An explicit formula for the inverse of a pentadiagonal Toeplitz matrix

Accepted Manuscript An explicit formula for the inverse of a pentadiagonal Toeplitz matrix Chaojie Wang, Hongyi Li, Di Zhao PII: DOI: Reference:

S0377-0427(14)00367-7 http://dx.doi.org/10.1016/j.cam.2014.08.010 CAM 9764

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Journal of Computational and Applied Mathematics

Received date: 29 December 2013 Revised date: 26 May 2014 Please cite this article as: C. Wang, H. Li, D. Zhao, An explicit formula for the inverse of a pentadiagonal Toeplitz matrix, Journal of Computational and Applied Mathematics (2014), http://dx.doi.org/10.1016/j.cam.2014.08.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

An explicit formula for the inverse of a pentadiagonal Toeplitz matrix 1 Chaojie Wanga , Hongyi Lia , Di Zhaoa,∗ a. LMIB, School of Mathematics and System Science, Beihang University , Beijing, P.R.China, 100191

Abstract In this paper, we mainly consider finding an explicit formula for the inverse of a pentadiagonal Toeplitz matrix. For that purpose, we first factorize the modified form of a pentadiagonal Toeplitz matrix by two tridiagonal Toeplitz matrices, and then use the Sherman-Morrison-Woodbury inversion formula. As a result, an explicit inverse of a pentadiagonal Toeplitz matrix is obtained under certain assumptions. And numerical experiments are given to show the effectiveness of our results. Keywords: Inverse, Pentadiagonal matrix, Toeplit matrix, Factorization, Sherman-Morrison-Woodbury inversion 1. Introduction The n-by-n pentadiagonal  x   z    w A=    

Toeplitz matrix takes the form  y v  ..  . x y  .. ..  . . z x  . .. .. .. .. . v  . . .   ... ... ... y  w z x

(1.1)

The project was supported by the National Natural Science Foundation of China (Grant No.61379001). ∗ Corresponding author: Di Zhao. E-mail address: [email protected] 1

Preprint submitted to Journal of Computational and Applied MathematicsSeptember 4, 2014

Pentadiagonal Toeplitz matrices often occur when solving partial differential equations numerically using finite difference method, finite element method, spectral method, and could be applied to the mathematical representation of high dimensional, nonlinear electromagnatic interference signals[17]. Pentadiagonal matrices is a certain class of special matrices, and other common types of special matrices are Jordan, Frobenius, generalized Vandermonde, Hermite, centrosymmetric, and arrowhead matrices[8-9, 26-28]. Usually, 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 differential equations. Thus, the inversion of pentadiagonal Toeplitz matrices is necessary to solve the partial differential equations in these cases. Recently, J. Jia, T. Sogabe and M. El-Mikkawy have proposed an explicit inverse of the tridiagonal Toeplitz matrix which is strictly row diagonally dominant[12]. However, this result can not 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. Based on the results presented in [12], we derive an explicit formula for the inverse of the pentadiagonal Toeplitz matrix under some assumptions by using the Sherman-Morrison-Woodbury inversion formula and the factorization of the modified pentadiagonal Toeplitz matrices [13] . The remainder of this paper is organized as follows: in Section 2, we perform a comparison with other methods. In Section 3, a method for obtaining the inverse of a pentadiagonal Toeplitz matrix is given. In Section 4, a numerical example is provided to show the effectiveness of our results. Finally, we present some concluding remarks in Section 5. 2. Comparison with other methods In recent years, many researchers have studied on the inversion of the pentadiagonal Toeplitz matrix or related matrices: • In [10] and [21], the authors proposed two explicit formulas for the inverse of general Toeplitz matrices, respectively. These two methods could be directly applied to inverting the pentadiagonal Toeplitz matrix, but they couldn’t make full use of the sparse structure of the pentadiagonal Toeplitz matrix.

2

• [22] gave a method to obtain the inverse of a general pentadiagonal matrix which was not specific to the pentadiagonal Toeplitz matrix and required general Doolittle factorization. • [20] presented a numerical method for computing the inverse of the pentadiagonal Toeplitz matrix that involves solving a system of linear equations. In summary, all the above methods are not very efficient for the inversion of the pentadiagonal Toeplitz matrix. This motivates us to find a better method for computing the inverse of the pentadiagonal Toeplitz matrix. 3. Main results In this section, we derive an explicit formula for computing the inverse of a pentadiagonal Toeplitz matrix. 3.1. The factorization of the modified pentadiagonal Toeplitz matrix by tridiagonal Toeplitz matrices It is easy to prove that the product of any two tridiagonal Toeplitz matrices is not a pentadiagonal Toeplitz matrix because the first and the last elements in the principle diagonal are different to the other ones [14-15]. Thus, it is necessary to consider the factorization of a pentadiagonal Toeplitz matrix modified in the first and the last elements of the principle diagonal. Let the modified form of A be an n-by-n pentadiagonal matrix as the following   x−α y v   ..  z  . x y   .. ..   . . z x  w  AM =  (3.1) . ... ... ... ...   v     . . . . . .   . . . y w z x−β

3

The real tridiagonal Toeplitz matrix of size  b c ..  .  a b T = . . .. .. c  a b Denote it by

n is given by    . 

