The determinants of the block band matrices based on the n-dimensional Fourier equation. Part 1

(1)

THE DETERMINANTS OF THE BLOCK BAND MATRICES BASED ON THE n-DIMENSIONAL FOURIER EQUATION

PART 1

Grzegorz Biernat, Sylwia Lara-Dziembek, Edyta Pawlak Institute of Mathematics, Czestochowa University of Technology

Czestochowa, Poland

grzegorz.biernat@im.pcz.pl, sylwia.lara@im.pcz.pl, edyta.pawlak@im.pcz.pl

Abstract. This paper contains the method of calculating the determinant of the block band matrix on the example of n-dimensional Fourier equation using the Finite Difference Method.

Keywords: block matrices, n-band matrices, determinant, Fourier equation

Introduction

We encounter the block band matrices, inter alia, considering the torsion bars in the theory of vibrations, determining the currents in the eyebolt method in the circuit theory, as well as issues of heat flow using the Finite Difference Method.

The Finite Difference Method (FDM) is based on the introduction of the modeled structure of the grid nodes and replacing a single differential equation (in our case the Fourier equation) by a set of differential algebraic equations.

The partial derivatives are approximated by the corresponding difference quotients.

The values of the search function (for example temperature values) are calculated only at points of discretization - mesh nodes. The algebraic systems of equations in which the main matrix has the band structure are obtained.

And it is the Fourier equation describing the heat conduction that will serve as an example to illustrate how to calculate the determinants of block band matrix.

1. Solution of the problem

We consider the n-dimensional Fourier equation

( ) ( ) ( )

( )

1

2 2 2

1 2 1 2 1 2

2 2 2

2

1 2

, , ..., , , , ..., , , , ..., ,

...

, , ..., ,

n

n n n

n

T x x x t T x x x t T x x x t

x x x

T x x x t

c

t λ

ρ

∂ ∂ ∂ 

 + + + =

 ∂ ∂ ∂ 

 

= ∂

∂

(1)

(2)

where λ is a thermal conductivity [W/mK], c is a specific heat [J/kgK], ρ is a mass density [kg/m³], T is temperature [K], x₁,x₂,...,x_n denote the geometrical coordinates and t is time [s].

For the dimension ₁ ₂ ...

I ×I × In and interval of the time L we get the spatiotem- poral grid

1 2.. . i i i ln

∆ and nodes

1i1 1 1

x = ∆i x for 1≤i₁≤m₁− , where 1 m₁∆x₁=I₁

2i2 2 2

x =i ∆x for 1≤i₂≤m₂− , where 1 m₂∆x₂=I₂

⋮

nin n n

x =i ∆x for 1≤i_n≤m_n− , where 1 m_n∆x_n=I_n for the spatial coordinates and constant increments ₁, ₂, ... ,

x x xn

∆ ∆ ∆

and

tl = ∆ for 1 l ql t ≤ ≤ , where q∆ = t L for coordinate of the time.

We assume the following designation 1 2...

(

11, 22, ... , ,

)

n n

i i i l i i ni l

T =T x x x t .

Then approximations of the second order partial derivatives using MRS are as follows:

( )

1 2 1 2 1 2

1

1 2 1 2 1 2

2

1 2 1 2 1 2

2

1, ,..., , , ,..., , 1, ,..., ,

1 1

2 2

1

2 , 1,..., , , ,..., , , 1,..., ,

2 2

2

, ,..., 1, , ,..., , , ,.

2

, 1 1

2

, 1 1

2

n n n

n n

i i i l i i i l i i i l

i i i l i i i l i i

n

T T T

T i m

x x

T T T

T i m

x x

T T T

T x

− +

−

− +

∂ = ≤ ≤ −

∂ ∆

− +

∂ = ≤ ≤ −

∂ ∆

− +

∂ =

∂

⋮

( )

.., 1,

2 ⁱⁿ ^l, 1 _n _n 1

n

i m

x

+ ≤ ≤ −

∆

(2)

However, the time derivative approximation takes the following form:

1,2,..., , 1,2,..., , 1

, 1

n n

i i i l i i i l

T T

T l q

t t

− −

∆ = ≤ ≤

∆ ∆ (3)

(3)

So, the internal iteration corresponding Fourier equation takes the form

2 2 2

1 2

...

n

T T T T

c

x x x t

λ ρ

∆ ∆ ∆  ∆

+ + + =

 

∆ ∆ ∆  ∆

 

