarsri 1 National School of Applied Sciences, Abdelmalek Essaadi University Tangier, Morocco 2 Regional Center of the Trades of Education and Training (CRMEF)-Tangier, Avenue My Abdelaziz Souani, BP: 3117, Tangier, Morocco [email protected], [email protected], [email protected] Received: 2 August 2018; Accepted: 27 November 2018 Abstract. In the current paper, we present a computationally efficient algorithm for obtain- ing the inverse of a pentadiogonal toeplitz matrix. Few conditions are required, and the al- gorithm is suited for implementation using computer algebra systems. MSC 2010: 15B05 65F40 33C45 Keywords: pentadiagonal matrix, Toeplitz matrix, K-Hessenberg matrix, inverse 1. Introduction Pentadiagonal Toeplitz matrices often occur when solving partial differential equations numerically using the finite difference method, the finite element method and the spectral method, and could be applied to the mathematical representation of high dimensional, nonlinear electromagnetic interference signals [1-7]. Pentadiagonal matrices are a certain class of special matrices, and other common types of special matrices are Jordan, Frobenius, generalized Vandermonde, Hermite, centrosymmetric, and arrowhead matrices [8-12]. Typically, after the original partial differential equations are processed with these numerical methods, we need to solve the pentadiagonal Toeplitz systems of linear equations for obtaining the numerical solutions of the original partial differ- ential equations. Thus, the inversion of pentadiagonal Toeplitz matrices is neces- sary to solve the partial differential equations in these cases. J. Jia, T. Sogabe and M. El-Mikkawy recently proposed an explicit inverse of the tridiagonal Toeplitz matrix which is strictly row diagonally dominant [13]. However, this result cannot be generalized to the case of the pentadiagonal Toeplitz matrix directly. This motivates us to investigate whether an explicit inverse of the pentadiagonal Toeplitz matrix also exists. 84 B. Talibi, A.D.A. Hadj, D. Sarsri In this paper, we present a new algorithm with a totally different approach for numerically inversing a pentadiagonal matrix. Few conditions are required, and the algorithm is suited for implementation using computer algebra systems. 2. Inverse of a Toeplitz pentadiagonal matrix algoritm 1 Definition: We consider the pentadiagonal Toeplitz matrix: a b c 0⋯ 0 a b c ⋱ ⋮ α β α ⋱ ⋱ ⋱ 0 Ρ = 0 ⋱ ⋱ ⋱ ⋱ c ⋮ ⋱ ⋱ ⋱ ⋱ b 0⋯ 0 β α a Ρ∈ Μ Κ a b c α Where: n, n ( ) . Also , , , and β are an arbitrary numbers. Assume that Ρ is non-singular and denote: −1 Ρ = (C1 , C 2 ,..., C n ) −1 Where (Ci ) 1≤ i ≤ n are the columns of the inverse Ρ . −1 From the relation Ρ Ρ = In . Where In denotes the identity matrix of order n. We deduce the relations: 1 C=( E − bC − aC ) n−2c nn − 1 n 1 C=( E − bC − aC − α C ) (1) n−3c nn −− 1 2 n − 1 n − 2 1 C=( E − bC −− aCα C − β C ) for j= n − 2,… ,3 j−2c jj − 1 jj ++ 12 j E =δ t ∈ k n kn Where j[( ij, ) 1 ≤ in ≤ ] is the vector of order j of the canonical basis of . Consider the sequence of numbers (Ai ) 1≤ i ≤ n and (Bi ) 1≤ i ≤ n characterized by a term recurrence relation: A0 = 0 A1 = 1 (2) aA0+ bA 1 + cA 2 = 0 α A0+ aA 1 + bA 2 + cA 3 = 0 A numerical algorithm for computing the inverse of a Toeplitz pentadiagonal matrix 85 And βAi−−−321+ α A i + aA i ++ bA ii cA + 1 = 0 for i ≥ 3 Also we have: B0 = 0 B1 = 1 (3) aB0+ bB 1 + cB 2 = 0 αB0+ aB 1 + bB 2 + cB 3 = 0 And βBi−−−321+ α B i + aB i ++ bB ii cB + 1 = 0 for i ≥ 3 We can give a matrix form to this term recurrence: ΡΑ = −AEnn−1 − A n + 1 E n (4) ΡΒ = −BEnn−1 − B n + 1 E n (5) t t Where Α = [,,,A0 A 1… A n− 1 ] and Β = [,,,B0 B 1… B n− 1 ] . Let’s define for each 0≤i ≤ n + 1 An+1 A i An A i X i = det and Yi = det (6) Bn+1 B i Bn B i Immediately we have: ΡΧ = − Xn E n −1 and ΡΥ = − Yn+1 E n (7) t t Where Χ = [,,,X0 X 1… X n− 1 ] and Υ = [,,,Y0 Y 1… Y