Vol 2014

Volume 2014, Article ID ama0157, 8 pages ISSN 2307-7743

http://scienceasia.asia

ON SUM CRITERION FOR THE EXISTENCE OF SOME ITERATIVE METHODS

K. RAUF, O. T. WAHAB, J. O. OMOLEHIN, I. ABDULLAHI, O. R. ZUBAIR, S. M. ALATA, I. F. USAMOT AND A. O. SANUSI

Abstract. In this paper, we evoke an existence for two iterative methods of the system of linear equations. Our results agree with existing results and suggest a strong procedural condition for the convergence of the system of linear equations.

1. _Introduction

We consider the system of linear equation

(1) Ax=b

where

A = 

  

a11 a12 . . . a1n a21 a22 . . . a2n

..

. ...

an1 an2 . . . ann



  

, x= 

  

x1 x2

.. .

xn



  

, b = 

  

b1 b2

.. .

bn



  

Problems concerning system of linear equations are often occurred in engineering and sciences. Many researchers have developed several methods for solving the problems. Noor et al. [2] had used decom-position method for solving the system (1). Babolian et al.[6] have used Adomian decomposition method to derive an iterative method which is similar to the Jacobi iteration. Allahviranloo [4] used Ado-mian decomposition method for fuzzy system. Keramati [7] and Liu [8] have used homotopy perturbation method to derive some iterative methods for solving the system. Improving Newton-Raphson method for non linear equations by modified Adomian decomposition method was discussed in [3]. In this paper, we evoke a strong procedural con-dition for the decomposition method in [2] which is sufficient for the existence of system (1).

2010Mathematics Subject Classification. 65H05, 65B99.

Key words and phrases. decomposition method, iterative methods, convergence.

c

2014 Science Asia

2. Preliminaries

We express system (1) as:

(2)

x1 =(1−a11)x1−a12x2−. . .−a1nxn+b1

x2 =−a21x1+ (1−a22)x2−. . .−a2nxn+b2

.. . ...

xn=−an1x1−an2x2−. . .+ (1−ann)xn+bn

If we set βij =δij −aij; 1 ≤ i, j ≤n, where δij is the kronecker delta

[]. Equation (2) can be written as

(3)

x1 =β11x1−β12x2−. . .−β1nxn+b1 x2 =−β21x1+β22x2−. . .−β2nxn+b2

.. . ...

xn =−βn1x1−βn2x2−. . .+βnnxn+bn

This implies

(4) xi =

n

X

j=1

βijxj+bj, i= 1,2, . . . , n

And in matrix form as

(5) x=Bx+b

where B = [βij], 1≤i, j ≤n

Definition 2.1. Let Rn_{, d(., .) be a metric space. Let T be a}

mapping such that T : Rn _{−→ R}n_{. T is called a contraction on R}n _if

there is a positive real number α <1 such that for all x, y ∈Rn

d(T x, T y)≤αd(x, y).

The point x∈Rn _{is fixed if it coincides with its image. i.ex=T x.}

Now, define a mappingT on (5) which map Rn _{into itself, then}

(6) T x=Bx+b

and we can write

(7) T xi =

n

X

j=1

2.1. Decomposed System. Let h 6= 0 be a parameter, for any s-plitting matrix Q and an auxiliary matrix H, we can decompose the system (1) as

(8) Qx+ (hHA−Q)x=hHb

Let Q=D=diag(aii),i= 1,2, . . ., then we can compare (8) with (5)

if we let

B0 =−D−1_{(hHA−D) and b}0

=D−1hHb

so that

(9) x(m+1)=B0x(m)+b0

where m is a positive integer representing the number of iterations. This is called the Adomian Jacobi method [5].

If we let Q=−L+D

whereLis the lower triangular matrix with principal diagonal all zero. From (8), we have

(10) x

(m+1)

= I−h(D−L)−1HAx(m)+h(D−L)−1Hb

x(m+1) ≡B00x(m)+b00

where B00 = I −h(D−L)−1_HA

and b00 =h(D−L)−1_Hb

This is called the Adomian Gauss-Seidel method [5].

