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


An = 








b
a
b
c
O
c
b
a
b
O
O
d
c
b
a
O
O
d
d
c
b
O
O
c
d
d
c
O
O
b
c
d
d
O
O
a
b
c
d
O
O
b
a
b
c
d
It means that A n = [aij ]n×n , where
a,
b,


aij = c,
d ,


0,
i− j =0
i − j =1
i− j =2
i− j =3
otherwise
O
c
b
a
b
c
d
c
b
a
b










d

c
b

a
(1)
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 processing. 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
this end we introduce two auxiliary heptadiagonal matrices A n and  n of the form
a
b

c
d


An = 








b
a
b
c
O
a
b

c
d


ˆ =
A
n








b
a
b
c
O
c
b
a
b
O
O
c
b
a
b
O
O
d
c
b
a
O
O
d
d
c
b
a
O
O
d
d
c
b
O
O
c
d
d
c
b
O
O
c
d
d
c
O
O
b
c
d
d
c
O
O
b
c
d
d
O
O
a
b
c
d
d
O
O
a
b
c
d
O
O
b
a
b
c
0
O
O
b
a
b
c
0
O
c
b
a
b
d
O
c
b
a
b
d
(2)
d
c
b
a
c










d

c
b

b
(3)
d
c
b
a
c










0

d
c

a
Recurrence form for determinant of a heptadiagonal symmetric Toeplitz matrix
21
It means that A n = [bij ]n×n , Â n = [cij ]n×n where
a,
c,

b, i = n, j = n
d ,
c, i = n, j = n − 1
0,


bij = d , i = n, j = n − 2 , cij = 
c,
0, i = n, j = n − 3


d ,

0,
aij , otherwise


aij
i = n, j = n
i = n − 1, j = n
i = n − 2, j = n
i = n − 3, j = n
i = n, j = n − 1
i = n, j = n − 2
i = n, j = n − 3
otherwise
By Wn , Wn , Wˆ n we denote determinants of matrices A n , A n , Â n , 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
(
)
(
)
Wn+7 = aWn+6 + bd bd − 2c 2 Wn+3 + d 2 2c 3 − 4bcd + b 2 c + ad 2 Wn+2 +

+ d 3 2c 2 d + b 2 d − bc 2 − d 3 Wn+1 − bcd 5Wn − bWn+6 + bcWn+5 +


+ d 2ac − b 2 Wn+ 4 + bd 2 (2c − a )Wn+3 + d 3 2bd − b 2 − 2c 2 Wn+ 2 +


+ cd 4 (b − 2d )Wn+1 + bd 6 Wn − c 2Wˆ n+5 + d (bc − da )Wˆ n+ 4


2
3 2
5
Wn+6 = bWn+5 − bcd Wn+2 + d c − bd Wn+1 + d cWn − cWn+5 +

+ bdWn+ 4 + ad 2Wn+3 + bd 3Wn+ 2 − cd 4Wn+1 − d 6Wn − cdWˆ n+4

Wˆ n+ 2 = aWn+1 − c 2Wn + 2cdWn − d 2Wˆ n

(
(
)
)
(
(
)
(4)
)
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
W1 = a , W2 = a 2 − b 2 , W3 = aW2 − bW2 + b 2 c − ac 2
W4 = aW3 − bW3 + bcW2 − c 2Wˆ2 + ad (3bc − ad ) + bd bd − b 2 − 2c 2
(
(
)
)
W5 = aW4 − bW4 + bcW3 + d 2ac − b 2 W2 − c 2Wˆ3 + d (bc − ad )Wˆ 2 +
(
+ bcd 3bd − 2ac − 4d
2
) + d (2c
2
3
+ ad 2
)
22
J. Borowska, L. Łacińska
(
)
(
)
W6 = aW5 + bd bd − 2c 2 W2 − bW5 + bcW4 + d 2ac − b 2 W3 +
+ bd 2 (2c − a )W2 − c 2Wˆ 4 + d (bc − ad )Wˆ3 + c 2 d 2 (2ac − 3bd ) +
(
)
(
)
+ abcd bd − 4d 2 + d 4 a 2 + 2c 2 + 3b 2 − d 2 − d 3b3
(
)
(
)
)
W7 = aW6 + bd bd − 2c 2 W3 + d 2 2c 3 − 4bcd + b 2 c + ad 2 W2 − bW6 + bcW5 +
+ d 2ac − b 2 W4 + bd 2 (2c − a )W3 + d 3 2bd − b 2 − 2c 2 W2 − c 2Wˆ5 +
(5)
+ d (bc − ad )Wˆ 4 + ad 3 2c 2 d + b 2 d − bc 2 − d 3 + bcd 4 (b − 3d )
(
)
(
(
)
(
)
(
)
W1 = b , W2 = ab − bc , W3 = bW2 − cW2 + d b 2 − ac
W4 = bW3 − cW3 + bdW2 − cdWˆ 2 + bd 2 (a − c )
(
W5 = bW4 − cW4 + bdW3 + ad 2W2 − cdWˆ3 + d 3 b 2 + c 2 − bd 2 d 2 + ac
)
W6 = bW5 − bcd 2W2 − cW5 + bdW4 + ad 2W3 + bd 3W2 +
− cdWˆ + cd 3 d 2 + ac − bd 4 (a + c )
4
(
)
Wˆ1 = a , Wˆ 2 = a 2 − c 2
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
An = 





