# Comparison Theorems of Spectral Radius for Splittings of Matrices

Authors: Cui-Xia Li; Su-Hua Li
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

A class of the iteration method from the double splitting of coefficient matrix for solving the linear system is further investigated. By structuring
a new matrix, the iteration matrix of the corresponding double splitting iteration method is presented. On the basis of convergence and comparison theorems for single splittings, we present some new convergence and comparison theorems on spectral radius for splittings of matrices.


---

## Body

## 1. Introduction

Let us consider the following linear system:(1)
A
x
=
b
,
where A
∈
R
n
×
n is a nonsingular matrix, b
∈
R
n
×
1 is a given vector, and x
∈
R
n
×
1 is an unknown vector. In order to solve the linear system (1) by iteration methods, the coefficient matrix A is split into
(2)
A
=
M
-
N
,
where M is nonsingular; then, an iterative formula for solving the linear system (1) is
(3)
x
k
+
1
=
M
-
1
N
x
k
+
M
-
1
b
≡
T
x
k
+
M
-
1
b
,
hhhhhhhhhhhhhhhhhhhh
k
=
0,1
,
2
,
…
,
where T
=
M
-
1
N is the iteration matrix in (3).The splitting (2) is called a (single) splitting of A and the iteration method (3) is called a (one-step) linear stationary iteration method. Obviously, the iteration method (3) converges to the unique solution of the linear system (1) if and only if the spectral radius ρ
(
T
) of the iteration matrix T is smaller than 1. The spectral radius of the iteration matrix is decisive for the convergence and stability, and the smaller it is, the faster the iteration method converges when the spectral radius is smaller than 1. So far, many comparison theorems of single splittings of matrices have been presented in some papers and books [1–8].Woźnicki [9] introduced the double splitting of A as
(4)
A
=
P
-
R
-
S
,
where P is a nonsingular matrix. The corresponding iterative scheme is spanned by three successive iterations:
(5)
x
k
+
1
=
P
-
1
R
x
k
+
P
-
1
S
x
k
-
1
+
P
-
1
b
,
hhhhhhhhhhhhhhhhh
k
=
0,1
,
2
,
…
,
which can be rewritten in the equivalent form
(6)
[
x
k
+
1
x
k
]
=
[
P
-
1
R
P
-
1
S
I
0
]
[
x
k
x
k
-
1
]
+
[
P
-
1
b
0
]
,
where I is the identity matrix. The iteration method given by (6) converges to the unique solution of (1) for all initial vectors x
0, x
1 if and only if the spectral radius of the iteration matrix
(7)
W
=
[
P
-
1
R
P
-
1
S
I
0
]
,
is less than one; that is, ρ
(
W
)
<
1.Recently, some convergence and comparison results for double splittings of matrices are presented. In [10], some convergence theorems for the double splitting of a monotone matrix or a Hermitian positive definite matrix are presented. Compared with the results in [10], some improved convergence and comparison results for the double splitting of a Hermitian positive definite matrix are proposed in [11]. In [12], some convergence results for the double splitting of a non-Hermitian positive semidefinite matrix are established. Further, some comparison theorems for double splittings of different monotone matrices are given in [13, 14] and some convergence and comparison results for nonnegative double splittings of matrices are given in [4, 15]. In this paper, by structuring a new matrix, the iteration matrix of the corresponding iteration method from double splitting of coefficient matrix is presented. On the basis of convergence and comparison theorems for single splittings, we present some new convergence and comparison theorems on spectral radius for splittings of matrices.


