# The Mixed Type Splitting Methods for Solving Fuzzy Linear Systems

Authors: H. Saberi Najafi; S. A. Edalatpanah; S. Shahabi
Journal: Applied Computational Intelligence and Soft Computing
 (2014)

---

## Abstract

We consider a class of fuzzy linear systems (FLS) and demonstrate some of the existing methods using the embedding approach for calculating the solution. The main aim in this paper is to design a class of mixed type splitting iterative methods for solving FLS. Furthermore, convergence analysis of the method is proved. Numerical example is illustrated to show the applicability of the methods and to show the efficiency of proposed algorithm.


---

## Body

## 1. Introduction

Solving fuzzy systems has been considered by many researchers, for example, [1–9] and the references therein. In [1, 2] Kandel et al. applied the embedding method for fuzzy linear system (hereafter denoted by FLS) and replaced the FLS by a 2
n
×
2
n crisp linear system. This model has been modified later by some other researchers; see [10–16] and the references therein.Here, based on mixed type splitting, we introduce a new iterative method to FLS. The mixed type splitting iterative method [17, 18] is given for the linear system of equations A
x
=
b, where A is positive real. Cheng et al. in [19] presented a class of the mixed type splitting iterative methods based on [17, 18] and some convergence conditions were given. They also proposed some sufficient and necessary conditions of convergence when coefficient matrix of the linear system is certain matrices such as M-matrix.In this paper, the mixed type splitting iterative method for FLS will be established, which is a generalization of mixed type splitting iterative method for linear system. Some sufficient conditions for convergence of the mixed type splitting iterative method will be considered. Moreover, we will discuss a comparison theorem, which describes the influences of the parameters on the convergence rates of the new methods.