b
a
b
O
c
b
a
O
O
c
b
O
O
c
c
O
O
b
c
O
O
a
b






c
b

a  n× n
(6)
23
Recurrence form for determinant of a heptadiagonal symmetric Toeplitz matrix
It can be seen that in this case the system of equations (4) can be reduced to
a system of two recurrence equations
Wn + 4 = aWn + 3 − ac 2Wn +1 + c 4Wn − bWn + 3 + bcWn + 2


Wn +1 = bWn − cWn
(7)
where n ∈ N .
At the same time initial conditions reduce to the form
W1 = a

2
2
W2 = a − b

2
2
W3 = aW2 − bW2 + b c − ac

2
4
2 2
W4 = aW3 − b W2 + 2bcW2 + c − a c
W1 = b

(8)
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


An = 
b a b


O O O 


b a b


b a  n× n

(9)
In this case from (4) we obtain finally one recurrence equation describing the
determinant of matrix (9)
Wn + 2 = aWn +1 − b 2Wn
(10)
where n ∈ N .
At the same time initial conditions reduce to the form
W1 = a , W2 = a 2 − b 2
(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 consider 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 . Bearing 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,
the formula for Wˆ n in (4) is transformed to the form
Wˆ n+5 = aWn+ 4 − c 2Wn +3 + 2cdWn+3 − d 2Wˆ n+3
and the additional initial conditions for Wˆ3 , Ŵ4 are imposed. Moreover let us denote
F = W , G = Wˆ . Bearing in mind (4) and the above assumptions we apply the following 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)
(
)
(
)
α1 := a : α 2 := bd bd − 2c 2 : α 3 := d 2 2c 3 − 4bcd + b 2 c + ad 2 :
3
(
2
2
2
3
)
5
α 4 := d 2c d + b d − bc − d : α 5 := −bcd : α 6 := −b: α 7 := bc:
(
(
)
)
α 8 := d 2ac − b 2 : α 9 := bd 2 (2c − a ) : α10 := d 3 2bd − b 2 − 2c 2 :
α11 := cd (b − 2d ) : α12 := bd : α13 := −c : α14 := d (bc − da ) :
4
6
2
(
)
β1 := b : β2 := −bcd 2 : β3 := d 3 c 2 − bd : β4 := d 5c : β5 := −c :
β6 := bd : β7 := ad 2: β8 := bd 3 : β9 := −cd 4 : β10 := −d 6 : β11 := −cd :
ξ1 := a : ξ 2 := −c 2 : ξ 3 := 2cd : ξ 4 := −d 2 :
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] + α 3W [n + 2] + α 4W [n + 1] + α 5W [n] +
+ α 6 F [n + 6] + α 7 F [n + 5] + α 8 F [n + 4] + α 9 F [n + 3] + α10 F [n + 2] + α11 F [n + 1] +
+ α12 F [n ] + α13G[n + 5] + α14G[n + 4]) :
F [n + 6] := evalf ( β1W [n + 5] + β2W [n + 2] + β3W [n + 1] + β4W [n] + β5 F [n + 5] +
+ β6 F [n + 4] + β7 F [n + 3] + β8 F [n + 2] + β9 F [n + 1] + β10 F [n] + β11G[n + 4]) :
G[n + 5] := evalf (ξ1W [n + 4] + ξ 2W [n + 3] + ξ 3 F [n + 3] + ξ 4G[n + 3]) :
end do:
print(W[100000])
At the end we get − 8.768970745 ⋅ 1061547 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.