(4)

It leads to the following band system of equations

( ) ( ) ( )

1 2 1 2 1 2

1, ,..., , , ,..., , 1, ,..., ,

2 2 2

1 1 1

, 1,..., , , ,..., , , 1,..., ,

2 2 2

, ,..., 1, , ,..., , , ,...,

2 2 2

2

n n n

n n

i i i l i i i l i i i l

i i i l i i i l i i

n n n

T T T

x x x

T T T

x x x

T T T

x x x

λ λ λ

− +

−

− + +

∆ ∆ ∆

+ − + +

∆ ∆ ∆

+ − +

∆ ∆ ∆

⋮

1 2 1 2

1,

, ,..., , , ,..., , 1

n

n n

i l

i i i l i i i l

c c

T T

t t

ρ ρ

+

−

=

= −

∆ ∆

(5)

for each time step l.

The main matrix of this system is a block matrix having the following structure

1 1

1 1 1

1 1

, 2

n n

n n n

n

n n n

n n

m m

A D

D A D

A n

D A D

D A

− −

− − −

− −

×

 

 

= ≥

 

 

… … … … …

… … … …

⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮

… … … …

… … … … …

(6)

while (n = 1)

1 1

1 1 1

1

1 1 1

1 1

m m

a b

b a b

A

b a b

b a

×

 

 

= 

 

 

… … … … …

… … … …

⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮

… … … …

… … … … …

(7)

where the elements of the matrix A₁ are as follows:

( ) ( ) ( ) ( )

1 2 2 2 1 2

1 2 1

2 2 2

... ,

n

a c b

x x x t x

λ λ λ ρ λ

= + + + + = −

∆ ∆ ∆ ∆ ∆

(8)

(4)

Then

1 1 1

1

1 1 1

2 4

1 1 1 1 1

1 1

1 1 1

2 4

1 2 1 4

1 1 1 1 1

1 2

det ...

1 2

...

1 2

m m m

m

m m m

a m a m a

A b

b b b

m m

a a b a b

− −

   −    −   

 

=   −   +   − =

− −

   

= −  +  +

   

(9)

The calculation of the above determinant is given in the article [1].

Then

2 2

1 1

1 1 1

2

1 1 1

1 1 m m

A D

D A D

A

D A D

D A

×

 

 

= 

 

 

… … … … …

… … … …

⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮

… … … …

… … … … …

(10)

where

1 1

1

0

0 0

,

0 0

0

m m

d d D

d d

×

 

 

= 

 

 

… … … … …

… … … …

⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮

… … … …

… … … … …

( )

1 2

2

d

x

= − λ

∆

(11)

Therefore

2 2 2

1 2

1

1 2

2 4

1 2 1 2 1

2 1

1 1 1

1 1,1 1 1, 2 1 1, 1

1 1 1

1

1 1,1

1

1 2

det det ...

1 2

det ...

det

m m m

m m

m

m m

A m A m A

A d

d d d

A A A

d p I p I p I

d d d

d A p

d

− −

   −    −   

 

=   −   +   − =

    

=  −  −     − =

= −

i

( ) ( ) ( )

1

1 1

1 1, 2 1 1, 1

1 1

1 1 1,1 1 1 1 1, 2 1 1 1 1, 1

det ... det

det det ... det

m

A A

I p I p I

d d

A d p I A d p I A d p I

     

⋅ − ⋅ ⋅ − =

     

     

= − ⋅ − ⋅ ⋅ −

(12)

where

1,_i1, 1 1 1

p ≤i ≤m are zeros of polynominal

(5)

( ) ( ) ( )

1 1 1

1

2 4

1 1

1

1, 1 1, 2 1,

1 2

( ) ...

1 2

...

m m m

m

m m

f x x x x

x p x p x p

− −

   

= −  +  + =

   

= − ⋅ − ⋅ ⋅ −

(13)

Consecutively

3 3

2 2

2 2 2

3

2 2 2

2 2

m m

A D

D A D

A

D A D

D A

×

 

 

= 

 

 

… … … … …

… … … …

⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮

… … … …

… … … … …

(14)

where

2 2

1

2

1

0

0 0

0

m m

C C D

C C

×

 

 

= 

 

 

… … … … …

… … … …

⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮

… … … …

… … … … …

(15)

1 1

2

1

2

0

0 0

,

0 0

0

m m

d d C

d d

×

 

 

= 

 

 

… … … … …

… … … …

⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮

… … … …

… … … … …

( )

2 2

3

d

x

= − λ

∆

(16)

Then we obtain

3 3 3

1 2 3

2

1 2 3

2 4

2 3 2 3 2

3 2

2 2 2

2 2,1 2 2, 2 2 2, 2

2 2 2

2

1 2

det det ...