## 2. Preliminaries

In this section we provide some basic notations and definitions of fuzzy number and fuzzy linear system.Definition 1.
An arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions(
u
_
(
r
)
,
u
¯
(
r
)
), 0
≤
r
≤
1, which satisfy the following requirements (see [2, 3]). (i)
u
_
(
r
) is a bounded monotonic increasing left continuous function over [
0,1
].
(ii)
u
¯
(
r
) is a bounded monotonic decreasing left continuous function over [
0,1
].
(iii)
u
_
(
r
)
≤
u
¯
(
r
), 0
≤
r
≤
1.A crisp numberα can be simply expressed as u
_
(
r
)
=
u
¯
(
r
)
=
α, 0
≤
r
≤
1. The addition and scalar multiplication of fuzzy numbers x
=
(
x
_
(
r
)
,
x
¯
(
r
)
) and y
=
(
y
_
(
r
)
,
y
¯
(
r
)
) can be described as follows:(i)
x
=
y if and only if x
_
(
r
)
=
y
_
(
r
) and x
¯
r
=
y
¯
r;
(ii)
x
+
y
=
x
_
r
+
y
_
r
,
x
¯
r
+
y
¯
r;
(iii)
for allK
∈
R; K
x
=
K
x
_
,
K
x
¯
,
K
≥
0
,
K
x
¯
,
K
x
_
,
K
<
0
.Definition 2.
Consider then
×
n linear system of the following equations:
(1)
a
11
x
1
+
a
12
x
2
+
⋯
+
a
1
n
x
n
=
b
1
,
a
21
x
1
+
a
22
x
2
+
⋯
+
a
2
n
x
n
=
b
2
,
⋮
a
n
1
x
1
+
a
n
2
x
2
+
⋯
+
a
n
n
x
n
=
b
n
,
where the coefficient matrix A
=
(
a
i
j
), 1
≤
i, j
≤
n, is a crisp matrix and b
i
∈
E
1, 1
≤
i
≤
n, is called a FLS.Definition 3.
A fuzzy number vectorX
=
(
x
1
,
x
2
,
…
,
x
n
)
T, given by the parametric form x
i
=
(
x
i
_
(
r
)
,
x
i
¯
(
r
)
), 1
≤
i
≤
n, 0
≤
r
≤
1, is called a solution of the FLS (1) if
(2)
∑
j
=
1
n
a
i
j
x
j
_
=
∑
j
=
1
n
a
i
j
x
j
_
=
b
i
_
,
\
d
i
s
p
l
a
y
s
t
y
l
e
∑
j
=
1
n
a
i
j
x
j
¯
=
∑
j
=
1
n
a
i
j
x
j
¯
=
b
i
¯
.
Friedman et al. [2, 3], in order to solve the system given by (2), have solved a 2
n
×
2
n crisp linear system as follows:
(3)
S
X
=
B
,
where S
=
(
s
i
j
) are determined as follows:
(4)
a
i
j
≥
0
⟶
s
i
j
=
a
i
j
,
s
i
+
n
,
j
+
n
=
a
i
j
,
a
i
j
<
0
⟶
s
i
,
j
+
n
=
-
a
i
j
,
s
i
+
n
,
j
=
-
a
i
j
,
and any (
s
i
j
) which is not determined by (4) is zero.
Then referring to [2, 3] we have
(5)
S
=
s
1
s
2
s
2
s
1
,
X
=
x
_
-
x
¯
,
B
=
b
_
-
b
¯
⟹
s
1
x
_
-
s
2
x
¯
=
b
_
,
s
2
x
_
-
s
1
x
¯
=
-
b
¯
.
Or
(6)
S
=
s
1
-
s
2
-
s
2
s
1
,
X
=
x
_
x
¯
,
B
=
b
_
b
¯
⟹
s
1
x
_
-
s
2
x
¯
=
b
