Block Preconditioned SSOR Methods for 𝐻-Matrices Linear Systems

Abstract

We present a block preconditioner and consider block preconditioned SSOR iterative methods for solving linear system 𝐴𝑥=𝑏. When 𝐴 is an 𝐻-matrix, the convergence and some comparison results of the spectral radius for our methods are given. Numerical examples are also given to illustrate that our methods are valid.

1. Introduction

For the linear system 𝐴𝑥=𝑏, (1) where 𝐴 is an 𝑛×𝑛 square matrix and 𝑥 and 𝑏 are 𝑛-dimensional vectors. The basic iterative method for solving (1) is 𝑀𝑥^𝑘+1=𝑁𝑥^𝑘+𝑏,𝑘=0,1,…, (2) where 𝐴=𝑀−𝑁 and 𝑀 is nonsingular. Thus (2) can be written as 𝑥^𝑘+1=𝑇𝑥^𝑘+𝑐,𝑘=0,1,…, (3) where 𝑇=𝑀⁻¹𝑁, 𝑐=𝑀⁻¹𝑏.

Let us consider the following partition of 𝐴: 𝐴𝐴=¹¹𝐴₁₂⋯𝐴_1𝑚𝐴₂₁𝐴₂₂⋯𝐴_2𝑚𝐴⋮⋮⋱⋮_𝑚1𝐴_𝑚2⋯𝐴_𝑚𝑚, (4) where the blocks 𝐴_𝑖𝑖∈𝐶^{𝑛𝑖×𝑛𝑖}, 𝑖=1,…,𝑚, are nonsingular and 𝑛₁+𝑛₂+⋯+𝑛_𝑚=𝑛.

Usually we split 𝐴 into 𝐴=𝐷−𝐿−𝑈, (5) where 𝐷=diag(𝐴₁₁,…,𝐴_𝑚𝑚), −𝐿 and −𝑈 are strictly block lower and strictly block upper triangular parts of 𝐴, respectively. Let 0<𝜔<2, and 1𝑀=𝜔()()𝐷2−𝜔𝐷−𝜔𝐿⁻¹(),1𝐷−𝜔𝑈𝑁=𝜔()))𝐷2−𝜔((1−𝜔𝐷+𝜔𝐿⁻¹((1−𝜔)𝐷+𝜔𝑈). (6) Then, the iteration matrix of the SSOR method for 𝐴 is given by ℒ_𝜔=𝑀⁻¹𝑁=(𝐷−𝜔𝑈)⁻¹𝐷(𝐷−𝜔𝐿)⁻¹×((1−𝜔)𝐷+𝜔𝐿)𝐷⁻¹×((1−𝜔)𝐷+𝜔𝑈). (7)

Transforming the original system (1) into the preconditioned form 𝑃𝐴𝑥=𝑃𝑏, (8) then we can define the basic iterative scheme: 𝑀_𝑝𝑥^𝑘+1=𝑁_𝑝𝑥^𝑘+𝑃𝑏,𝑘=0,1,…, (9) where 𝑃𝐴=𝑀_𝑝−𝑁_𝑝 and 𝑀_𝑝 is nonsingular. Thus (9) can also be written as 𝑥^𝑘+1=𝑇𝑥^𝑘+𝑐,𝑘=0,1,…, (10) where 𝑇=𝑀_𝑝⁻¹𝑁_𝑝,𝑐=𝑀_𝑝⁻¹𝑃𝑏. Similar to the original system (1), we call the basic iterative methods corresponding to the preconditioned system the preconditioned iterative methods.

When 𝐴 is an 𝑀-matrix, Alanelli and Hadjidimosin [1] considered the preconditioner 𝑃=𝑄+𝑆, where 𝑄=diag(𝐿⁻¹₁₁,𝐼₂₂,…,𝐼_𝑚𝑚) and 𝑆 is given by 𝑂𝑆=¹¹𝑂₁₂⋯𝑂_1𝑚−𝐴₂₁𝐿⁻¹₁₁𝑂₂₂⋯𝑂_2𝑚⋮⋮⋱⋮−𝐴_𝑚1𝐿⁻¹₁₁𝑂_𝑚2⋯𝑂_𝑚𝑚, (11) with 𝐿₁₁ being the lower triangular matrix in the LU triangular decomposition of 𝐴₁₁.