(3.2)

T = tridn [a, b, c]. Factorize AM by two tridiagonal Toeplitz matrices as AM = T1 T2

(3.3)

T1 = tridn [a1 , b1 , c1 ], T2 = tridn [a2 , b2 , c2 ]

(3.4)

where It is easy to verify that the factorization AM = T1 T2 exists if and only if the following equations are satisfied b 1 b 2 + c 1 a2 + a1 c 2 b1 c 2 + c 1 b2 b1 a2 + a1 b2 c1 c2 a1 a2 a1 c 2 c 1 a2

= = = = = = =

x, y, z, v, w, α, β.

(3.5)

A method for solving (3.5) has been proposed in [14]: let s = α + β, then the following two equations are satisfied αβ = vw.

(3.6)

f (s) = 0.

(3.7)

where f (s) = s3 − xs2 + (yz − 4vw)s − (y 2 w + z 2 v − 4xvw). Remark 3.1. Denote the coefficients of the equation (3.7) by ca = 1, cb = −x, cc = yz − 4vw, cd = −(y 2 w + z 2 v − 4xvw). 4

Let ∆1 = c2b − 3ca cc , ∆2 = cb cc − 9ca cd , ∆3 = c2c − 3cb cd , ∆ = ∆22 − 4∆1 ∆3 . Then we have 1) If ∆1 = ∆2 = 0 , equation (3.7) has only one real triple root; 2) If ∆ > 0 , equation (3.7) has one real root and a pair of conjugate imaginary roots; 3) If ∆ = 0 , equation (3.7) has three real roots: one simple and the other double; 4) If ∆ < 0 , equation (3.7) has three different real roots. And we can obtain the roots of the cubic equation (3.7) using Shengjin formulas given in [18]. Then we can get α and β by solving the following quadratic equation h2 − sh + vw = 0.

(3.8)

After α, β and s are obtained, let us study the unknown entries of the tridiagonal Toeplitz matrices T1 and T2 . First, we can easily find from (3.5) that c1 : a1 = v : α, c2 : a2 = v : β. Second, let m = b1 a2 , n = a1 b2 , then m + n = z, mn = w(x − s). So we can obtain m and n by solving the following quadratic equation h2 − zh + w(x − s) = 0.

(3.9)

Thus b1 : a1 = m : w, b2 : a2 = n : w. And then b 1 : c 1 : a1 =

n v m v : : 1, b2 : c2 : a2 = : : 1. w α w β

(3.10)

As a result, we have AM = w · tridn [1,

m v n v , ] · tridn [1, , ]. w α w β 5

(3.11)

Hence we have gained the factorization of the pentadiagonal matrix AM by tridiagonal Toeplitz matrices T1 and T2 by solving one cubic equation and two quadratic equations. 3.2. A method for the inverse of the modified pentadiagonal Toeplitz matrix According to (3.3), if T1 and T2 are nonsingular, then −1 −1 A−1 M = T2 T1

(3.12)

On the basis of (3.11), we have −1 A−1 · tridn [w, m, β]−1 M = w · tridn [w, n, α]

(3.13)

Lemma 3.1. Let T be a tridiagonal Toeplitz matrix as in (3.2) and T −1 = (tij ) . If T is strictly row diagonal dominant, then c j−i tij = ( ) 2 · (ω|i−j| − ω|i+j| ) a

(3.14)

where

and

sign(a)(γ m+1 + γ 2n−m+3 ) , m = 0, 1, · · ·, 2n, ωm = √ ac(γ 2 − 1)(1 − γ 2(n+1) )

√ −sign(a)b + b2 − 4ac sign(a)b √ , √ > 2, ac 2 ac √ γ=  −sign(a)b − b2 − 4ac sign(a)b  √  < −2. , √ ac 2 ac    

Proof. See [12].

¤

Denote T T1 = tridn [w, n, α]. and T T2 = tridn [w, m, β], then we have the following proposition:

6

−1 M Proposition 3.1. Let A−1 = (t1ij ), T T2−1 = (t2ij ), if |m| > M = (aij ), T T1 |w| + |β| and |n| > |w| + |α|, then

aM ij

=w·

n X k=1

t1ik · t2kj

(3.15)

where t1ik and t2kj are defined in (3.14). Proof. We can derive the result immediately according to Lemma 3.1 and the matrix multiplication algorithms. ¤ Therefore we have obtained the explicit inverse of the modified pentadiagonal Toeplitz matrix AM as in (3.1). 3.3. The explicit inverse of a pentadiagonal Toeplitz matrix As for the pentadiagonal Toeplitz matrix defined in (1.1), we can write it as the sum of two matrices A = AM + B where

Moreover,



   B=  

(3.16) 

α 0 ..