_
,
-
s
2
x
_
+
s
1
x
¯
=
b
¯
,
where s
1
,
s
2
≥
0 and A
=
s
1
-
s
2.Definition 4 (see [20]).
(a) A matrixA
=
a
i
j is called a Z-matrix if, for any i
≠
j, a
i
j
≤
0.
(b) AZ-matrix is an L-matrix, if a
i
i
>
0.
(c) AZ-matrix is an M-matrix, if A is nonsingular and A
-
1
≥
0.
(d) For any matrixA the comparison matrix 〈
A
〉
=
(
m
i
j
)
∈
R
n
×
n is defined by
(7)
m
i
i
=
a
i
i
,
m
i
j
=
-
a
i
j
,
i
≠
j
,
1
≤
i
,
j
≤
n
.
(e) A complex matrix A is an H-matrix if 〈
A
〉
=
(
m
i
j
)
∈
R
n
×
n is an M-matrix.
(f) MatrixA is called a Generalized Diagonally Dominant Matrix (GDDM) if there exists a positive diagonal matrix W such that A
W is a Strictly Diagonally Dominant Matrix (S
D
D
M).Lemma 5 (see [19]).
LetA be the coefficient matrix of the linear system A
x
=
b. If A is an M-matrix and conditions of (19) are satisfied, then the mixed type splitting iterative method is convergent.Remark 6.
For any splitting,A
=
M
-
N, where M is nonsingular, the iterative method for solving linear systems of A
x
=
b is
(8)
x
(
i
+
1
)
=
M
-
1
N
x
(
i
)
+
M
-
1
b
,
i
=
0,1
,
…
.
This iterative process converges to the unique solution x
=
A
-
1
b for initial vector value x
0
∈
R
n if and only if the spectral radius ρ
(
M
-
1
N
)
<
1, where T
=
M
-
1
N is called the iteration matrix [20]. For example, suppose diag
⁡
(
A
)
=
I and A
=
I
-
L
-
U, where L and U are strictly lower and strictly upper triangular part of A, respectively. Then for classical AOR (see; [10]) we have
(9)
M
r
,
w
=
1
w
I
-
r
L
,
N
r
,
w
=
1
w
1
-
w
I
+
w
-
r
L
+
w
U
⟹
l
r
,
w
=
M
r
,
w
-
1
N
r
,
w
=
I
-
r
L
-
1
1
-
w
I
+
w
-
r
L
+
w
U
.
And when w
=
r, we have SOR method [20]; that is,
(10)
M
w
=
1
w
I
-
w
L
,
N
w
=
1
w
1
-
w
I
+
w
U
⟹
l
w
=
M
w
-
1
N
w
.Lemma 7 (see [19]).
LetA be the coefficient matrix of the linear system A
x
=
b. If A is an L-matrix, conditions of (19) are satisfied and
(11)
0
≤
D
d
≤
1
w
1
-
w
D
,
0
≤
L
l
≤
1
-
r
w
,
0
≤
r
≤
w
≤
1
.
Then (i)
Ifρ
(
l
r
,
w
)
<
1
⇒
ρ
(
T
)
≤
ρ
(
L
r
·
w
).
(ii)
Ifρ
(
l
r
,
w
)
=
1
⇒
ρ
(
T
)
=
ρ
(
L
r
·
w
).
(iii)
Ifρ
(
l
r
,
w
)
>
1
⇒
ρ
(
T
)
≥
ρ
(
L
r
·
w
).Lemma 8 (see [20]).
MatrixA is a GDDM matrix if and only if A is an H-matrix.