n− 1 ] . Lemma: If X n = 0 the matrix Ρ is singular. Proof: If X n = 0. The Χ is non-null and by (7) we have ΡΧ = 0. We conclude that Ρ is singular. THEOREM 2.1: We supposed that X n ≠ 0 then Ρ is invertible. And: t −Y0 − Y 1 − Y n− 1 Cn = , ,… , Yn+1 Y n + 1 Y n + 1 t (8) −X0 − X 1 − X n− 1 Cn−1 = , ,… , Xn X n X n 86 B. Talibi, A.D.A. Hadj, D. Sarsri n−2 Proof: We have det()Ρ =c X n ≠ 0 [14]. So Ρ is invertible from the relations Equations (7) and (8) we have ΡCn= E n and ΡCn−1= E n − 1 . The proof is completed. Algorithm: New efficient computational algorithm for computing the inverse of the pentadiagonal matrix. INPUT: The dimension n. The arbitrary numbers: β, α ,a , b and c. OUTPUT: −1 The inverse matrix Ρ = (C1 , C 2 ,… , C n ). Step 1: t −Y0 − Y 1 − Y n− 1 Cn = , ,… , Yn+1 Y n + 1 Y n + 1 t −X0 − X 1 − X n− 1 Cn−1 = , ,… , Xn X n X n Step 2: 1 C=( E − bC − aC ), n−2c nn − 1 n 1 C=( E − bC − aC − α C ), n−3c nn − 1 − 2 n − 1 n Step 3: 1 C=( E − bC −− aCα C − β C ) for j= n − 2,… ,3 j−2c jj − 1 jj ++ 12 j 3. Inverse of a Toeplitz pentadiagonal matrix algoritm 2 H= h Definition: Hessenberg (or lower Hessenberg) matrices are the matrices [ij ] h = 0 − > 1 satisfying the condition ij for j i . More generally, the matrix is H = 0 − > k-Hessenberg if and only if ij for j i k . THEOREM [15]: Let H be a strict-Hessenberg matrix with the block decomposi- tion: B A H A∈Μ F B∈Μ F C∈ Μ F D∈ Μ F = , n− k ( ) , (n− k ) × k ( ) , k×( n − k ) ( ) , k ( ) . (9) D C A numerical algorithm for computing the inverse of a Toeplitz pentadiagonal matrix 87 The H is invertible if and only if CA−1 B− D is invertible and if H is invertible we have: 0 0 I H−1 = −k CA−1 B −− D − 1 CA − 1 I −1 − 1 ( ) k (10) A0 − A B R. Slowik [16] worked in the particular case of the Toeplitz-Hessenberg matrix. Our work is applied on pentdiagonal matrix Ρ of the form: a b c 0⋯ 0 a b c ⋱ ⋮ α β α ⋱ ⋱ ⋱ 0 Ρ = (11) 0 ⋱ ⋱ ⋱ ⋱ c ⋮ ⋱ ⋱ ⋱ ⋱ b 0⋯ 0 β α a Then we find: 0 0 I −1 Ρ−1 = −2 CA−1 B −− D CA − 1 I −1 − 1 () 2 (12) A0 − A B The blocks B, C, D and A are given as: a b a α β α B 0 = β (13) 0 0 ⋮ ⋮ 0 0 Also: 0⋯ 0 β α a b C = (14) 0⋯ ⋯ 0 β α a 0 0 D = (15) 0 0 And: 88 B. Talibi, A.D.A. Hadj, D. Sarsri c 0 0⋯ ⋯ ⋯ ⋯ 0 b c 0⋯ ⋯ ⋯ ⋯ 0 a b c 0⋯ ⋯ ⋯ 0 α a b c 0⋯ ⋯ 0 A = (16) β α a b c 0 ⋯ ⋮ 0β α a b c 0 ⋮ ⋮ ⋱ ⋱ ⋱⋱ ⋱ ⋱ 0 0⋯ 0 β α a b c Then the inverse of the matrix A will be: 1 n−2 n − 1 − k A−1 = l E (17) c ∑ ∑ k r+ kr, k=0 r = 1 1 Where l =1 and l=−( bl + al +α l + β l ) for k ≥ 1. 0 kc kk−1 − 2 k − 3 k − 4 −1 Proof: Since A is a triangular matrix, A as well. We prove (17) inductively on n. We have: 1 1 1 −b 01 1 = − l 1 (18) c 1 −b l2 l 1 1 0 1 −a c 0 1 c With: 2 b b a 1 l1 = − ; l2= −=−( bl 1 + al 0 ) (19) c c c c The first step of induction holds. Consider now n > 4. We have 1 b 1 − 1 1 c 0 1 l 1 a 1 0l 1 − 1 =l l 1 1 c 2 1 (20) ⋮⋮⋱ ⋮⋮⋱ 0 1 0ll⋯ 1 lll⋯ 1 n−3 1 ⋮ ⋱ n−2 n − 3 1 0 1 A numerical algorithm for computing the inverse of a Toeplitz pentadiagonal matrix 89 1 Then: l=− (bl + al +α l + β l ) n−2c nnnn −−−− 3456 4. Example In this section, we give a numerical example to illustrate the effectiveness of our algorithm. Our algorithm is tested by MATLAB R2014a. Consider the following 7-by-7 pentadiagonal matrix 2340000 5234000 6523400 Ρ 0652340 = (21) 0065234 0006523 0000652 −1 The columns of the inverse Ρ are: −1.0787 −0.1948 0.6886 −3.4400 −1.2309 2.1724 3.3693 1.0205 −1.9736 C = 0.5414 C = 0.3435 C = 0.4667 1 , 2 , 3 − 3.8273 1.0629 −2.1615 −2.1929 −0.3983 1.0629 −5.9997 −2.1929 3.8273 0.0305 0.5616 −0.6926 −0.0843 0.4866 1.7791 −1.9104 −0.6926 −0.3802 −1.6151 1.7791 0.5616 C = 0.0037 C = 0.3802 C = 0.4866 C = 0.0305 4 , 5 − , 6 , 7 −0.4667 −1.9736 2.1724 0.6886 0.3435 1.0205 −1.2309 −0.1948 0.5414 3.3693 −3.4400 −1.0787 Let’s employ the same example and give the blocks A, B, C, and D of the matrix Ρ 90 B.
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