## 2. Preliminaries

For convenience, we give some notations, definitions, and lemmas which will be used in the sequel.The matrixA is called nonnegative and is denoted by A
≥
0 if a
i
j
≥
0 for i, j
=
1,2
,
…
,
n. We write A
≥
B  (
A
>
B
) if a
i
j
≥
b
i
j
(
a
i
j
>
b
i
j
) for i, j
=
1,2
,
…
,
n. The matrix A is called a monotone matrix if A
-
1
≥
0.Definition 1.
LetA be a nonsingular matrix. Then, A
=
M
-
N is called (i)
regular ifM
-
1
≥
0 and N
≥
0;
(ii)
weak regular ifM
-
1
≥
0 and M
-
1
N
≥
0;
(iii)
nonnegative ifM
-
1
N
≥
0;
(iv)
M-splitting if M is an M-matrix and N
≥
0.Definition 2 (see [4, 10, 15]).
LetA be a nonsingular matrix. Then, the double splitting A
=
P
-
R
-
S is (i)
convergent if and only ifρ
(
W
)
<
1;
(ii)
a regular double splitting ifP
-
1
≥
0, R
≥
0, and S
≥
0;
(iii)
a weak regular double splitting ifP
-
1
≥
0, P
-
1
R
≥
0, and P
-
1
S
≥
0;
(iv)
a nonnegative splitting ifP
-
1
R
≥
0 and P
-
1
S
≥
0;
(v)
anM-double splitting if P is an M-matrix and R
≥
0 and S
≥
0.Lemma 3 (see [2]).
LetA
≥
0. Then
(8)
α
x
≤
A
x
,
x
≥
0
,
i
m
p
l
i
e
s
α
≤
ρ
(
A
)
,
A
x
≤
β
x
,
x
>
0
,
i
m
p
l
i
e
s
ρ
(
A
)
≤
β
.Lemma 4 (see [16]).
LetA
∈
R
n
×
n and A
=
M
1
-
N
1
=
M
2
-
N
2 be M-splittings of A (i.e., M
i are M-matrices; N
i
≥
0, i
=
1,2) and
(9)
N
1
≥
N
2
,
N
1
≠
N
2
,
N
2
≠
0
.
Then, exactly one of the following statements holds: (1)
0
≤
ρ
(
M
2
-
1
N
2
)
<
ρ
(
M
1
-
1
N
1
)
<
1. In addition, if A is irreducible, the first inequality is also strict;
(2)
ρ
(
M
2
-
1
N
2
)
=
ρ
(
M
1
-
1
N
1
)
=
1;
(3)
ρ
(
M
2
-
1
N
2
)
>
ρ
(
M
1
-
1
N
1
)
>
1.Lemma 5 (see [17]).
LetA
∈
R
n
×
n and A
=
M
1
-
N
1
=
M
2
-
N
2 be nonnegative and convergent. (1)
If eitherM
1
-
1
M
2
≥
I or M
2
-
1
M
1
≤
I, then ρ
(
M
1
-
1
N
1
)
<
ρ
(
M
2
-
1
N
2
)
<
1.
(2)
If there existα, 0
<
α
<
1, such that M
1
-
1
M
2
≥
(
1
/
α
)
I or M
2
-
1
M
1
≤
α
I, then ρ
(
M
1
-
1
N
1
)
<
ρ
(
M
2
-
1
N
2
)
<
1.