## 3. The Mixed Type Splitting Iterative Methods for Fuzzy Linear Systems

LetS be nonsingular and S
=
D
-
L
-
U, where
(12)
s
1
-
s
2
-
s
2
s
1
x
_
x
¯
=
b
_
b
-
,
D
=
D
1
0
0
D
1
,
L
=
L
1
0
s
2
L
1
,
U
=
U
1
s
2
0
U
1
,
D
1
=
diag
⁡
s
1
>
0
,
D
1
-
L
1
-
U
1
=
S
1
,
and -
L
1, -
U
1 are strictly lower and upper triangular matrices of s
1, respectively. The iterative method for S
X
=
B is
(13)
X
(
i
+
1
)
=
M
-
1
N
X
(
i
)
+
M
-
1
B
,
i
=
0,1
,
…
,
where S
=
M
-
N, det (
M
)
≠
0, and X
(
0
) is any initial vector. There are several well-known point iterative methods and block numerical iterative methods for FLS such as Jacobi, Gauss-Seidel, SOR, and AOR; see [9–14]. As a matter of fact, these methods are generalization of iterative methods for crisp linear systems A
x
=
b. For instance, in AOR method for FLS [12] we have
(14)
x
(
i
+
1
)
=
T
r
,
w
x
(
i
)
+
D
-
r
L
-
1
w
b
,
i
=
0,1
,
…
,
where the iterative matrix is
(15)
T
r
,
w
=
D
-
r
L
-
1
1
-
w
D
+
w
-
r
L
+
w
U
.
Therefore, we obtain (16)
M
=
D
-
r
L
=
D
1
-
r
L
1
0
-
r
S
2
D
1
-
r
L
1
⟹
M
-
1
=
D
1
-
r
L
1
-
1
︸
k
0
︸
β
r
D
1
-
r
L
1
-
1
S
2
D
1
-
r
L
1
-
1
︸
η
D
1
-
r
L
1
-
1
︸
γ
. And we have (17)
T
r
,
w
=
M
-
1
N
=
k
0
r
k
S
2
k
k
1
-
w
D
1
0
0
1
-
w
D
1
+
w
-
r
L
1
0
w
-
r
S
2
w
-
r
L
1
+
w
U
1
w
S
2
0
w
U
1
⟹
T
r
,
w
=
k
1
-
w
D
1
+
w
-
r
L
1
+
w
U
1
︸
Φ
w
k
S
2
︸
Γ
w
k
S
2
k
1
-
r
D
1
+
r
U
1
︸
Ω
k
1
-
w
D
1
+
w
-
r
L
1
+
r
w
S
2
k
S
2
+
w
U
1
︸
Ψ
.Other methods are the same and we know that, by choosing special parameters, the similar results can be obtained, for example,(1)
Jacobi method forw
=
1, r
=
0;
(2)
JOR (Jacobi Overrelaxation) method forr
=
0;
(3)
Gauss-Seidel method forr
=
w
=
1;
(4)
SOR method forr
=
w.Now, fromX
(
i
)
=
x
_
(
i
)
x
¯
(
i
) we have the following algorithms (mentioned in [12] for the first time).Algorithm 9.
AOR iterative method for FLS.
Step
1. Choose an initial vector X
(
0
)
=
x
_
(
0
)
x
¯
(
0
) and parameters r and w.
Step
2. For k
=
0,1
,
2
,
… do
(18)
x
_
(
i
+
1
)
=
Φ
x
_
(
i
)
+
Γ
x
¯
(
i
)
+
w
(
k
b
_
+
β
b
¯
)
x
¯
(
i
+
1
)
=
Ψ
x
¯
(
i
)
+
Ω
x
_
(
i
)
+
w
η
b
_
+
γ
b
¯
.
Step
3. If x
¯
(
i
+
1
)
-
x
¯
(
i
)
/
x
¯
(
i
+
1
)
<
ε or x
_
(
i
+
1
)
-
x
_
(
i
)
/
x
_
(
i
+
1
)
<
ε, then stop; otherwise set k
=
k
+
1 and go to Step 2.
Step
4. End for.
Step
5. Return to X
(
k
+
1
).Next, we will establish the mixed type splitting iterative method for FLS. The mixed type splitting iterative methods [19] for solving A
x
=
b are given by the following:
(19)
D
+
D
d
+
L
s
-
L
x
i
+
1
=
D
d
+
L
s
+
U
x
i
+
b
,
i
=
0,1
,
