Recurrence Form for Determinant of a Heptadiagonal Symmetric Toeplitz Matrix

Home , Heptadiagonal matrix, Pentadiagonal matrix, Precision (statistics)

Journal of Applied Mathematics and Computational Mechanics 2014, 13(4), 19-26

RECURRENCE FORM FOR DETERMINANT OF A HEPTADIAGONAL SYMMETRIC TOEPLITZ MATRIX

Jolanta Borowska, Lena Łacińska Institute of Mathematics, Czestochowa University of Technology Częstochowa, Poland [email protected], [email protected]

Abstract. In this paper we show that the determinant of heptadiagonal symmetric Toeplitz matrix can be represented by a particular solution of the system of three homogeneous linear recurrence equations. The general considerations are illustrated by certain numerical example implementated in the Maple system.

Keywords: Toeplitz matrix, heptadiagonal matrix, determinant, linear recurrence equations

Introduction

The subject of consideration is a heptadiagonal symmetric Toeplitz matrix of the form a b c d  b a b c d    c b a b c d  d c b a b c d     O O O O O O O  A =  O O O O O O O  (1) n    d c b a b c d   d c b a b c d     d c b a b c  d c b a b    d c b a A = a It means that n [ ] ×nnij , where

a, i − j = 0  b, i − j =1  aij = c, i − j = 2  , − = 3 d i j  ,0 otherwise 20 J. Borowska, L. Łacińska

Heptadiagonal Toeplitz matrices are a special class of band Toeplitz matrices which play an important role in physics, mathematics, statistics, and signal process- ing. The numerical algorithms for computing determinants or eigenvalues of heptadiagonal Toeplitz matrices are proposed for example in [1-3]. In the present paper we are to show that the determinant of heptadiagonal symmetric Toeplitz matrix can be described by a system of three linear recurrence equations. This approach enables us to construct a simple and effective numerical algorithm for calculating determinant of considered matrix with fixed dimension and fixed elements on bands.

1. The main results

In this section we present a recurrence relation for determinant of matrix (1). To ˆ this end we introduce two auxiliary heptadiagonal matrices An and An of the form

a b c d  b a b c d    c b a b c d  d c b a b c d     O O O O O O O  A =  O O O O O O O  (2) n    d c b a b c d   d c b a b c d    d c b a b c  d c b a b    0 d c b

a b c d  b a b c d    c b a b c d  d c b a b c d     O O O O O O O  Aˆ =  O O O O O O O  (3) n    d c b a b c d   d c b a b c 0    d c b a b d   d c b a c    0 d c a Recurrence form for determinant of a heptadiagonal symmetric Toeplitz matrix 21

A = b Aˆ = c It means that n [ ] ×nnij , n [ ] ×nnij where

a, i = n, j = n  c, i = n − ,1 j = n b, i = n, j = n   d, i = n − ,2 j = n c, i = n, j = n −1   ,0 i = n − ,3 j = n bij = d, i = n, j = n − 2 , c =  ij c, i = n, j = n −1  ,0 i = n, j = n − 3   d, i = n, j = n − 2 a , otherwise  ij  = = −  ,0 i n, j n 3 aij otherwise ˆ ˆ By Wn , Wn , Wn we denote determinants of matrices An , An , An , respectively.

In order to obtain the determinant Wn we adapt the approach proposed in [4]. After repeated use of the theorem of Laplace expansion we finally obtain a system of three linear recurrence equations

2 2 3 2 2 Wn+7 = aWn+6 + bd(bd − 2c )Wn+3 + d (2c − 4bcd + cb + ad )Wn+2 +  3 2 2 2 3 5  + d ()2 dc + db − bc − d Wn+1 − bcd Wn − Wb n+6 + bcWn+5 +  d 2ac b2 W bd 2 2c a W d 3 2bd b2 2c2 W  + ()− n+4 + ()− n+3 + ()− − n+2 +  4 6 2 ˆ ˆ  + cd ()b − 2d Wn+1 + bd Wn − c Wn+5 + d()bc − da Wn+4 (4) W bW bcd 2W d 3 c2 bd W d 5cW Wc  n+6 = n+5 − n+2 + ()− n+1 + n − n+5 +  + bdW + ad 2W + bd 3W − cd 4W − d 6W − cdWˆ  n+4 n+3 n+2 n+1 n n+4 Wˆ aW c2W 2cdW d 2Wˆ  n+2 = n+1 − n + n − n where n∈ N.