We consider the preconditioner 𝑃₁=𝐼+𝑆₁, where 𝐼+𝑆₁=𝐼₁₁−𝛼₁𝐴⁻¹₁₁𝐴_1𝑠𝐼₂₂−𝛼₂𝐴⁻¹₂₂𝐴_2𝑡⋱⋱−𝛼_𝑚𝐴⁻¹_𝑚𝑚𝐴_𝑚𝑢𝐼_𝑚𝑚. (12)

Let 𝑃₁𝐴=𝐼+𝑆₁(𝐷−𝐿−𝑈)=𝐷−𝐿−𝑈, (13) where 𝐷, −𝐿 and −𝑈 are block diagonally, strictly block lower, and strictly block upper triangular parts of 𝑃₁𝐴, respectively. If 𝐷 is nonsingular, then (𝐷−𝜔𝐿)⁻¹ and (𝐷−𝜔𝑈)⁻¹ exist and it is possible to define the SSOR iteration matrix for 𝑃₁𝐴. Namely, ℒ_𝜔=𝐷−𝜔𝑈⁻¹𝐷𝐷−𝜔𝐿⁻¹×(1−𝜔)𝐷+𝜔𝐿𝐷⁻¹(1−𝜔)𝐷+𝜔𝑈. (14)

Alanelli and Hadjidimos in [1] showed that the preconditioned Gauss-Seidel, the preconditioned SOR, and the preconditioned Jacobi methods with preconditioner 𝑃 are better than original methods. Our work in the presentation is to prove convergence of the block preconditioned SSOR method with preconditioner 𝑃₁ and give some comparison results of the spectral radius for the case when 𝐴 is an 𝐻-matrix.

Let |𝐴| denote the matrix whose elements are the moduli of the elements of the given matrix. We call ⟨𝐴⟩=(𝑎_𝑖𝑗) to comparison matrix if 𝑎_𝑖𝑗=|𝑎_𝑖𝑗| for 𝑖=𝑗, if 𝑎_𝑖𝑗=−|𝑎_𝑖𝑗| for 𝑖≠𝑗. For (4), under the previous definition, we have ⟨𝐴⟩=𝐴₁₁−𝐴₁₂⋯−𝐴_1𝑚−𝐴₂₁𝐴₂₂⋯−𝐴_2𝑚−𝐴⋮⋮⋱⋮_𝑚1−𝐴_𝑚2⋯𝐴_𝑚𝑚. (15) Let ⟨𝐴⟩=⟨𝐷⟩−|𝐿|−|𝑈|, where ⟨𝐷⟩, −|𝐿|, and −|𝑈| are block diagonally, strictly block lower, and strictly block upper triangular parts of ⟨𝐴⟩, respectively.

Notice that the preconditioner of the matrix ⟨𝐴⟩ corresponding to 𝑃₁ is 𝑃₂=𝐼+𝑆₂; namely, 𝐼+𝑆₂=𝐼₁₁𝛼₁𝐴₁₁⁻¹𝐴_1𝑠𝐼₂₂𝛼₂𝐴₂₂⁻¹𝐴_2𝑡⋱⋱𝛼_𝑚𝐴_𝑚𝑚⁻¹𝐴_𝑚𝑢𝐼_𝑚𝑚. (16)

Let 𝑃₂⟨𝐴⟩=(𝐼+𝑆₂)⟨𝐴⟩=𝐷−𝐿−𝑈, where 𝐷, −𝐿, and −𝑈 are block diagonally, strictly block lower, and strictly block upper triangular parts of 𝑃₂⟨𝐴⟩, respectively.