1 2

det ...

det

m m m

m

m m m

A m A m A

A d

d d d

A A A

d p I p I p I

d d d

d

− −

   −    −   

 

=   −   +   − =

    

=  −  −     − =

=

i i

( ) ( ) ( )

2

2 2 2

2,1 2 2, 2 2 2, 2

2 2 2

2 2 2,1 2 2 2 2, 2 2 2 2 2, 2

det ... det

det det ... det

m

A A A

p I p I p I

d d d

A d p I A d p I A d p I

     

− ⋅ − ⋅ ⋅ − =

     

     

= − ⋅ − ⋅ ⋅ −

(17)

(6)

where

2,_i2, 1 2 2

p ≤i ≤m are zeros of polynomial

2 2 2 2 2 2 4

2

1 2

( ) ...

1 2

m m m m m

f x x  − x ⁻  − x ⁻

= −  +  +

    (18)

where

( )

¹ ²

1 2

1

2 2

2 2 2,1 2 1 2,1 2

1 1

1 2 1 2

1 1,1 1 2,1 1 1, 2 1 2,1 1

1 1 1 1

1 2

1, 1 2,1 1

1 1

1 1 1,1

det det

m m

m

A d

A d p I d p I

d d

A d A d

d p I p I p I p I

d d d d

A d

p I p I

d d

A d p

 

− =  − =

 

     

=  − − ⋅  − − ⋅ ⋅

  

⋅  − − =

= −

i

…

( ) ( )

(

1

)

2 2,1 1 1 1 1, 2 2 2,1 1

1 1 1, 2 2,1 1

det ...

det

m

d p I A d p d p I

A d p d p I

 +   − + ⋅ ⋅

   

 

⋅  − + 

(19)

and analogously

(

2

)

¹ ² 2

1 2

2 2

1 2

2 2

2 2 2, 2 1 2, 2

1 1

1 2 1 2

1 1,1 1 2, 1 1, 2 1 2, 1

1 1 1 1

1 2

1, 1 2, 1

1 1

1

det det

m m

A d

A d p I d p I

d d

A d A d

d p I p I p I p I

d d d d

A d

p I p I

d d

A d

 

− =  − =

 

     

=  − − ⋅  − − ⋅ ⋅

  

⋅  − − =

= −

i

…

( ) ( )

( )

2 2

1 2

1 1,1 2 2, 1 1 1 1, 2 2 2, 1

1 1 1, 2 2, 1

det ...

det

m m

p d p I A d p d p I

A d p d p I

 + ⋅  − + ⋅ ⋅

   

 

⋅ − +

 

(20)

Generally

(7)

1 2 3

1

2 4

1 1 1

1

1 1 1

1,1 1 1, 2 1

1 1

det

1 2

det ...

1 2

det ...

n

n n n

n

m m m m

n

m m m

n n n n n

n n n

m m m m

n

n n n

n n n n

n n

A d

A m A m A

d d d

d

A A A

p I p I

d d

−

− −

− − −

−

− − −

− − − −

− −

=

= ⋅

   −    −   

 

⋅   −   +   − =

        

 

= ⋅

  

⋅  −  − ⋅ ⋅

  

i i i…i

( ) ( )

1

1 2 3

1

1, 1

1

1 1 1

1,1 1 1, 2 1 1, 1

1 1 1

1 1 1,1 1 1 1 1, 2 1

det det ... det

det det ...

d

n

n m n

n

m m m m

n

n n n

n n n n n m n

n n n

n n n n n n n n

p I

d d

A A A

p I p I p I

d d d

A d p I A d p I

−

− −

−

− − −

− − − − − −

− − −

− − − − − − − −

  

− =

  

  

 

= ⋅

     

⋅  − ⋅  − ⋅ ⋅  − =

     

= − ⋅ − ⋅ ⋅

⋅

i i i…i

(

1 1 1, 1 1

)

et

n n n mn n

A d p I

− − − − − −

(21)

where

1, 1, 1 1 1

n in n n

p i m

− − ≤ − ≤ − are zeros of polynomial

1 1 1 2 1 1 4

1

1 2

( ) ...

