# Comparison Theorems for Single and Double Splittings of Matrices

Authors: Cui-Xia Li; Qun-Fa Cui; Shi-Liang Wu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Some comparison theorems for the spectral radius of double splittings of different matrices under suitable conditions are presented, which are superior to the corresponding results in the recent paper by Miao and Zheng (2009). Some comparison theorems between the spectral radius of single and double splittings of matrices are established and are applied to the Jacobi and Gauss-Seidel double SOR method.


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## Body

## 1. Introduction

Consider the linear system(1)Ax=b,
where A∈ℝn×n is nonsingular, b∈ℝn×1 is given, and x∈ℝn×1 is unknown. The splitting of the coefficient matrix
(2)A=M-N,
where M is nonsingular, is called a single splitting of A in [1]; the basic iterative method for solving (1) is
(3)xk+1=M-1Nxk+M-1b≡Txk+M-1b,k=0,1,2,…,
where matrix T=M-1N is the iteration matrix in (3). Obviously, the iterative method (3) converges to the unique solution of the linear system (1) if and only if the spectral radius ρ(M-1N) of the iteration matrix 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 iterative method converges when the spectral radius is smaller than 1. So far, many comparison theorems of single splitting of matrices have been arisen in some papers and books [2–8].The double splitting ofA was introduced by Woźnicki [1] and can be described as follows. Splitting the matrix A in the form
(4)A=P-R-S
is called the double splitting of A, where P is a nonsingular matrix; the corresponding iterative scheme is spanned by three successive iterations:
(5)xk+1=P-1Rxk+P-1Sxk-1+P-1b,k=0,1,2,….
Following the idea of Golub and Varga [9], Woźnicki wrote (5) in the following equivalent form:
(6)[xk+1xk]=[P-1RP-1SI0][xkxk-1]+[P-1b0],
where I is the identity matrix. Then, the iterative method (6) converges to the unique solution of (1) for all initial vectors x0, x1 if and only if the spectral radius of the iteration matrix
(7)W=[P-1RP-1SI0]
is less than one, that is, ρ(W)<1.Recently, some comparison theorems for double splittings of monotone matrices and Hermitian positive definite matrices were presented in [8, 10–13]. Elsner et al. [14] presented some comparison theorems of single splittings of different monotone matrices, that is, matrices with nonnegative inverses. Our basic purpose here is to derive some new comparison theorems for the spectral radius of double splittings of different matrices. Under suitable conditions, new comparison theorems are superior to the corresponding results in the recent paper [12]. Some comparison theorems between the spectral radius of single and double splittings of matrices are also established and are applied to the Jacobi and Gauss-Seidel double SOR method.


## 2. Preliminaries

For convenience, we give some of the notations, definitions, and lemmas which will be used in the sequel.The matrixA is called nonnegative and denoted by A≥0 if aij≥0 for i,j=1,2,…,n. We write A≥B(A>B) if aij≥bij(aij>bij) for i,j=1,2,…,n. The matrix A is called a monotone matrix if A-1≥0. Matrix A is an L-matrix if aii>0(i=1,…,n) and aij<0 for all i,j=1,…,n; i≠j.Definition 1.
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-1R≥0, and P-1S≥0;(iv)
a nonnegative splitting ifP-1R≥0 and P-1S≥0.Lemma 2 (see [3]).
LetA≥0. Then, αx≤Ax,x≥0, implies α≤ρ(A) and Ax≤βx,x>0, implies ρ(A)≤β.Lemma 3 (see [10]).
LetA-1≥0 and A=P-R-S be a weak regular double splitting. Then, ρ(W)<1.


## 3. Comparison Theorem

In [12], Miao and Zheng gave a comparison theorem for the spectral radius of double splittings of different monotone matrices. That is, [12, Theorem 3.1] is a major result and is described as follows.LetA1 and A2 be two monotone matrices, and let A1=P1-R1-S1, and let A2=P2-R2-S2 be double splittings of A1 and A2, respectively. Consequently,
(8)W1=[P1-1R1P1-1S1I0],W2=[P2-1R2P2-1S2I0].Theorem 4 (see [12]).
LetA1 and A2 be two nonsingular matrices with A1-1≥0 and A2-1≥0, A1=P1-R1-S1, and let A2=P2-R2-S2 be weak regular double splittings. If P1-1A1≥P2-1A2 and P1-1R1≥P2-1R2, then ρ(W1)≤ρ(W2)<1.Based on the forms ofW1 and W2, we have the following theorem.Theorem 5.
LetA1 and A2 be two nonsingular matrices, and let A1=P1-R1-S1 and A2=P2-R2-S2 be nonnegative splittings. If P1-1R1≤P2-1R2 and P1-1S1≤P2-1S2, then ρ(W1)≤ρ(W2)<1 for ρ(W2)<1.Proof.
Obviously, ifP1-1R1≤P2-1R2 and P1-1S1≤P2-1S2, then 0≤W1≤W2. Therefore, we obtain that ρ(W1)≤ρ(W2)<1 for ρ(W2)<1.Based on Definition1, we obtain the following Theorem 6, which is superior to Theorem 4 [12].Theorem 6.
LetA1 and A2 be two nonsingular matrices, and let A1=P1-R1-S1 and A2=P2-R2-S2 be nonnegative splittings. If P1-1A1≥P2-1A2 and P1-1R1≥P2-1R2, then ρ(W1)≤ρ(W2)<1 for 0<ρ(W2)<1.Proof.
Obviously,W1≥0 and W2≥0. By the Perron-Frobenius theorem [3], there exists a vector
(9)x=[x1x2]≥0,x≠0
such that W2x=ρ(W2)x; that is,
(10)P2-1R2x1+P2-1S2x2=ρ(W2)x1,x1=ρ(W2)x2.
Then, we have
(11)W1x-ρ(W2)x=[P1-1R1x1+P1-1S1x2-ρ(W2)x1x1-ρ(W2)x2]=[(P1-1R1-P2-1R2)x1+1ρ(W2)(P1-1S1-P2-1S2)x1x1-ρ(W1)x2]≤1ρ(W2)[(P1-1R1-P2-1R2)x1+(P1-1S1-P2-1S2)x10]=1ρ(W2)[(P1-1R1+P1-1S1)x1-(P2-1R2+P2-1S2)x10]=1ρ(W2)[P1-1(R1+S1)x1-P2-1(R2+S2)x10]=1ρ(W2)[P1-1(P1-A1)x1-P2-1(P2-A2)x10]=1ρ(W2)[(P2-1A2-P1-1A1)x10]≤0.
From Lemma 2, we obtain that ρ(W1)≤ρ(W2)<1 for 0<ρ(W2)<1.By investigating Theorem6, it is easy to see that the conditioners, Theorem 6 are weaker than those of Theorem 4 [12]. That is, the result of Theorem 6 holds without A1-1≥0 and A2-1≥0.Similarly, we have the following result.Theorem 7.
LetA1 and A2 be two nonsingular matrices, and let A1=P1-R1-S1 and A2=P2-R2-S2 be nonnegative splittings. If P1-1A1≥P2-1A2 and P1-1S1≤P2-1S2, then ρ(W1)≤ρ(W2)<1 for 0<ρ(W2)<1.