If 𝐷 is nonsingular, then (𝐷−𝜔𝐿)⁻¹ and (𝐷−𝜔𝑈)⁻¹ exist and the SSOR iteration matrix for 𝑃₂⟨𝐴⟩ is as follows: ℒ_𝜔=𝐷−𝜔𝑈⁻¹𝐷𝐷−𝜔𝐿⁻¹×(1−𝜔)𝐷+𝜔𝐿𝐷⁻¹(1−𝜔)𝐷+𝜔𝑈. (17)

2. Preliminaries

A matrix 𝐴 is called nonnegative (positive) if each entry of 𝐴 is nonnegative (positive). We denote it by 𝐴≥0 (𝐴>0). Similarly, for 𝑛-dimensional vector 𝑥, we can also define 𝑥≥0 (𝑥>0). Additionally, we denote the spectral radius of 𝐴 by 𝜌(𝐴). 𝐴^𝑇 denotes the transpose of 𝐴. A matrix 𝐴=(𝑎_𝑖𝑗) is called a 𝑍-matrix if for any 𝑖≠𝑗, 𝑎_𝑖𝑗≤0. A 𝑍-matrix is a nonsingular 𝑀-matrix if 𝐴 is nonsingular and 𝐴⁻¹≥0, If ⟨𝐴⟩ is a nonsingular 𝑀-matrix , then 𝐴 is called an 𝐻-matrix. 𝐴=𝑀−𝑁 is said to be a splitting of 𝐴 if 𝑀 is nonsingular, 𝐴=𝑀−𝑁 is said to be regular if 𝑀⁻¹≥0 and 𝑁≥0, and weak regular if 𝑀⁻¹≥0 and 𝑀⁻¹𝑁≥0, respectively.

Some basic properties on special matrices introduced previously are given to be used in this paper.

Lemma 1 (see [2]). Let A be a 𝑍-matrix. Then the following statements are equivalent.(a)𝐴is an 𝑀-matrix.(b)There is a positive vector 𝑥 such that 𝐴𝑥>0.(c)𝐴⁻¹≥0.(d)All principal submatrices of 𝐴 are 𝑀-matrices.(e)All principal minors are positive.

Lemma 2 (see [3, 4]). Let 𝐴 be an 𝑀-matrix and let 𝐴=𝑀−𝑁 be a weak regular splitting. Then 𝜌(𝑀⁻¹𝑁)<1.

Lemma 3 (see [2]). Let 𝐴 and 𝐵 be two 𝑛×𝑛 matrices with 0≤𝐵≤𝐴. Then 𝜌(𝐵)≤𝜌(𝐴).

Lemma 4 (see [5]). If 𝐴 is an 𝐻-matrix, then |𝐴⁻¹|≤⟨𝐴⟩⁻¹.

Lemma 5 (see [6]). Suppose that 𝐴₁=𝑀₁−𝑁₁ and 𝐴₂=𝑀₂−𝑁₂ are weak regular splitting of monotone matrices 𝐴₁ and 𝐴₂, respectively, such that 𝑀₂⁻¹≥𝑀₁⁻¹. If there exists a positive vector 𝑥 such that 0≤𝐴₁𝑥≤𝐴₂𝑥, then for the monotone norm associated with 𝑥, 𝑀₁⁻¹𝑁_1𝑥≤𝑀₂⁻¹𝑁_2𝑥. (18) In particular, if 𝑀₁⁻¹𝑁₁ has a positive Perron vector, then 𝜌𝑀₁⁻¹𝑁₁≤𝜌𝑀₂⁻¹𝑁₂. (19) Moreover if 𝑥 is a Perron vector of 𝑀₁⁻¹𝑁₁ and strict inequality holds in (18), then strict inequality holds in (19).

Lemma 6. If 𝐴 and 𝐵 are two 𝑛×𝑛 matrices, then ⟨𝐴−𝐵⟩≥⟨𝐴⟩−|𝐵|.

Proof. It is easy to see that |𝑎_𝑖𝑗−𝑏_𝑖𝑗|≥|𝑎_𝑖𝑗|−|𝑏_𝑖𝑗|, for 𝑖=𝑗, and −|𝑎_𝑖𝑗−𝑏_𝑖𝑗|≥−|𝑎_𝑖𝑗|−|𝑏_𝑖𝑗|, for 𝑖≠𝑗. Therefore, ⟨𝐴−𝐵⟩≥⟨𝐴⟩−|𝐵| is true.

Lemma 7. If 𝐴 is an 𝐻-matrix with unit diagonal elements, then ‖⟨𝐴⟩⁻¹‖_∞>1.