## 3. Comparison Theorems

Let(10)
A
=
P
1
-
R
1
-
S
1
=
P
2
-
R
2
-
S
2
,
be two double splittings of A. Then, we define
(11)
W
1
=
[
P
1
-
1
R
1
P
1
-
1
S
1
I
0
]
,
W
2
=
[
P
2
-
1
R
2
P
2
-
1
S
2
I
0
]
.
Let
(12)
A
=
[
A
0
-
I
I
]
.
Then, A
∈
R
2
n
×
2
n and
(13)
A
-
1
=
[
A
-
1
0
A
-
1
I
]
.
This shows that A is nonsingular whenever A is nonsingular. Let A be split as
(14)
A
=
M
-
N
,
with
(15)
M
=
[
P
S
0
I
]
,
N
=
[
R
+
S
S
I
0
]
.
Then
(16)
W
=
M
-
1
N
=
[
P
-
1
R
P
-
1
S
I
0
]
.In [4], some comparison theorems for the double splitting (4) through investigating the matrix splitting defined by (14) were obtained, which were described as follows.Theorem 6 (see [4]).
LetA
-
1
≥
0, and let the two double splittings (10) be nonnegative and convergent. Suppose
(17)
P
1
≤
P
2
,
S
1
≤
S
2
;
then
(18)
ρ
(
W
1
)
≤
ρ
(
W
2
)
.Corollary 7 (see [4]).
LetA
-
1
≥
0, and let the two double splittings (10) be nonnegative and convergent. Suppose
(19)
R
1
≤
R
2
,
S
1
≤
S
2
;
then
(20)
ρ
(
W
1
)
≤
ρ
(
W
2
)
.Theorem 8 (see [4]).
LetA
-
1
≥
0, A
=
P
1
-
R
1
-
S
1 be regular double splitting, and let A
=
P
2
-
R
2
-
S
2 be nonnegative and convergent double splitting. Suppose
(21)
P
1
-
1
≥
P
2
-
1
,
P
1
-
1
S
1
≤
P
2
-
1
S
2
;
then ρ
(
W
1
)
≤
ρ
(
W
2
).In [4], they claimed that A is nonsingular whenever A is nonsingular. In fact, we make use of the following strategy to make A nonsingular. That is to say,
(22)
M
-
1
A
=
I
-
M
-
1
N
.
Obviously, if I
-
M
-
1
N is nonsingular, then we immediately obtain that matrix A is nonsingular too. When one discusses the convergence properties of the iteration scheme (6), it is expected that the spectral radius ρ
(
W
) of the iteration matrix W
=
M
-
1
N is less than one. In this case, the iteration scheme (6) is convergent. In this meanwhile, we also know that A is nonsingular. Further, comparison theorems discussed are more meaningful as the spectral radius ρ
(
W
) of the iteration matrix W
=
M
-
1
N is less than one. Based on this idea, we can consider the choice of matrixes M and N as
(23)
M
=
[
P
0
0
I
]
,
N
=
[
R
S
I
0
]
.
In light of this choice, we also have the same as the iteration matrix W,
(24)
W
=
M
-
1
N
=
[
P
-
1
R
P
-
1
S
I
0
]
.
In this case, the matrix A is not
(25)
A
=
[
A
0
-
I
I
]
but is
(26)
A
=
[
P
-
R
-
S
-
I
I
]
.
Then, we have
(27)
A
=
M
-
N
≡
[
P
0
0
I
]
-
[
R
S
I
0
]
.Based on Lemma4, we have the following results.Theorem 9.
Let(28)
A
=
M
i
-
N
i
=
[
P
i
0
0
I
]
-
[
R
i
S
I
0
]
(
i
=
1,2
)
,
be M-splittings of A. If R
1
≤
R
2, then exactly one of the following statements holds: (1)
0
≤
ρ
(
W
1
)
<
ρ
(
W
2
)
<
1. In addition, if A is irreducible, the first inequality is also strict;
(2)
ρ
(
W
1
)
=
ρ
(
W
2
)
=
1;
(3)
ρ
(
W
1
)
>
ρ
(
W
2
)
>
1.Proof.
Fori
=
1,2, let
(29)
M
i
=
[
P
i
0
0
I
]
,
N
i
=
[
R
i
S
0
I
]
.
Then
(30)
A
=
M
i
-
N
i
,
W
i
=
M
i
-
1
N
i
.
Since R
1
≤
R
2, then N
1
≤
N
2. That is, from Lemma 4, the results in Theorem 9 hold true.Theorem 10.
Let(31)
A
=
M
i
-
N
i
=
[
P
i
0
0
I
]
-
[
R
i
S
I
0
]
(
i
=
1,2
)
,
be nonnegative and convergent. If either P
1
-
1
P
2
≥
I or P
2
-
1
P
1
≤
I, then ρ
(
W
1
)
<
ρ
(
W
2
)
<
1.Proof.
Let(32)
A
=
M
i
-
N
i
=
[
P
i
0
0
I
]
-
[
R
i
S
I
0
]
(
i
=
1,2
)
,
be nonnegative. By direct operation, we obtain
(33)
M
1
-
1
M
2
=
[
P
1
-
1
P
2
0
0
I
]
,
M
2
-
1
M
1
=
[
P
2
-
1
P
1
0
0
I
]
.
Since P
1
-
1
P
2
≥
I or P
2
-
1
P
1
≤
I, we have M
1
-
1
M
2
≥
I or M
2
-
1
M
1
≤
I. From Lemma 5, the results of Theorem 10 hold true.Obviously, from Lemma5, we have the following result.Theorem 11.
Let(34)
A
=
M
i
-
N
i
=
[
P
i
0
0
I
]
-
[
R
i
S
I
0
]
(
i
=
1,2
)
,
be nonnegative and convergent. If there exist α, 0
<
α
<
1, such that P
1
-
1
P
2
≥
(
1
/
α
)
I or P
2
-
1
P
1
≤
α
I, then ρ
(
W
1
)
<
ρ
(
W
2
)
<
1.Compared with Corollary7 and Theorem 8, the condition A
-
1
≥
0 in Theorems 9, 10, and 11 is not necessary.Theorem 12.
Let(35)
A
-
1
=
[
P
-
R
-
S
-
I
I
]
-
1
≥
0
,
A
=
M
i
-
N
i
=
[
P
i
0
0
I
]
-
[
R
i
S
I
0
]
(
i
=
1,2
)
,
be nonnegative and convergent. If P
1
≤
P
2, then ρ
(
W
1
)
<
ρ
(
W
2
)
<
1.Proof.
Obviously, ifP
1
≤
P
2, then M
1
≤
M
2. From Theorem  2.11 in [17], the results of Theorem 12 hold true.Theorem 13.
Let(36)
A
-
1
=
[
P
-
R
-
S
-
I
I
]
-
1
≥
0
,
let A
=
M
1
-
N
1 be regular, and let A
=
M
2
-
N
2 be nonnegative and convergent. If P
2
-
1
≤
P
1
-
1, then ρ
(
W
1
)
<
ρ
(
W
2
)
<
1.Proof.
Assume thatρ
(
W
1
)
=
0; the result is trivial. Assume that ρ
(
W
1
)
>
0. Then, the splitting A
=
M
1
-
N
1
=
M
2
-
N
2 is nonnegative.
By the Perron-Frobenius Theorem in [1], there exists a vector x
≥
0 and x
≠
0, such that W
1
x
=
ρ
(
W
1
)
x. From N
1
≥
0, we derive N
1
x
≥
0. Since P
2
-
1
≤
P
1
-
1, then M
2
-
1
≤
M
1
-
1. From Theorem 2.17 in [17], the results of Theorem 13 hold true.