## 4. Convergence for the Jacobi and Gauss-Seidel Double SOR Method

To establish some comparison theorems between the spectral radius of single and double splittings of matrices, based on (3) and (5), we obtain that M=P and N=R+S. Here and now, T=P-1(R+S).The result for comparingρ(W) with ρ(T) is stated as in the following theorem.Theorem 8.
LetA=P-R-S be a nonnegative splitting. Then,(1)
ρ(T)≤ρ(W)<1 for ρ(W)<1;(2)
ρ(T)≥ρ(W)>1 for ρ(W)>1.Proof.
By Definition1, obviously, matrix W≥0. Based on the Perron-Frobenius theorem [3], there exists a vector
(12)x=[x1x2]≥0,x≠0
such that Wx=ρ(W)x, that is,
(13)[P-1RP-1SI0][x1x2]=ρ(W)[x1x2].
The above equation is equivalent to
(14)P-1Rx1+P-1Sx2=ρ(W)x1,(15)x1=ρ(W)x2.
From (15), we get that x2=(1/ρ(W))x1. Substituting it into (14) yields
(16)ρ(W)x1=P-1Rx1+1ρ(W)P-1Sx1.
If ρ(W)<1, then
(17)ρ(W)x1≥P-1Rx1+P-1Sx1.
That is, ρ(W)x1≥Tx1. By Lemma 2, it is easy to obtain that ρ(T)≤ρ(W)<1.
Obviously, we also obtain thatρ(T)≥ρ(W)>1 for ρ(W)>1.Example 9.
Let(18)A=[1-2-21]=[0-2-20]-[-1000]-[000-1]≡P-R-S.
Then,
(19)P-1=[0-12-120]≤0,P-1R=[00120]≥0,P-1S=[01200]≥0.
By the simple computations, we have ρ(W)=0.63 and ρ(T)=0.5. Clearly, ρ(T)≤ρ(W)<1 holds.
Let the matrixA be split as
(20)A=D-L-U,
where D=diag(A), and L, U are strictly lower and upper triangular matrices, respectively, for ω≠0 and σ≠0. Let
(21)PJ=1ωσD,RJ=1ωσ[ωσ(L+U)-(ω-1)D-(σ-1)D],SJ=(ω-1)(σ-1)ωσD.
Then, the iterative method (5) corresponding to the double splitting
(22)A=PJ-RJ-SJ
is called the Jacobi double SOR method [1, 15].
Based on (21), we have the following lemma.Lemma 10.
Let A be an L-matrix, and let the double splittings be defined by (21) and (22). Suppose
(23)2-σ-ω≥0,(ω-1)(σ-1)≥0.
Then, the double splitting defined by (22) is regular.Let(24)WJ=[PJ-1RJPJ-1SJI0],TJ=PJ-1(RJ+SJ).
Then, we have the following result.Theorem 11.
Under the conditions of Lemma10, then  (1)
ρ(TJ)≤ρ(WJ)<1 for ρ(WJ)<1;(2)
ρ(TJ)≥ρ(WJ)>1 for ρ(WJ)>1.Proof.
From Theorem8, it is easy to see that Theorem 11 holds.Let(25)PG=1ωσD(I-σD-1U),SG=(ω-1)(σ-1)ωσD,RG=1ωσ[ωσL-(ω-1)D(I-σD-1U)-(σ-1)D].
Then, the iterative method (5) corresponding to the double splitting
(26)A=PG-RG-SG
is called the Gauss-Seidel double SOR method [1, 15].Let(27)WG=[PG-1RGPG-1SGI0],TG=PG-1(RG+SG).
Similarly, we have the following result.Theorem 12.
Let A be an L-matrix, and let the double splittings be defined by (25) and (26), for 0<σ≤1 and 0<ω≤1; then (1)
ρ(TG)≤ρ(WG)<1 for ρ(WG)<1;(2)
ρ(TG)≥ρ(WG)>1 for ρ(WG)>1.From Theorems8, 11, and 12, it is easy to see that the spectral radius of single splitting method is less than the spectral radius of double splitting method under suitable conditions. That is, the efficiency of the single splitting method maybe be superior to that of the double splitting method under suitable conditions.



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