2
,
…
;
whose iteration matrix is
(20)
T
=
D
+
D
d
+
L
s
-
L
-
1
D
d
+
L
s
+
U
,
A
=
D
-
L
-
U
,
D
d
≥
0
,
0
≤
L
s
≤
L
.
Now, we consider the mixed type splitting iterative methods for solving FLS. Based on above demonstration, we have
(21)
D
1
+
D
d
︸
E
+
L
s
-
L
1
0
L
2
s
-
s
2
D
1
+
D
d
+
L
s
-
L
1
︸
M
x
_
i
+
1
x
¯
i
+
1
=
D
d
+
L
s
+
U
1
s
2
L
2
s
D
d
+
L
s
+
U
1
︸
N
x
_
(
i
)
x
¯
(
i
)
+
b
_
b
¯
,
where 0
≤
L
s
≤
L and 0
≤
L
2
s
≤
s
2. Now, let
(22)
k
=
E
+
L
s
-
L
1
-
1
.
Therefore, we obtain
(23)
M
-
1
=
k
0
-
k
(
L
2
s
-
s
2
)
k
︸
η
^
k
.
And we have(24)
T
=
M
-
1
N
=
k
D
d
+
L
s
+
U
1
︸
Φ
^
k
s
2
︸
Γ
^
-
k
L
2
s
-
s
2
k
D
d
+
L
s
+
U
1
+
k
L
2
s
︸
Ω
^
-
k
L
2
s
-
s
2
k
s
2
+
k
D
d
+
L
s
+
U
1
︸
Ψ
^
. Now, from X
(
i
)
=
x
_
(
i
)
x
¯
(
i
) we have the following algorithms (mentioned in [12] for the first time).Algorithm 10.
Mixed type splitting iterative methods_1 for FLS.
Step
1. Choose an initial vector X
(
0
)
=
x
_
(
0
)
x
¯
(
0
) and parameters r and w.
Step
2. For k
=
0,1
,
2
,
… do
(25)
x
_
(
i
+
1
)
=
Φ
^
x
_
(
i
)
+
Γ
^
x
¯
(
i
)
+
(
k
b
_
)
x
¯
(
i
+
1
)
=
Ψ
^
x
¯
(
i
)
+
Ω
^
x
_
(
i
)
+
η
^
b
_
+
k
b
¯
.
Step
3. If x
¯
(
i
+
1
)
-
x
¯
(
i
)
/
x
¯
(
i
+
1
)
<
ε and x
_
(
i
+
1
)
-
x
_
(
i
)
/
x
_
(
i
+
1
)
<
ε, then stop; otherwise set k
=
k
+
1 and go to Step 2.
Step
4. End for.
Step
5. Return to X
(
k
+
1
).Note that when(26)
D
d
=
1
w
(
1
-
w
)
D
1
,
L
s
=
0
,
L
2
s
=
0
,
we have SOR method for FLS (see [11]) and when
(27)
D
d
=
1
w
1
-
w
D
1
,
L
s
=
1
w
w
-
r
L
1
,
L
2
s
=
0
,
where (
w
,
r
) are real parameters with w
≠
0, we have AOR method for FLS (see [12]).Theorem 11.
LetA be an H-matrix and let conditions of Lemma 7 be satisfied.
Then we have(28)
(
i
)
:
ρ
(
T
)
≤
ρ
(
〈
T
〉
)
≤
ρ
(
〈
l
r
,
w
〉
)
<
1
,
(
i
i
)
:
ρ
(
T
)
≤
ρ
(
〈
T
〉
)
≤
ρ
(
〈
l
w
〉
)
<
1
.Proof.
We only prove (i); (ii) can be similarly verified.
LetA be an H-matrix. Then 〈
A
〉 is an M-matrix and by Lemmas 5 and 7,
(29)
ρ
T
≤
ρ
l
r
,
w
<
1
.
By definition of mixed splitting method, we have
(30)
T
=
D
+
D
d
+
L
l
-
L
-
1
D
d
+
L
l
+
U
=
E
I
-
E
-
1
L
-
L
l
-
1
n
n
n
×
E
E
-
1
D
d
+
E
-
1
L
l
+
E
-
1
U
I
-
E
-
1
L
-
L
l
-
1
E
I
-
E
-
1
L
-
L
l
-
1
=
I
-
E
-
1
L
-
L
l
-
1
E
-
1
D
d
+
E
-
1
L
l
+
E
-
1
U
=
E
-
1
L
-
L
l
2
I
+
E
-
1
L
-
L
l
n
n
n
+
E
-
1
L
-
L
l
2
+
⋯
E
-
1
D
d
+
E
-
1
L
l
+
E
-
1
U
=
E
-
1
L
-
L
l
+
E
-
1
L
-
L
l
2
+
⋯
nnn
×
E
-
1
D
d
+
E
-
1
L
l
+
E
-
1
U
nnn
+
E
-
1
D
d
+
E
-
1
L
l
+
E
-
1
U
E
-
1
L
-
L
l
+
E
-
1
L
-
L
l
2
+
…
≤
E
-
1
L
-
L
l
+
E
-
1
L
-
L
l
2
+
⋯
n
n
n
×
E
-
1
D
d
+
E
-
1
L
l
+
E
-
1
U

