Comparison Theorems for Single and Double Splittings of Matrices

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.

1. Introduction

Consider the linear system 𝐴𝑥=𝑏, (1) where 𝐴∈ℝ^𝑛×𝑛 is nonsingular, 𝑏∈ℝ^𝑛×1 is given, and 𝑥∈ℝ^𝑛×1 is unknown. The splitting of the coefficient matrix 𝐴=𝑀−𝑁, (2) where 𝑀 is nonsingular, is called a single splitting of 𝐴 in [1]; the basic iterative method for solving (1) is 𝑥^𝑘+1=𝑀⁻¹𝑁𝑥^𝑘+𝑀⁻¹𝑏≡𝑇𝑥^𝑘+𝑀⁻¹𝑏,𝑘=0,1,2,…, (3) 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 of 𝐴 was introduced by Woźnicki [1] and can be described as follows. Splitting the matrix 𝐴 in the form 𝐴=𝑃−𝑅−𝑆 (4) is called the double splitting of 𝐴, where 𝑃 is a nonsingular matrix; the corresponding iterative scheme is spanned by three successive iterations: 𝑥^𝑘+1=𝑃⁻¹𝑅𝑥^𝑘+𝑃⁻¹𝑆𝑥^𝑘−1+𝑃⁻¹𝑏,𝑘=0,1,2,…. (5) Following the idea of Golub and Varga [9], Woźnicki wrote (5) in the following equivalent form: 𝑥^𝑘+1𝑥^𝑘=𝑃⁻¹𝑅𝑃⁻¹𝑆𝑥𝐼0𝑥^𝑘𝑘−1+𝑃⁻¹𝑏0, (6) 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 𝑃𝑊=⁻¹𝑅𝑃⁻¹𝑆𝐼0 (7) 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 matrix 𝐴 is called nonnegative and denoted by 𝐴≥0 if 𝑎_𝑖𝑗≥0 for 𝑖,𝑗=1,2,…,𝑛. We write 𝐴≥𝐵(𝐴>𝐵) if 𝑎_𝑖𝑗≥𝑏_𝑖𝑗(𝑎_𝑖𝑗>𝑏_𝑖𝑗) 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 𝑎_𝑖𝑖>0(𝑖=1,…,𝑛) and 𝑎_𝑖𝑗<0 for all 𝑖,𝑗=1,…,𝑛; 𝑖≠𝑗.

Definition 1. Let 𝐴 be a nonsingular matrix. Then, the double splitting 𝐴=𝑃−𝑅−𝑆 is (i)convergent if and only if 𝜌(𝑊)<1; (ii)a regular double splitting if 𝑃⁻¹≥0, 𝑅≥0 and 𝑆≥0; (iii)a weak regular double splitting if 𝑃⁻¹≥0, 𝑃⁻¹𝑅≥0, and 𝑃⁻¹𝑆≥0; (iv)a nonnegative splitting if 𝑃⁻¹𝑅≥0 and 𝑃⁻¹𝑆≥0.

Lemma 2 (see [3]). Let 𝐴≥0. Then, 𝛼𝑥≤𝐴𝑥,𝑥≥0, implies 𝛼≤𝜌(𝐴) and 𝐴𝑥≤𝛽𝑥,𝑥>0, implies 𝜌(𝐴)≤𝛽.

Lemma 3 (see [10]). Let 𝐴⁻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.

Let 𝐴₁ 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, 𝑊₁=𝑃₁⁻¹𝑅₁𝑃₁⁻¹𝑆₁𝐼0,𝑊₂=𝑃₂⁻¹𝑅₂𝑃₂⁻¹𝑆₂𝐼0. (8)

Theorem 4 (see [12]). Let 𝐴₁ 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 of 𝑊₁ and 𝑊₂, we have the following theorem.

Theorem 5. Let 𝐴₁ 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, if 𝑃₁⁻¹𝑅₁≤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 Definition 1, we obtain the following Theorem 6, which is superior to Theorem 4 [12].