In order to obtain value of determinant Wn we must take into account the system of equations (4) together with initial conditions of the form

2 2 2 2 W1 = a, W2 = a − b , W3 = aW2 − Wb 2 + cb − ac 2 ˆ 2 2 W4 = aW3 − Wb 3 + bcW2 − c W2 + ad(3bc − ad )+ bd(bd − b − 2c )

2 2 ˆ ˆ W5 = aW4 − Wb 4 + bcW3 + d(2ac − b )W2 − c W3 + d(bc − ad )W2 +

+ bcd()()3bd − 2ac − 4d 2 + d 2 2c3 + ad 2 22 J. Borowska, L. Łacińska

2 2 W6 = aW5 + bd(bd − 2c )W2 − Wb 5 + bcW4 + d(2ac − b )W3 + 2 2 ˆ ˆ 22 + bd ()2c − a W2 − c W4 + d()()bc − ad W3 + dc 2ac − 3bd + + abcd()()bd − 4d 2 + d 4 a2 + 2c2 + 3b2 − d 2 − bd 33

2 2 3 2 2 W7 = aW6 + bd(bd − 2c )W3 + d (2c − 4bcd + cb + ad )W2 − Wb 6 + bcW5 + 2 2 3 2 2 2 ˆ + d()2ac − b W4 + bd ()2c − a W3 + d ()2bd − b − 2c W2 − c W5 + (5) ˆ 3 2 2 2 3 4 + d()bc − ad W4 + ad ()2 dc + db − bc − d + bcd ()b − 3d

2 W1 = b, W2 = ab − bc, W3 = bW2 − Wc 2 + (bd − ac)

ˆ 2 W4 = bW3 − Wc 3 + bdW2 − cdW2 + bd (a − c)

2 ˆ 3 2 2 2 2 W5 = bW4 − Wc 4 + bdW3 + ad W2 − cdW3 + d (b + c )− bd (d + ac)

2 2 3 W6 = bW5 − bcd W2 − Wc 5 + bdW4 + ad W3 + bd W2 +

ˆ 3 2 4 − cdW4 + cd ()d + ac − bd ()a + c

ˆ ˆ 2 2 W1 = a , W2 = a − c

The above initial conditions are obtained by calculation of pertinent determinants. Hence the value of determinant of heptadiagonal matrix (1) is the particular solution of the system of equations (4) fulfilling the initial conditions (5). It can be observed that the direct solution of the system of equations (4) can be obtained only in some very special cases. However, for an arbitrary but fixed n∈N the proposed system of equations is suitable for constructing an effective algorithm for computation of determinant Wn .

Remark 1. If d = 0 then we deal with pentadiagonal symmetric Toeplitz matrix of the form

a b c  b a b c    c b a b c  A =  O O O O O  (6) n    O O O O c  c b a b   c b a   ×nn Recurrence form for determinant of a heptadiagonal symmetric Toeplitz matrix 23

It can be seen that in this case the system of equations (4) can be reduced to a system of two recurrence equations

2 4 Wn+4 = aWn+3 − ac Wn+1 + c Wn − Wb n+3 + bcWn+2  (7) Wn+1 = bWn − Wc n where n∈N. At the same time initial conditions reduce to the form

W1 = a  2 2 W2 = a − b  2 2 W3 = aW2 − Wb 2 + cb − ac (8) W aW b2W 2bcW c4 ca 22  4 = 3 − 2 + 2 + − W b  1 =

Hence the determinant of matrix (6) is the particular solution of the system of equations (7) fulfilling the initial conditions (8), compare [4].

Remark 2. If c = d = 0 then we deal with tridiagonal symmetric Toeplitz matrix of the form a b  b a b     b a b  A =  b a b  (9) n    O O O   b a b   b a   ×nn In this case from (4) we obtain finally one recurrence equation describing the determinant of matrix (9)

2 Wn+2 = aWn+1 − b Wn (10) where n∈N. At the same time initial conditions reduce to the form

2 2 W1 = a , W2 = a − b (11)

Hence the determinant of tridiagonal matrix (9) is the particular solution of the equation (10) fulfilling the initial conditions (11). 24 J. Borowska, L. Łacińska

2. Illustrative example

In order to illustrate the general results obtained in the previous section we con- sider a special form of heptadiagonal matrix (1) in which a =1, b = 2, c = 3, d = 4. Moreover, assuming that matrix under considerations has the order 105 ×105 . Bear- ing in mind the above assumptions and formula (4) determinant of this matrix is given by the system of three linear recurrence equations with constant coefficients. It can be observed that it is impossible to solve such a system of recurrence equations using known analytical methods, [5]. Therefore, in order to calculate determinant of matrix under considerations, we construct an algorithm which will be implemented in the Maple algebra system. In order to speed up Maple calculations, ˆ the formula for Wn in (4) is transformed to the form

ˆ 2 2 ˆ Wn+5 = aWn+4 − c Wn+3 + 2cdWn+3 − d Wn+3

ˆ ˆ and the additional initial conditions for W3, W4 are imposed. Moreover let us denote F =W , G = Wˆ . Bearing in mind (4) and the above assumptions we apply the follow- ing syntax in Maple:

Step 1 (generating of initial conditions) with (LinearAlgebra ): h(i, j ):= piecewise (i=j , a, i=j+ 1, b , i=j–1, b, i=j+ 2, c, i=j-2, c, i=j+ 3, d, i=j–3, d): p(i, j ):= piecewise (i=n and j=n , b, i=n and j=n–1, c, i=n and j=n–2, d, i=n and j=n–3, 0, h(i, j )): l(i, j):= piecewise (i=n and j=n , a, i=n–1 and j=n , c, i=n–2 and j=n , d, i=n–3 and j=n , 0, i=n and j=n–, c, i=n and j=n–2, d, i=n and j=n–3, 0, h(i, j)): for i from 1 to 7 do W[i]:= Determinant (Matrix (h,i)) end do : for i from 1 to 6 do n:= i: F[i]:= Determinant (Matrix (p,i)) end do : for i from 1 to 5 do n:= i: G[i]:= Determinant (Matrix (l,i)) end do :

Recurrence form for determinant of a heptadiagonal symmetric Toeplitz matrix 25

Step 2 (generating coefficients of equations)

2 2 3 2 2 α1 := a : α 2 := bd(bd − 2c ): α3 := d (2c − 4bcd + cb + ad ): 3 2 2 2 3 5 α 4 := d ()2 dc + db − bc − d : α5 : −= bcd : α6 : −= b: α7 := bc: 2 2 3 2 2 α8 := d()2ac − b : α9 := bd ()2c − a : α10 := d ()2bd − b − 2c : 4 6 2 α11 := cd ()()b − 2d : α12 := bd : α13 : −= c : α14 := d bc − da :

2 23 5 β1 := b: β2 : −= bcd : β3 := d (c − bd ): β4 := cd : β5 : −= c: 2 3 4 6 β6 := bd : β7 := ad : β8 := bd : β9 : −= cd : β10 : −= d : β11 : −= cd :

2 2 ξ1 := a: ξ2 : −= c : ξ3 := 2cd : ξ4 : −= d :

Step 3 (assigning fixed data to matrix elements) a = :1: b := :2 c := :3 d := :4

Step 4 (main loop for determinant Wn ) for n from 1 to 99993 do W [n + ]:7 = evalf (α1W [n + 6]+α 2W [n + 3]+α3W [n + 2]+α 4W [n +1]+α5W [n]+

+ α6F[][][][][][]n + 6 + α7 F n + 5 +α8 nF + 4 + α9 nF + 3 + α10 nF + 2 +α11 nF +1 +

+ α12 [][][]nF +α13G n + 5 +α14G n + :)4

F[n + ]:6 = evalf (β1W [n + 5]+ β2W [n + 2]+ β3W [n +1]+ β4W [n]+ β5F[n + 5]+

+ β6 [][][][][][]nF + 4 + β7 F n + 3 + β8F n + 2 + β9 nF +1 + β10F n + β11G n + :)4

G[n + ]:5 = evalf (ξ1W [n + 4]+ ξ2W [n + 3]+ ξ3 [nF + 3]+ ξ4G[n + ] :)3 end do : print (W[100000])

61547 At the end we get − .8 768970745⋅10 as the value of determinant of the matrix under considerations. It can be emphasized that all calculations are performed with Maple default precision (Digits = 10).

Conclusions

It was shown that the determinant of the heptadiagonal symmetric Toeplitz matrix can be obtained as a particular solution of the system of three homogeneous linear recurrence equations. Moreover, it was presented that in the case of determinant of pentadiagonal symmetric Toeplitz matrix the above system reduces into two linear recurrence equations. Whilst in the case of determinant of pertinent 26 J. Borowska, L. Łacińska tridiagonal matrix we finally obtain one equation. The general considerations are illustrated by the example in which the implementation of the proposed approach to the Maple system was presented.

References

[1] Elouafi M., A note for an explicit formula for the determinant of pentadiagonal and heptadiagonal symmetric Toeplitz matrices, Applied Mathematics and Computation 2013, 219, 4789-4791. [2] Elouafi M., On a relationship between Chebyshev polynomials and Toeplitz determinants, Applied Mathematics and Computation 2014, 229, 27-33. [3] Solary M.S., Finding eigenvalues for heptadiagonal symmetric Toeplitz matrices, J. Math. Anal. Appl. 2013, 402, 719-730. [4] Borowska J., Łacińska L., Rychlewska J., A system of linear recurrence equations for determinant of pentadiagonal matrix, Journal of Applied Mathematics and Computational Mechanics 2014, 13(2), 5-12. [5] Elaydi S., An Introduction to Difference Equations, Springer, 2005.