3. Main Results

In this section, we obtain the generalization of sum criterion and deduce some results which are sufficient for the existence of (1). The following Lemma is significant to our main results.

Lemma 3.1. Let Rn be a metric space with metric dp on Rn andT

be a mapping of Rn into itself. Then, T is a contraction if

(11)

n

X

i=1 n

X

j=1 |βij|

p

<1, 1≤p < ∞

Proof

LetRn_{be a metric space, for x, y ∈R}n_{, the metricd}

p onRn is defined

by

(12) dp(x, y) = n

X

|xi−yi| p

!1_p

LetT :Rn−→Rn and x, y ∈Rn. Using (6), we have

dp(T x, T y) = n

X

i=1

|T xi−T yi| p

!_p1

= n X i=1 n X j=1

βij(xj−yj)

p!1_p

≤ n X i=1 n X j=1 βij n X j=1

(xj −yj)

p!1_p

= n X i=1 n X j=1 βij p n X j=1

(xj −yj)

p!1_p

≤ n X i=1 n X j=1 |βij|

p

!1_{p n} X

j=1

|xj −yj| p

!1_p

(Schwarz’s inequality) = n X i=1 n X j=1 |βij|

p

!_p1

dp(x, y)

Choose α=Pn i=1

Pn

j=1|βij|p

1_p hence,

dp(T x, T y)≤αdp(x, y)

Therefore, the mapping T on Rn is a contraction.

Remark. If the condition (11) is applied on either (9) or (10), then we obtain (13) n X i=1 n X j=1

|aij|p <|aii|p

Inequality (13) implies that the elements in the principal diagonal of matrix A must be large, and it will be called p-th sum criterion. The condition (11) is sufficient for the convergence of (6) while (13) is sufficient for the convergence of (9) and (10) but the manner at which they converge differ depending on the choice of p.

We consider the following cases

Case I: Supposep= 1 in equation (13), we obtain

n

X

i=1

forj = 1,2, . . . , nand it is called column sum criterion

Case II: Suppose p = 2 in equation (13), we have the euclidean metric and the condition is called the square sum criterion given by

n

X

i=1 n

X

j=1

a2_ij < a2_ii

See [1] for similar work.

Case III: Whenp= 3 with the metric defined onRn_{. For (13), we obtain}

the condition

n

X

i=1 n

X

j=1

|aij|3 <|aii|3

which is called the cubic sum criterion

Case IV: Ifp=∞with the metric d∞ onRn, we obtain the condition

sup

i n

X

j=1

|aij|<|aii|

This is called the row sum criterion.

We justify the aforementioned cases with the following examples:

Example 1:

Consider the system of linear equation 4x1−x2−x4 = 0

−x1+ 4x2−x3−x5 = 5 −x2+ 4x3−x6 = 0 −x1+ 4x4−x5 = 6

−x2−x4+ 4x5−x6 =−2 −x3−x5+ 4x6 = 6

Solution

j = 1, 6

X

i=1

|aij|=|a21|+|a31|+|a41|+|a51|+|a61|=

3 4

|aii|>

3

4, for i= 1,2,3,4,5,6. Case I is satisfied.

6

X

i=1 6

X

j=1

a2_ij =a2₁₂+a2₂₁+a₂₃2 +a2₂₅+a2₃₂+a2₃₆+a2₄₁+a2₄₅+a2₅₂+a2₅₄+a2₅₆+a2₆₃+a2₆₄= 7 8

|aii|>

7

Case II is satisfied

6

X

i=1 6

X

j=1

a3_ij =a3₁₂+a3₂₁+a₂₃3 +a3₂₅+a3₃₂+a3₃₆+a3₄₁+a3₄₅+a3₅₂+a3₅₄+a3₅₆+a3₆₃+a3₆₄= 7 32

|aii|>

7

32, for i= 1,2,3,4,5,6. Case III is satisfied

sup

i 6

X

j=1

|aij|=

1 4 <1

|aii|>

1

4, for i= 1,2,3,4,5,6. Case IV is satisfied

Example 2.