Proof. Let ⟨𝐴⟩=𝐼−𝐵, from ⟨𝐴⟩ being an 𝑀-matrix; then 𝐵≥0 and 𝜌(𝐵)<1, and thus, we have ⟨𝐴⟩⁻¹=^∞_𝐾=0𝐵^𝑘≥𝐼 (20) and then ‖⟨𝐴⟩⁻¹‖_∞>1.

3. Convergence Results

Let 𝑒_𝑖=(1,…,1)^𝑇∈𝑅^𝑛𝑖, 𝑖=1,2,…,𝑚, 𝑒=(𝑒^𝑇₁,…,𝑒^𝑇_𝑚)^𝑇, 𝑟=(𝑟^𝑇₁,…,𝑟^𝑇_𝑚)^𝑇=⟨𝐴⟩⁻¹𝑒, 𝑂_𝑖=(0,…,0)^𝑇∈𝑅^𝑛𝑖, where 𝑟 and 𝑒 are partitioned in accordance with the block partitioning of the matrix 𝐴, and let 𝑠_𝑖=𝐴_𝑖𝑖⁻¹𝐴_𝑖𝑘𝑒_𝑘∞,ℎ_𝑖=1𝑠_𝑖2⟨𝐴⟩⁻¹_∞,−1𝑖=1,2,…,𝑚. (21)

Theorem 8. Let 𝐴 be a nonsingular H-matrix; if |𝛼_𝑖|<ℎ_𝑖, 𝑖=1,2,…,𝑚, then 𝑃₁𝐴 is also an H-matrix.

Proof. From 𝐴 being an 𝐻-matrix, we have 𝑟>0, and 𝑟_𝑘≤‖⟨𝐴⟩⁻¹‖_∞𝑒_𝑘. Let 𝑃₁𝐴_𝑖𝑗=𝐴_𝑖𝑗−𝛼_𝑖𝐴⁻¹_𝑖𝑖𝐴_𝑖𝑘𝐴_𝑘𝑗𝐴,𝑖≠𝑗,𝑖,𝑗=1,2,…,𝑚,𝑘≠𝑖,_𝑖𝑖−𝛼_𝑖𝐴⁻¹_𝑖𝑖𝐴_𝑖𝑘𝐴_𝑘𝑖,𝑖=𝑗,𝑖,𝑗=1,2,…,𝑚,𝑘≠𝑖. (22) Then 𝑃₁𝐴𝑟_𝑖=𝐴_𝑖𝑖−𝛼_𝑖𝐴⁻¹_𝑖𝑖𝐴_𝑖𝑘𝐴_𝑘𝑖𝑟_𝑖−^𝑚_{𝑗≠𝑖,𝑘}𝐴_𝑖𝑗−𝛼_𝑖𝐴⁻¹_𝑖𝑖𝐴_𝑖𝑘𝐴_𝑘𝑗𝑟_𝑗−𝐴_𝑖𝑘−𝛼_𝑖𝐴⁻¹_𝑖𝑖𝐴_𝑖𝑘𝐴_𝑘𝑘𝑟_𝑘≥𝐴_𝑖𝑖𝑟_𝑖−𝛼_𝑖𝐴⁻¹_𝑖𝑖𝐴_𝑖𝑘𝐴_𝑘𝑖𝑟_𝑖−^𝑚_{𝑗≠𝑖,𝑘}𝐴_𝑖𝑗−^𝑚_{𝑗≠𝑖,𝑘}𝛼_𝑖𝐴⁻¹_𝑖𝑖𝐴_𝑖𝑘𝐴_𝑘𝑗𝑟_𝑗−𝐴_𝑖𝑘𝑟_𝑘−𝛼_𝑖𝐴⁻¹_𝑖𝑖𝐴_𝑖𝑘𝐴_𝑘𝑘𝑟_𝑘≥𝑒_𝑖−𝛼_𝑖𝐴_𝑖𝑖⁻¹𝐴_𝑖𝑘𝐴_𝑘𝑖𝑟_𝑖−^𝑚_{𝑗≠𝑖,𝑘}𝛼_𝑖𝐴_𝑖𝑖⁻¹𝐴_𝑖𝑘𝐴_𝑘𝑗𝑟_𝑗−𝛼_𝑖𝐴_𝑖𝑖⁻¹𝐴_𝑖𝑘𝐴_𝑘𝑘𝑟_𝑘=𝑒_𝑖+𝛼_𝑖𝐴_𝑖𝑖⁻¹𝐴_𝑖𝑘×−^𝑚_𝑗≠𝑘𝐴_𝑘𝑗𝑟_𝑗−𝐴_𝑘𝑘𝑟_𝑘+𝐴_𝑘𝑘𝑟_𝑘−𝐴_𝑘𝑘𝑟_𝑘=𝑒_𝑖+𝛼_𝑖𝐴_𝑖𝑖⁻¹𝐴_𝑖𝑘𝑒_𝑘−2𝑟_𝑘≥𝑒_𝑖−𝛼_𝑖2⟨𝐴⟩⁻¹_∞−1𝐴_𝑖𝑖⁻¹𝐴_𝑖𝑘𝑒_𝑘≥𝑒_𝑖−𝛼_𝑖𝑠_𝑖2⟨𝐴⟩⁻¹_∞−1𝑒_𝑖>𝑂_𝑖. (23) Therefore, ⟨𝑃₁𝐴⟩ is an 𝑀-matrix, and then 𝑃₁𝐴 is an 𝐻-matrix.