## 4. Numerical Example

In this section, we make use of an example to illustrate Theorems9, 10, 12, and 13.Example 1.
Assume that(37)
A
=
[
3
0
-
1
-
1
0
4
-
1
-
1
-
1
0
1
0
0
-
1
0
1
]
.
Then
(38)
A
-
1
=
[
0.6
0.2
0.8
0.8
0.2
0.4
0.6
0.6
0.6
0.2
1.8
0.8
0.2
0.4
0.6
1.6
]
≥
0
.
Let
(39)
M
1
=
[
4
0
0
0
0
5
0
0
0
0
1
0
0
0
0
1
]
,
N
1
=
[
1
0
-
1
-
1
0
1
-
1
-
1
1
0
0
0
0
1
0
0
]
,
M
2
=
[
5
0
0
0
0
6
0
0
0
0
1
0
0
0
0
1
]
,
N
2
=
[
2
0
-
1
-
1
0
2
-
1
-
1
1
0
0
0
0
1
0
0
]
.
Then
(40)
P
1
-
1
=
[
1
4
0
0
1
5
]
,
P
2
-
1
=
[
1
5
0
0
1
6
]
.
Therefore, we have the following facts. That is to say,
(41)
R
1
≤
R
2
;
P
1
-
1
P
2
≥
I
,
P
2
-
1
P
1
≤
I
;
P
1
≤
P
2
;
P
2
-
1
≤
P
1
-
1
,
which, respectively, satisfy the conditions of Theorems 9, 10, 12, and 13. In this case, by the simple computations, we have
(42)
ρ
(
W
1
)
=
0.7949
,
ρ
(
W
2
)
=
0.8195
.
Clearly, ρ
(
W
1
)
≤
ρ
(
W
2
)
<
1. That is to say, Theorems 9, 10, 12, and 13 hold true.



---

*Source: 573024-2014-07-08.xml*