1 2

n n n

m n m n m

n

m m

f x x ⁻ ⁻ x ⁻⁻ ⁻ x ⁻⁻

−

− −

   

= −  +  +

    (22)

where the first and the second element of the product (21) are as follows:

( )

¹ ² ¹

1 2 1

1 1

1 1 1,1 1 2 1,1 1

2 2

2 1

2 1,1 2 1,1 2

2 2

2 1

1, 2 2 1,1 2

2 2

det det

det

n

m m m n n

n n n n n n n

n n

m m m n n

n n n n

n n

n n n

n n

A d

A d p I d p I

d d

A d

d p I p I

d d

A d

p I p I

d d

−

− −

− − − − − − −

− −

− − − −

− −

− − −

− −

 

− =  − =

 

  

=  − − ⋅

  

⋅  − −

 

i i…i

( )

2

2 1

1, 2 1,1 2

2 2

2 2 1,1 1 1,1 2

2 2 1, 2 1 1,1 2

2 2 1, 1 1,1 2

det det

det ...

det

n

n n

m n n n

n n

n n n n n

n n m n n n

A d

p I p I

d d

A d p d p I

−

− −

− − −

− −

− − − − −

⋅ ⋅

 

 

  

⋅  − − =

 

=  − + ⋅

 

⋅  − + ⋅ ⋅

 

⋅ − +

 

…

(23)

(8)

( )

¹ ² ¹

1 2 1

1 1

1 1 1, 2 1 2 1, 2 1

2 2

2 1

2 1,1 2 1, 2 2

2 2

2 1

1, 2 2 1, 2 2

2 2

det det

det

n

m m m n n

n n n n n n n

n n

m m m n n

n n n n

n n

n n n

n n

A d

A d p I d p I

d d

A d

d p I p I

d d

A d

p I p I

d d

−

− −

− − − − − − −

− −

− − − −

− −

− − −

− −

 

− =  − =

 

  

=  − − ⋅

  

⋅  − −

 

i i…i

( )

2

2 1

1, 2 1, 2 2

2 2

2 2 1,1 1 1, 2 2

2 2 1, 2 1 1, 2 2

2 2 1, 1 1, 2 2

det det

det ...

det

n

n n

m n n n

n n

n n n n n

n n m n n n

A d

p I p I

d d

A d p d p I

−

− −

− − −

− −

− − − − −

⋅ ⋅

 

 

  

⋅  − − =

 

=  − + ⋅

 

⋅  − + ⋅ ⋅

 

⋅ − +

 

…

(24)

and the last element is in the form

(

1

)

¹ ² ¹ 1

1 2 1

1

1 1

1 1 1, 1 2 1, 1

2 2

2 1

2 1,1 2 1, 2

2 2

2 1

1, 2 2

2 2

det det

det

n

n n

n

m m m n n

n n n m n n n m n

n n

m m m n n

n n n m n

n n

A d

A d p I d p I

d d

A d

d p I p I

d d

A d

p I p

d d

−

− −

−

− −

− − − − − − −

− −

− − − −

− −

−

− −

 

− =  − =

 

  

=  − − ⋅

 

⋅  − −

 

i i…i

( )

1

2 1

1

2

1, 2

2 1

1, 2 1, 2

2 2

2 2 1,1 1 1, 2

2 2 1, 2 1 1, 2

2 2 1, 1 1,

det

det ...

det

n

n n

n

m n

n n

m n n m n

n n

n n n n m n

n n m n n m

I

A d

p I p I

d d

A d p d p I

A d p d p

−

− −

−

− −

− − −

− −

− − − − −

− − − −

 

⋅ ⋅

 

 

  

⋅  − − =

 

= − + ⋅

 

 

⋅  − + ⋅ ⋅

⋅ − +

…

(

_n1

)

In 2

− −

 

 

(25)

(9)

Conclusion

This paper is the presentation of the algebraic method for solving a large-size system of equations occurring, for example, in the heat flow. In the article, the method of calculating the determinant of the main matrix of the system is given.

References

[1] Biernat G., Lara-Dziembek S., Pawlak E., The determinants of the three-band block matrices, Scientific Research of the Institute of Mathematics and Computer Science 2012, 3(11), 5-8.

[2] Lang S., Algebra, Springer Science Business Media Inc., 2002.

[3] Mostowski A., Stark M., Elementy algebry wyższej, PWN, Warszawa 1975.