Theorem 9. If 𝐴 is a nonsingular 𝐻-matrix with unit diagonal elements, 0<𝜔≤1 and |𝛼_𝑖|<ℎ_𝑖,𝑖=1,2,…,𝑚. Then 𝜌(ℒ_𝜔)<1.

Proof. From Theorem 8, we know ⟨𝑃₁𝐴⟩=⟨𝐷⟩−|𝐿|−|𝑈| is an 𝑀-matrix; if we let 𝑃₁𝐴=1𝜔()𝐷𝐿𝐷2−𝜔−𝜔⁻¹𝐷𝑈−1−𝜔𝜔()(𝐷𝐿𝐷2−𝜔1−𝜔)+𝜔⁻¹×(1−𝜔)𝐷+𝜔𝑈, (24) then the SSOR iteration matrix for ⟨𝑃₁𝐴⟩ is as follows: ℒ̈_𝜔=𝐷−𝜔𝑈⁻¹𝐷𝐷−𝜔𝐿⁻¹×(1−𝜔)𝐷+𝜔𝐿𝐷⁻¹×(1−𝜔)𝐷+𝜔𝑈. (25) Since ⟨𝑃₁𝐴⟩ is an 𝑀-matrix; we have ⟨𝐷⟩, ⟨𝐷⟩−𝜔|𝐿| and ⟨𝐷⟩−𝜔|𝑈| are 𝑀-matrices; by simple calculation, we obtain that (24) is a weak regular splitting; from Lemma 2, we know that 𝜌(ℒ̈_𝜔)<1. Since ℒ_𝜔=𝐷−𝜔𝑈⁻¹𝐷𝐷−𝜔𝐿⁻¹×(1−𝜔)𝐷+𝜔𝐿𝐷⁻¹)𝑈=𝐷(1−𝜔𝐷+𝜔𝐼−𝜔⁻¹𝑈⁻¹𝐼−𝜔𝐷⁻¹𝐿⁻¹×(1−𝜔)𝐼+𝜔𝐷⁻¹𝐿(1−𝜔)𝐼+𝜔𝐷⁻¹𝑈≤𝐼−𝜔𝐷⁻¹𝑈⁻¹𝐼−𝜔𝐷⁻¹𝐿⁻¹×(1−𝜔)𝐼+𝜔𝐷⁻¹𝐿(1−𝜔)𝐼+𝜔𝐷⁻¹𝑈≤𝐼−𝜔𝐷⁻¹|𝑈|⁻¹𝐼−𝜔𝐷⁻¹𝐿⁻¹×(1−𝜔)𝐼+𝜔𝐷⁻¹𝐿(1−𝜔)𝐼+𝜔𝐷⁻¹𝑈=𝜌ℒ̈_𝜔 (26) then, by Lemma 3, 𝜌(ℒ_𝜔)≤𝜌(|ℒ_𝜔|)≤𝜌(ℒ̈_𝜔)<1.