. 0 β

   .  

B = UV T where



α 0 .. .

0 0 .. .

   U =   0 0 0 1

(3.17) 



      , V =      

1 0 0 0 .. .. . . 0 0 0 β



   .  

Lemma 3.2.(Sherman-Morrison-Woodbury formula) Let C be an n-by-n 7

nonsingular matrix, X and Y be n-by-m (n ≥ m) matrices. If Im + Y T C −1 X is nonsingular, then C + XY T is nonsingular. In addition, the inverse of the matrix C + XY T can be explicitly given by (C + XY T )−1 = C −1 − C −1 X(Im + Y T C −1 X)−1 Y T C −1

(3.18) ¤

Proof. See [16].

Proposition 3.2. If I2 + V T A−1 M U is nonsingular, we have the explicit inverse of the pentadiagonal Toeplitz matrix A of (1.1) as the following −1 T −1 −1 T −1 A−1 = A−1 M − AM U (I2 + V AM U ) V AM

(3.19)

Proof. The result is obvious after we substitute A = AM + U V T into the Sherman-Morrison-Woodbury formula. ¤ Consequently, we have derived an explicit inverse for the pentadiagonal Toeplitz matrix under some assumptions. 4. Numerical results In this section, a numerical example is presented to illustrate the conclusion of Proposition 3.2. All computations are carried out in Matlab R2010a using double precision arithmetic. Example 1. We consider the fourth-order model problem α1 u − α2 uxx + uxxxx = f, x ∈ R+ u(0) = ux (0) = lim u(x) = lim ux (x) = 0. x→+∞

(4.1)

x→+∞

Here the coefficients α1 and α2 are constant. When the Laguerre-Galerkin spectral method is applied[5], a system with the coefficient matrix A = α1 M + α2 Q + S

8

(4.2)

can be obtained, where M = (mij )n×n , Q = (qij )n×n and S = (sij )n×n are all symmetric pentadiagonal Toeplitz matrices as the following  6, i = j;          −4, |i − j| = 1; mij =   1, |i − j| = 2;        0, others,  1   , i = j;   2     1 qij = − , |i − j| = 2;   4       0, others, and

 3   − ,   8       1    − , 4 sij =    1   − ,    16      0,

i = j;

|i − j| = 1; |i − j| = 2; others.

Thus, the coefficient matrix A = (aij )n×n in (4.2) is also a symmetric pentadiagonal Toeplitz matrix and its elements are  1 3   6α1 + α2 + , i = j;   2 8         −4α1 + 1 ,  |i − j| = 1; 4 aij =    1 1   α1 − α2 + , |i − j| = 2;    4 16      0, others. 9

For example, if we take α1  65.375 −39.75   −39.75 65.375    7.5625 −39.75 A= ...    

= α2 = 10, the coefficient matrix A will be  7.5625  ..  . −39.75  .. ..  . . 65.375  . ... ... ... 7.5625    ... ... ... −39.75  7.5625 −39.75 65.375

Then, let us compute the inverse of the coefficient matrix A by using our new method. First, consider equation (3.7), ∆ = −1728000 < 0, so it has three different roots according to Remark 3.1. And we can use Shengjin formulas to obtain the three real roots as the following: s1 = 15.125, s2 = 31.4496, s3 = 18.8004. Second, solve equation (3.8) and equation (3.9), and note that T T1 , T T2 should be strictly row diagonal dominant, we have the following three groups of solution (i) s1 = 15.125, α = 7.5625, β = 7.5625, m = −23.7480, n = −16.0020; (ii) s1 = 15.125, α = 7.5625, β = 7.5625, m = −16.0020, n = −23.7480; (iii) s3 = 18.8004, α = 14.9835, β = 3.8170, m = −13.3332, n = −26.4168. Finally, we use Lemma 3.1, Proposition 3.1 and Proposition 3.2 to comkA−1 A − IkF pute A−1 . The residual norms of the inverse corresponding kIkF to the three groups of solution are provided in Table 1. Here, I denote the identity matrix and k · kF is the Frobenius norm. Table 1 The residual norms of inverse for Example 1

n (i) (ii) (iii)

100 200 6.1081e-015 5.5398e-015 6.0316e-015 5.7866e-015 7.7457e-015 7.6055e-015

500 800 1000 6.7348e-015 7.1951e-015 6.6435e-015 5.9913e-015 6.1924e-015 6.4783e-015 8.2541e-015 8.8657e-015 9.1655e-015

The numerical result of Example 1 shows that our conclusions have a good validity. 10

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