E
-
1
L
-
L
l
2
+
E
-
1
D
d
+
E
-
1
L
l
+
E
-
1
U
=
I
-
E
-
1
L
-
L
l
-
1
E
-
1
D
d
+
E
-
1
L
l
+
E
-
1
U
=
T
⟹
ρ
T
≤
ρ
(
T
)
.
Therefore we have
(31)
ρ
T
≤
ρ
T
≤
ρ
T
.
And the proof is completed.Lemma 12.
The MatrixS in (5) or (6) is an H
-matrix if and only if A in (1) is H
-matrix.Proof.
LetA be an H-matrix; then by Lemma 8, there exists a positive diagonal matrix W such that A
W is strictly diagonally dominant matrix. Without loss of generality, let A
W be row strictly diagonally dominant; that is,
(32)
∑
j
=
1
j
≠
i
n
a
i
j
w
j
j
<
a
i
i
w
i
i
.
Now, let
(33)
j
+
=
a
i
j
∣
a
i
j
≥
0
,
j
-
=
a
i
j
∣
a
i
j
<
0
,
1
≤
i
,
j
≤
n
;
then we have
(34)
∑
j
=
1
j
≠
i
n
a
i
j
w
j
j
=
∑
j
∈
j
+
j
≠
i
a
i
j
w
j
j
+
∑
j
∈
j
-
j
≠
i
a
i
j
w
j
j
<
a
i
i
w
i
i
.
By considering the structure of S and since, for all i
=
1
,
…
,
n, a
i
i
≥
0, we have
(35)
∑
j
=
1
j
≠
i
n
s
i
j
=
∑
j
∈
j
+
j
≠
i
a
i
j
,
∑
j
=
n
+
1
2
n
s
i
j
=
∑
j
∈
j
-
a
i
j
,
1
≤
i
≤
n
,
∑
j
=
1
n
s
i
j
=
∑
j
∈
j
-
a
i
j
,
∑
j
=
n
+
1
j
≠
i
2
n
s
i
j
=
∑
j
∈
j
+
j
≠
i
a
i
j
,
n
+
1
≤
i
≤
2
n
.
Therefore
(36)
∑
j
=
1
j
≠
i
n
s
i
j
w
j
j
=
∑
j
∈
j
+
j
≠
i
a
i
j
w
j
j
+
∑
j
∈
j
-
j
≠
i
a
i
j
w
j
j
<
a
i
i
w
i
i
nnnnnnnnnnnnnnnnnnn
1
≤
i
≤
n
,
∑
j
=
1
j
≠
i
n
s
i
j
w
j
j
=
∑
j
∈
j
-
j
≠
i
a
i
j
w
j
j
+
∑
j
∈
j
+
j
≠
i
a
i
j
w
j
j
<
a
i
i
w
i
i
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
+
1
≤
i
≤
2
n
,
⟹
∑
j
=
1
j
≠
i
2
n
s
i
j
w
j
j
<
a
i
i
w
i
i
=
s
i
i
w
i
i
.
Then, by choice of
(37)
T
=
W
0
0
W
2
n
×
2
n
S
T is row S
D
D
M. Therefore, by Lemma 8, S is also an H-matrix. Conversely, if S is an H-matrix, then by reasoning similar to that above, it can be seen that A is an H-matrix too.Therefore, we can obtain the following theorem.Theorem 13.
LetA be an H
-matrix and the following conditions are satisfied:
(38)
0
≤
D
d
≤
1
w
1
-
w
D
1
,
0
≤
L
l
≤
1
w
w
-
r
L
1
,
0
≤
L
2
l
≤
s
2
.
Then the speed of convergence of Algorithm 9 is faster than the speed of convergence of SOR and AOR methods for FLS.Proof.
LetA be an H-matrix; then by Lemma 12, S is an H-matrix too. Furthermore, by Theorem 11, ρ
(
〈
l
r
,
w
〉
)
<
1. Moreover, similar to the proving process of Theorem 11 and (31), we can obtain
(39)
ρ
l
r
,
w
≤
ρ
l
r
,
w
≤
ρ
l
r
,
w
.
Therefore,
(40)
ρ
(
l
r
,
w
)
<
1
,
and by Lemma 7 the proof is completed.In the sequel, we will improve Algorithm10 by the following updating technique.(I)
First, usex
_
i and x
¯
i from our previous information to compute the new x
_
(
i
+
1
) exactly as in the first line of Step 2.
(II)
Then, replacex
_
(
i
) in the second line of Step 2 with x
_
(
i
+
1
), obtained just as in the above, to compute the new x
¯
(
i
+
1
).Algorithm 14.
Mixed type splitting iterative methods_2 for FLS.
Step
1. Choose an initial vector X
(
0
)
=
x
_
(
0
)
x
¯
(
0
) and parameters r and w.
Step
2. For k
=
0,1
,
2
,
… do
(41)
x
_
i
+
1
=
Φ
^
x
_
i
+
Γ
^
x
¯
i
+
k
b
_
,
u
p
d
=
Ω
^
x
_
(
i
+
1
)
x
¯
(
i
+
1
)
=
Ψ
^
x
¯
(
i
)
+
u
p
d
+
η
^
b
_
+
k
b
¯
.
Step
3. If x
¯
(
i
+
1
)
-
x
¯
(
i
)
/
x
¯
(
i
+
1
)
<
ε or x
_
(
i
+
1
)
-
x
_
(
i
)
/
x
_
(
i
+
1
)
<
ε, then stop; otherwise set k
=
k
+
1 and go to Step 2.
Step
4. End for.
Step
5. Return to X
(
k
+
1
).