4. Comparison Results of Spectral Radius

Theorem 10. Let 𝐴 be a nonsingular 𝐻-matrix with unit diagonal elements, 0<𝜔≤1 and |𝛼_𝑖|<ℎ_𝑖,𝑖=1,2,…,𝑚. Then 𝑃₂⟨𝐴⟩ is an 𝑀-matrix and 𝜌(ℒ_𝜔)<1.

Proof. Similar to the proof of Theorems 8 and 9, it is easy to get the proof of this theorem.

In what follows we will give some comparison results on the spectral radius of preconditioned SSOR iteration matrices with different preconditioner.

Let ⟨𝐴⟩=𝑁=1𝑀−𝜔()𝐷⟩|𝐿𝐷⟩2−𝜔(⟨−𝜔|)⟨⁻¹𝐷⟩|𝑈−1(⟨−𝜔|)𝜔()𝐷⟩|𝐿𝐷⟩2−𝜔((1−𝜔)⟨+𝜔|)⟨⁻¹×((1−𝜔)⟨𝐷⟩+𝜔|𝑈|), (27) where 1𝑀=𝜔()𝐷⟩|𝐿𝐷⟩2−𝜔(⟨−𝜔|)⟨⁻¹𝐷⟩|𝑈,1(⟨−𝜔|)𝑁=𝜔()𝐷⟩|𝐿𝐷⟩2−𝜔((1−𝜔)⟨+𝜔|)⟨⁻¹×((1−𝜔)⟨𝐷⟩+𝜔|𝑈|). (28) Then the SSOR iteration matrix for ⟨𝐴⟩ is as follows: ℒ_𝜔=𝑀⁻¹𝑁=(⟨𝐷⟩−𝜔|𝑈|)⁻¹⟨𝐷⟩(⟨𝐷⟩−𝜔|𝐿|)⁻¹×((1−𝜔)⟨𝐷⟩+𝜔|𝐿|)⟨𝐷⟩⁻¹((1−𝜔)⟨𝐷⟩+𝜔|𝑈|), (29) and let 𝑃₂⟨𝐴⟩=𝑀−𝑁=1𝜔(2−𝜔)𝐷−𝜔𝐿𝐷⁻¹𝐷−𝜔𝑈−1𝜔()()2−𝜔1−𝜔𝐷+𝜔𝐿𝐷⁻¹×(1−𝜔)𝐷+𝜔𝑈, (30) where 1𝑀=𝜔(2−𝜔)𝐷−𝜔𝐿𝐷⁻¹𝐷−𝜔𝑈,1𝑁=𝜔()()2−𝜔1−𝜔𝐷+𝜔𝐿𝐷⁻¹(1−𝜔)𝐷+𝜔𝑈. (31) Then the AOR iteration matrix for 𝑃₂⟨𝐴⟩ is (17).

Theorem 11. If 𝐴 is a nonsingular 𝐻-matrix with unit diagonal elements, 0<𝜔≤1 and |𝛼_𝑖|<ℎ_𝑖,𝑖=1,2,…,𝑚. Then 𝜌(ℒ_𝜔)≤𝜌(ℒ_𝜔).

