A New Algorithm for the Determinant and the Inverse of Banded Matrices

(1)

http://dx.doi.org/10.4236/oalib.1100543

A New Algorithm for the Determinant

and the Inverse of Banded Matrices

Mohamed Elouafi1_{, Driss Aiat Hadj Ahmed}2

1_{Classes Préparatoites aux Grandes Ecoles d'Ingénieurs, Lycée My Alhassan, Tangier, Morocco} 2_{Regional Center for Career Education and Training (CRMEF)-Tangier, Tangier, Morocco}

Email: [email protected], [email protected]

Received **** 2014

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

In the current article, the authors present a new recurrence formula for the determinant of a banded matrix. An algorithm for inverting general banded matrices is derived.

Keywords

Banded Matrices, Determinants

Subject Areas: Mathematical Analysis, Numerical Mathematics

1. Introduction

The aim of this work is to extend the algorithm presented in [1] to obtain the inverse of the banded matrix 1,1 1, 1

2, 2

1,1 2,2

0 0

,

0

p

q

a a

a

+ +

… …

 

 

=  

 

 _{… …} 

 

  

 

 



    



B

(2)

is much less than n.

Standard method to compute the inverse of the matrix B is to use the LU based methods [3] and there are special algorithms taking into account the special form of the matrix [4]. Other methods were proposed more recently [5].

For the n n× banded matrix B, where we assume that ai p i, + ≠0 for i=1,,n−p, we associate the se-quence

( )

Ai k, ₁_{≤ ≤ +}_{i n p}_,1_{≤ ≤}_{k p} defined by the following relations for k=1,,p:

, , for 1, , , i k i k

A =δ i=  p (1)

and for i=1,,n−p:

1

, , , ,

1 , p i

i p i p i k i s s k s

a A a A

+ −

+ + =

− =

_∑

(2)

and for i= − +n p 1,, :n

, , , 1

. n

p i k i s s k s A ₊ a A

=

∑

(3)

Here δi k, is the Kronecker symbol and we put ai j, =0 if i− >j q. The relations above can be written in the matrix form

1,

2,

, 0

0

n k k

n k

n p k A A

A

A +

+

 

 

=  

 

 

B



 ,

where

T 1, , 2, , , ,

k k k n k

A = A A  A _ . We shall note Q the p×p matrix

(

_,

)

1 , . n i j _{i j p} A₊

≤ ≤

The determinant of the matrix Q is related to the determinant of the matrix B. This is the purpose of the next section.

2. Remark

Suppose that we have ai p i, + =0 for some i=1,,n−p,. Let B

( )

ε be the matrix obtained from B by re-placing ai p i, + such that ai p i, + =ε with ε0. We have limε→0B

( )

ε =B.

Then, we can compute B−1 tending ε e to 0 in B

( )

ε .

3. Determinant of a Banded Matrix

Theorem. Let the banded matrix B and the associated sequence

( )

_,

1 ,1 . i k _{i n p} _{k p}

A _{≤ ≤ +} _{≤ ≤} Then

( ) ( )

( )

_{( )}

, 1

det 1 det .

n p p n p

i i p i

a − −

+ =

 

= − _ _



∏



B Q

Proof. Let T the n n× triangular matrix

p

n p−

 

=  

 

I T

D I

(3)

where Ir denotes the r r× identity matrix and D is the

(

n− ×p

)

p matrix

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

,1 ,2 ,

.

p p p p

n n n p

A A A

+ + +

…

 

 _… 

 

 

   



Report that

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

,1 ,2 ,

,

p

p p

p p p p

A A A

…

 

 _… 

 _{ =}

 

 

I

   



and thus using the relation Equation 1 we get

,

 

=  

 

C BT

Q R

0

where C is the

(

n− × −p

) (

n p

)

triangular submatrix

1, 1

2, 1 2, 2

, 1 , 2 ,

0 0

0

p

p p

n p p n p p n p n a

a a

a a a

+

+ +

− + − + −

…

 

 _… 

 

 



  



and R is a p× −

(

n p

)

matrix.

Obviously, det

( )

T =1 and det

(

BT

) ( )

= −1 p n p( − )det

( ) ( )

C det Q . The result follows.

Example. For p= =q 2, formula of the determinant of the pentadiagonal matrix is presented in [6].