Consider the system 2x1+x2 +x3 = 4

x1+ 2x2 +x3 = 4

x1+x2+ 2x3 = 4

Solution

Using the cases above, we have

3

X

j=1

|aij|= 1 =|aii|, for i= 1,2,3.

This gives a non-expansive type.

3

X

i=1 3

X

j=1

a2_ij =a2₁₂+a2₁₃+a₂₁2 +a2₂₃+a2₃₁+a2₃₂ = 6 4

|aii|<

6

4, for i= 1,2,3. Case II is not satisfied.

3

X

i=1 3

X

j=1

a3_ij =a3₁₂+a3₁₃+a₂₁3 +a3₂₃+a3₃₁+a3₃₂ = 3 4

|aii|>

3

4, for i= 1,2,3. Case III is satisfied.

sup

i 3

X

j=1

|aij|=

|aii|>

1

2, for i= 1,2,3. Case IV is also satisfied.

Example 3:

Consider the system

x1 +x3 = 2 −x1+x2 = 0

x1+ 2x2−3x3 = 0

Solution

j = 1, 3

X

i=1

|aij|=|a21|+|a31|= 4 3

|aii|< 4

3, for i= 1,2,3. Case I is not satisfied.

3

X

i=1 3

X

j=1

a2_ij =a2₁₂+a2₁₃+a₂₁2 +a2₂₃+a2₃₁+a2₃₂= 23 9

|aii|< 23

9 , for i= 1,2,3. Case II is not satisfied

3

X

i=1 3

X

j=1

a3_ij =a3₁₂+a3₁₃+a₂₁3 +a3₂₃+a3₃₁+a3₃₂= 47 27

|aii|< 47

27, for i= 1,2,3. Case III is also not satisfied

sup

i 3

X

j=1

|aij|= 1

|aii|= 1, fori= 1,2,3.

This is non-expansive.

3.1. Conclusion. It is generally known that if the system (5) of n linear equations satisfies at least one of the cases I, II, III and IV, then, it has precisely one solution.

seidel iteration diverges. On this note, the Jacobi and the Gauss-seidel iteration are not comparable, though the Gauss-Gauss-seidel iteration converges faster than the Jacobi iteration but we are concerned on the criterion at which the iterative methods converge.

References

[1] Erwin Kreyszig,Introductory Functional Analysis with Applications, John Wi-ley & Sons. Inc. New York, Santa Barbara, London, Sydney, Toronto: (1978). [2] M. A. Noor, K. I. Noor and M. Waseem, Decomposition Method for Solving System of Linear Equations, Engineering Mathematics Letter,2No. 1 (2013): 34-41.

[3] S. Abbasbandy, Improving Newton-Raphson method for non linear equations by modified Adomian decomposition method, Appl. Math. Comput.145(2003), 887-893.

[4] T. Allahviranloo,The Adomian Decomposition method for fuzzy system of lin-ear equations, Appl. Math. Comput.163(2005), 553-563.

[5] G. Adomian, Solving Frontier Problems of Physics: The Decomposition method, Kluwer Academic Publishers Dordrecht, (1994).

[6] E. Babolian, J. Biazar and A. R. Vahidi, On the decomposition method for system of linear equations and system of linear Volterra integral equations, Appl. Math. Comput.,147(2004), 19-27.

[7] B. Keramati,An approach to the solution of linear system of equations by He’s homotopy perturbation, Chaos, solution and Fractals,41(2009), 152-156. [8] H. K. Liu,Application of homotopy perturbation methods for solving system of

linear equations, Appl. Math. Comput. Doi: 10.1016/j.amc.2010.11.024.