Proof. Since ⟨𝐴⟩ is a nonsingular 𝑀-matrix, by Theorem 10, 𝑃₂⟨𝐴⟩ is a nonsingular 𝑀-matrix, and thus ⟨𝐴⟩ and 𝑃₂⟨𝐴⟩ are two monotone matrices.
From ⟨𝐴⟩ and 𝑃₂⟨𝐴⟩ being 𝑀-matrices, we can get ⟨𝐷⟩, 𝐷, 𝑀, and 𝑀 are 𝑀-matrices, together with (1−𝜔)𝐼+𝜔⟨𝐷⟩⁻¹|𝐿|⟨𝐷⟩⁻¹(1−𝜔)𝐼+𝜔⟨𝐷⟩⁻¹|𝑈()|>0,1−𝜔𝐼+𝜔𝐷⁻¹𝐿𝐷⁻¹(1−𝜔)𝐼+𝜔𝐷⁻¹𝑈>0. (32) We obtain that ⟨𝐴⟩=𝑀−𝑁 and 𝑃₂⟨𝐴⟩=𝑀−𝑁 are two weak regular splittings. By simple calculation, we have 1𝑀=𝜔(2−𝜔)𝐷−𝜔𝐿𝐷⁻¹𝐷−𝜔𝑈≤1𝜔()𝐷⟩|𝐿𝐷⟩2−𝜔(⟨−𝜔|)⟨⁻¹(⟨𝐷⟩−𝜔|𝑈|)=𝑀 (33) and thus 𝑀⁻¹≥𝑀⁻¹≥0; letting 𝑥=⟨𝐴⟩⁻¹𝑒>0, then (𝑃₂⟨𝐴⟩−⟨𝐴⟩)𝑥=(𝐼+𝑆₂)𝑒>0; since 𝑀⁻¹≥𝑀⁻¹≥0, we have 𝑀⁻¹𝑃₂⟨𝐴⟩𝑥=𝐼−𝑀⁻¹𝑁𝑥≥𝑀⁻¹⟨𝐴⟩𝑥=𝐼−𝑀⁻¹𝑁𝑥. (34) It follows that 𝑀⁻¹𝑁_𝑥≤𝑀⁻¹𝑁_𝑥. (35)
As ⟨𝐴⟩=𝑀−𝑁 is a weak regular splitting, there exists a positive perron vector 𝑦; by Lemma 5, the following inequality holds: 𝜌𝑀⁻¹𝑁≤𝜌𝑀⁻¹𝑁, (36) that is, 𝜌ℒ_𝜔≤𝜌ℒ_𝜔. (37)

When 𝐴 is a nonsingular 𝑀-matrix, we have 𝐴=⟨𝐴⟩. If 𝛼_𝑖>0, 𝑖=1,2,…,𝑚, then 𝑃₂=𝑃₁. Furthermore, we have ℒ_𝜔=ℒ_𝜔 and ℒ_𝜔=ℒ_𝜔; therefore, we get the following result.

Corollary 12. Let 𝐴 be a nonsingular 𝑀-matrix with unit diagonal elements, 0<𝛼_𝑖<|ℎ_𝑖|, 𝑖=1,2,…,𝑚, and 0<𝜔≤1. Then 𝜌ℒ_𝜔=𝜌ℒ_𝜔≤𝜌ℒ_𝜔=𝜌ℒ_𝜔. (38)

Theorem 13. Let 𝐴 be a nonsingular 𝐻-matrix with unit diagonal elements, 0<𝜔≤1 and |𝛼_𝑖|<ℎ_𝑖, 𝑖=1,2,…,𝑚. Then 𝜌(ℒ̈_𝜔)≤𝜌(ℒ_𝜔).

Proof . Let 𝑃₁𝐴=1𝜔()𝐷𝐿𝐷2−𝜔−𝜔⁻¹𝐷𝑈−1−𝜔𝜔()(𝐷𝐿𝐷2−𝜔1−𝜔)+𝜔⁻¹×(1−𝜔)𝐷+𝜔𝑈. (39) Then the SSOR iteration matrix for ⟨𝑃₁𝐴⟩ is ℒ̈_𝜔 which is defined in the proof of Theorem 9, and let 𝑃₂⟨𝐴⟩=1𝜔(2−𝜔)𝐷−𝜔𝐿𝐷⁻¹𝐷−𝜔𝑈−1𝜔()()2−𝜔1−𝜔𝐷+𝜔𝐿𝐷⁻¹×(1−𝜔)𝐷+𝜔𝑈. (40) Then the SSOR iteration matrix for 𝑃₂⟨𝐴⟩ is (17). It is easy to know that the previous two splittings are weak regular splittings. Furthermore, by Lemma 6, we have the following result, for any 𝑖, 𝑖=1,2,…,𝑚, 𝐷_𝑖𝑖=𝐴_𝑖𝑖−𝛼_𝑖𝐴⁻¹_𝑖𝑖𝐴_𝑖𝑘𝐴_𝑘𝑖≥𝐴_𝑖𝑖−𝛼_𝑖𝐴_𝑖𝑖⁻¹𝐴_𝑖𝑘𝐴_𝑘𝑖=𝐷_𝑖𝑖. (41)
From ⟨𝑃₁𝐴⟩ and 𝑃₂⟨𝐴⟩ being two 𝑀-matrices, we have 0≤𝐷⁻¹≤𝐷⁻¹ (42) and then ℒ̈_𝜔=𝐷−𝜔𝑈⁻¹𝐷𝐷−𝜔𝐿⁻¹×(1−𝜔)𝐷+𝜔𝐿𝐷⁻¹(𝐷𝑈=𝐷1−𝜔)+𝜔𝐼−𝜔⁻¹𝑈⁻¹𝐼−𝜔𝐷⁻¹𝐿⁻¹×(1−𝜔)𝐼+𝜔𝐷⁻¹𝐿(1−𝜔)𝐼+𝜔𝐷⁻¹𝑈≤𝐼−𝜔𝐷⁻¹𝑈⁻¹𝐼−𝜔𝐷⁻¹𝐿⁻¹×(1−𝜔)𝐼+𝜔𝐷⁻¹𝐿(1−𝜔)𝐼+𝜔𝐷⁻¹𝑈=ℒ_𝜔. (43) Therefore, by Lemma 3, 𝜌(ℒ̈_𝜔)≤𝜌(ℒ_𝜔).