## 4. Numerical Example

In this section, we give an example of FLS to illustrate the results obtained in previous sections.Example 1.
Consider then
×
n fuzzy system
(42)
2
x
1
-
x
3
=
2
+
r
,
4
-
r
,
x
2
+
2
x
3
-
x
4
=
(
2
+
r
,
4
-
r
)
,
x
3
+
2
x
4
-
x
5
=
2
+
r
,
4
-
r
,
⋮
x
n
-
3
+
2
x
n
-
2
=
(
2
+
r
,
4
-
r
)
,
x
n
-
2
+
2
x
n
-
1
=
(
2
+
r
,
4
-
r
)
,
x
n
-
1
+
2
x
n
-
x
1
=
(
2
+
r
,
4
-
r
)
.
The extended 2
n
×
2
n matrix is
(43)
S
=
s
1
-
s
2
-
s
2
s
1
,
where
(44)
s
1
=
2
0
⋯
0
0
0
0
1
2
⋯
0
0
0
0
⋮
⋱
⋱
⋯
⋱
⋱
⋮
0
⋯
1
2
0
⋯
0
0
⋯
0
1
2
0
0
0
⋯
⋱
0
1
2
0
0
⋯
⋯
0
0
1
2
,
s
2
=
0
0
0
-
1
0
⋯
0
0
0
0
0
-
1
⋱
⋮
⋮
⋮
⋱
⋱
⋱
⋱
0
0
⋯
⋯
0
0
⋱
-
1
0
⋯
⋯
0
0
0
0
0
⋯
⋯
0
0
0
0
-
1
0
⋯
0
0
0
0
.
Evidently, A is an H-matrix and therefore S is also an H-matrix. Table 1 shows the numerical results of the above example with the tolerance ε
=
1
0
-
10. Moreover, the initial approximation is zero vector. In the Table 1, we reported the number of iterations (Iter), Elapsed time (ELP), and associated spectral radii (ρ) for the iterative methods with different n, w, and r. This results show that our algorithms can be applied to a large class of FLS. Furthermore, from Table 1, we can see that for solving FLS, Algorithm 14 is superior to the other methods from point of view rate of convergence.
This example is computed with MATLAB7 on a personal computer Pentium 4-256 MHZ.Table 1
(D
d
=
0.1
(
-
1
+
1
/
w
)
D
1
;
L
s
=
0.5
(
1
-
r
/
w
)
L
1
;
L
2
s
=
0.01
s
2).
Method
AOR or SOR  iterative methods
Algorithm10
Algorithm14
n
w
r
Iter.
ELP
ρ
Iter.
ELP
ρ
Iter.
ELP
ρ
5
0.5
0.1
64
0.003005
0.6739
20
0.001933
0.3764
20
0.001228
0.3228
0.5
0.5
62
0.003032
0.6322
17
0.001876
0.2278
15
0.001183
0.1476
0.7
0.3
34
0.001592
0.5191
16
0.001545
0.2530
16
0.001199
0.1953
0.9
0.6
21
0.001523
0.3192
13
0.001109
0.1399
12
0.001056
0.1084
50
0.5
0.1
528
0.034716
0.9173
219
0.024478
0.9261
188
0.019920
0.5229
0.5
0.5
460
0.032090
0.8419
142
0.020052
0.8847
126
0.015032
0.6006
0.7
0.3
359
0.029076
0.8241
175
0.019944
0.9072
152
0.018738
0.4894
0.9
0.6
189
0.019587
0.9227
140
0.015078
0.8831
125
0.013490
0.6094
100
0.5
0.1
1123
0.069947
0.9290
403
0.053815
0.9611
343
0.069539
0.5275
0.5
0.5
652
0.060993
0.8534
264
0.041211
0.9392
233
0.060378
0.6449
0.7
0.3
753
0.057825
0.8544
324
0.046366
0.9511
279
0.053829
0.5134
0.9
0.6
452
0.051150
0.6884
260
0.037807
0.9383
232
0.053185
0.6554
250
0.5
0.1
2578
3.042835
0.9672
920
0.898741
0.9840
780
0.795855
0.5320
0.5
0.5
1993
1.967367
0.9302
619
0.635806
0.9749
549
0.582322
0.6738
0.7
0.3
1553
1.566464
0.9419
746
0.761868
0.9798
644
0.656939
0.5342
0.9
0.6
882
0.885911
0.8819
607
0.617928
0.9745
543
0.571853
0.6854


## 5. Conclusion

In this paper, we have proposed a class of splitting methods, called mixed type splitting iterative method for fuzzy linear systems that contains auxiliary matrices. Furthermore, our method with some other iterative methods is compared in the frame of fuzzy linear systems and it is shown that by proper choice of these auxiliary matrices, the new iterative method is faster than these mentioned methods form point of view of the convergence speed.



---

*Source: 960795-2014-11-18.xml*