4. Inverse of a Banded Matrix

Assume that the matrix B is invertible and let us denote by Cj the j-th column vector of the inverse ma-trix B−1.

From the relation B B−1 =I_n, where In denotes the identity matrix of order n, we get the relations:

, 1 ,

1

,

n

n j p n j i n j i

i n j p n j p n j

C E a C

a

− − − −

= − − + − − −

 

=  − 



∑



for j=0,,n− −p 1, where

( )

T , ₁

n j i j _{i n}

E =_δ _{≤ ≤} _ ∈ is the vector of order j of the canonical basis of n.

It follow from r1 that knowing the p last columns C Cn, n−1,,Cn p− +1 determine recursively all other col-umns Cn p− ,Cn p− −1,,C1. We give a straightforward recurrence formulae for computing C Cn, n−1,,Cn p− +1.

Since the matrix B is invertible, we obtain from the previous section that the matrix Q is invertible too. We shall denotes Xj = x1,j,,xp j, T the j-th column vector of the matrix

1

.

− Q

Theorem. The j-th vector column of the inverse matrix 1 , −

B 1≤ ≤j p, is given by

, 1 p

n p j k j k k C− + x

=

∑

A

(4)

, 1, 1 , 1 , 2, 1 , , 1 0 0 p

k j n k p

k k j k _p k

k j n k k

p

k j n p k k x A x x A x A + = = + = + =                 ⋅ =                

∑

B A   .

The result follows from the fact that

( )

, 1, 1 1, T 2, , 2,

1 , ₁

,

, , 1

. p

k j n k k

j p

j k j n k

k i j _{i p}

p j p

k j n p k k x A x x x A x x A δ + = + = _{≤ ≤} + =       _ _   _ _  ₌ _ _{ }₌ _         _ _     _ _      

∑

Q  

We will use the relations Equation (2) and r1 to explicit the coefficients cs j, for the inverse 1

. −

B we have for 1≤ ≤j p and 1≤ ≤s n:

, , ,

1 p

s n p j k j s k k c − + x A

=

∑

(4)

and for j=0,,n− −p 1:

, , , , 1 , 1 . n

s n j p s n j i n j s i i n j p

n j p n j

c a c

a δ − − − − = − − + − − −   = _ − _



∑

 (5)

Here are the different steps of our algorithm. The implementation is left to the readers’ choice. • Compute Ai k, by the relations (1), (2) and (3). If det

(

An i j+,

)

_{1 ,}_{≤ ≤}_{i j p}

)

=0 then B is non invertible.

• Compute the inverse

( )

xi j, _{1 ,}_≤_{i j p}_≤ of the matrix

(

An i j+,

)

_{1 ,}_{≤ ≤}_{i j p}. • Compute cs n j p, + − for 1≤ ≤j p and 1≤ ≤s n by the relation (4). • Compute cs n j p, − − for j=0,,n− −p 1 and 1≤ ≤s n by the relation (5).

5. Conclusion

If we fix the bandwidth p+q, one can show easily that the complexity of the algorithm is _{O n}

( )

2 _._{Of course,} other algorithms of similar complexity exist (methods based on LU decomposition for example). However, the new method provides more benefits to others: First, the recursive formulas are simple and can be imple-mented effectively in a parallel machine to reduce the cost. Also, we can solve these relations for some matrices to get the explicit determinant of those matrices.

References

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[2] Gravvanis, G.A. (2003) On the Solution of Boundary Value Problems by Using Fast Generalized Approximate Inverse Banded Matrix Techniques. The Journal of Supercomputing, 25, 119-129.http://dx.doi.org/10.1023/A:1023936410006 [3] Ran, R.S. and Huang, T.Z. (2009) An Inversion Algorithm for a Banded Matrix. Computers and Mathematics with

Ap-plications, 58, 1699-1710. http://dx.doi.org/10.1016/j.camwa.2009.07.069

[4] Trench, W.F. (1974) Inversion of Toeplitz Band Matrices. Mathematics of Computation, 28, 1089-1095.

http://dx.doi.org/10.1090/S0025-5718-1974-0347066-8

[5] Kratz, W. (2001) Banded Matrices and Difference Equations. Linear Algebra and its Applications, 337, 1-20.

http://dx.doi.org/10.1016/s0024-3795(01)00328-7