Combining the previous Theorems, we can obtain the following conclusion.

Theorem 14. Let 𝐴 be a nonsingular 𝐻-matrix with unit diagonal elements, 0<𝜔≤1 and |𝛼_𝑖|<ℎ_𝑖, 𝑖=1,2,…,𝑚. Then 𝜌ℒ_𝜔≤𝜌ℒ̈_𝜔≤𝜌ℒ_𝜔≤𝜌ℒ_𝜔<1. (44)

5. Numerical Example

For randomly generated nonsingular 𝐻-matrices for 𝑛=100,200,500,1000 with 𝑛₁=𝑛₂=⋯=𝑛_𝑚=5, we have determined the spectral radius of the iteration matrices of SSOR method mentioned previously with preconditioner 𝑃₁. We report the spectral radius of the corresponding iteration matrix by 𝜌. The 𝑚 parameters 𝛼_𝑖, 𝑖=1,2,...,𝑚, are taken from the 𝑚 equal-partitioned points of the interval [0,1]. We take 𝑃₁=𝐴⁻¹₁₁𝐴⁻¹₁₁𝐴₁₂𝑂₁₃⋯𝑂_1𝑚𝑂₂₁𝐴⁻¹₂₂𝐴⁻¹₁₁𝐴₂₃𝑂⋱⋮⋮⋱⋱⋱⋮_𝑚−1,1⋯𝑂_{𝑚−1,𝑚−2}𝐴⁻¹_{𝑚−1,𝑚−1}𝐴⁻¹_{𝑚−1,𝑚−1}𝐴_{𝑚−1,𝑚}𝐴⁻¹_𝑚𝑚𝐴_𝑚1𝑂_𝑚2⋯𝑂_{𝑚,𝑚−1}𝐴⁻¹_𝑚𝑚. (45)

For 𝑃₁, we make two groups of experiments. In Figure 1, we test the relation between 𝜔 and 𝜌, when 𝑁=100, 𝜔=0.6, where “×”, “+”, “∗”, “⋅” and “∘” denote the spectral radius of ⟨𝐴⟩, 𝑃₂⟨𝐴⟩, ⟨𝑃₁𝐴⟩, 𝐴, and 𝑃₁𝐴, respectively. In Table 1, the meaning of notations 𝜌(ℒ_𝜔), 𝜌(ℒ_𝜔), 𝜌(ℒ̈_𝜔), 𝜌(ℒ_𝜔), and 𝜌(ℒ_𝜔) denotes the spectral radius of 𝑃₁𝐴, 𝑃₂⟨𝐴⟩, ⟨𝑃₁𝐴⟩, ⟨𝐴⟩, and 𝐴, respectively.

Table 1 

Comparison of spectral radius with preconditioner 𝑃₁.

Figure 1 

The relation between 𝜔 and 𝜌, when 𝑁=100, 𝜔=0.6.

From Figure 1 and Table 1, we can conclude that the spectral radius of the preconditioned SSOR method with preconditioner 𝑃₁ is the best among others, which further illustrates that, Theorem 14 is true.

Acknowledgments

The authors express their thanks to the editor Professor Hak-Keung Lam and the anonymous referees who made much useful and detailed suggestions that helped them to correct some minor errors and improve the quality of the paper.