Theorem 6. Let 𝐴₁ 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 𝑥𝑥=¹𝑥₂≥0,𝑥≠0 (9) such that 𝑊₂𝑥=𝜌(𝑊₂)𝑥; that is, 𝑃₂⁻¹𝑅₂𝑥₁+𝑃₂⁻¹𝑆₂𝑥₂=𝜌𝑊₂𝑥₁,𝑥₁=𝜌𝑊₂𝑥₂. (10) Then, we have 𝑊₁𝑥−𝜌𝑊₂𝑥=𝑃₁⁻¹𝑅₁𝑥₁+𝑃₁⁻¹𝑆₁𝑥₂−𝜌𝑊₂𝑥₁𝑥₁−𝜌𝑊₂𝑥₂=𝑃₁⁻¹𝑅₁−𝑃₂⁻¹𝑅₂𝑥₁+1𝜌𝑊₂𝑃₁⁻¹𝑆₁−𝑃₂⁻¹𝑆₂𝑥₁𝑥₁−𝜌𝑊₁𝑥₂≤1𝜌𝑊₂𝑃₁⁻¹𝑅₁−𝑃₂⁻¹𝑅₂𝑥₁+𝑃₁⁻¹𝑆₁−𝑃₂⁻¹𝑆₂𝑥₁0=1𝜌𝑊₂𝑃₁⁻¹𝑅₁+𝑃₁⁻¹𝑆₁𝑥₁−𝑃₂⁻¹𝑅₂+𝑃₂⁻¹𝑆₂𝑥₁0=1𝜌𝑊₂𝑃₁⁻¹𝑅₁+𝑆₁𝑥₁−𝑃₂⁻¹𝑅₂+𝑆₂𝑥₁0=1𝜌𝑊₂𝑃₁⁻¹𝑃₁−𝐴₁𝑥₁−𝑃₂⁻¹𝑃₂−𝐴₂𝑥₁0=1𝜌𝑊₂𝑃₂⁻¹𝐴₂−𝑃₁⁻¹𝐴₁𝑥₁0≤0. (11) From Lemma 2, we obtain that 𝜌(𝑊₁)≤𝜌(𝑊₂)<1 for 0<𝜌(𝑊₂)<1.

By investigating Theorem 6, 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. Let 𝐴₁ 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 comparing 𝜌(𝑊) with 𝜌(𝑇) is stated as in the following theorem.

Theorem 8. Let 𝐴=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 Definition 1, obviously, matrix 𝑊≥0. Based on the Perron-Frobenius theorem [3], there exists a vector 𝑥𝑥=¹𝑥₂≥0,𝑥≠0 (12) such that 𝑊𝑥=𝜌(𝑊)𝑥, that is, 𝑃⁻¹𝑅𝑃⁻¹𝑆𝑥𝐼0𝑥¹₂(𝑊)𝑥=𝜌¹𝑥₂. (13) The above equation is equivalent to 𝑃⁻¹𝑅𝑥₁+𝑃⁻¹𝑆𝑥₂=𝜌(𝑊)𝑥₁,𝑥₁=𝜌(𝑊)𝑥₂. (14)(15) From (15), we get that 𝑥₂=(1/𝜌(𝑊))𝑥₁. Substituting it into (14) yields 𝜌(𝑊)𝑥₁=𝑃⁻¹𝑅𝑥₁+1𝜌(𝑊)𝑃⁻¹𝑆𝑥₁. (16) If 𝜌(𝑊)<1, then 𝜌(𝑊)𝑥₁≥𝑃⁻¹𝑅𝑥₁+𝑃⁻¹𝑆𝑥₁. (17) That is, 𝜌(𝑊)𝑥₁≥𝑇𝑥₁. By Lemma 2, it is easy to obtain that 𝜌(𝑇)≤𝜌(𝑊)<1.
Obviously, we also obtain that 𝜌(𝑇)≥𝜌(𝑊)>1 for 𝜌(𝑊)>1.

Example 9. Let =−−𝐴=1−2−210−2−20−1000000−1≡𝑃−𝑅−𝑆. (18) Then, 𝑃⁻¹=10−2−120𝑃≤0,₋₁1𝑅=0020𝑃≥0,₋₁01𝑆=200≥0. (19) By the simple computations, we have 𝜌(𝑊)=0.63 and 𝜌(𝑇)=0.5. Clearly, 𝜌(𝑇)≤𝜌(𝑊)<1 holds.
Let the matrix 𝐴 be split as 𝐴=𝐷−𝐿−𝑈, (20) where 𝐷=diag(𝐴), and 𝐿, 𝑈 are strictly lower and upper triangular matrices, respectively, for 𝜔≠0 and 𝜎≠0. Let 𝑃_𝐽=1𝑅𝜔𝜎𝐷,_𝐽=1[()−()()𝐷],𝑆𝜔𝜎𝜔𝜎𝐿+𝑈𝜔−1𝐷−𝜎−1_𝐽=(𝜔−1)(𝜎−1)𝜔𝜎𝐷. (21) Then, the iterative method (5) corresponding to the double splitting 𝐴=𝑃_𝐽−𝑅_𝐽−𝑆_𝐽 (22) 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 2−𝜎−𝜔≥0,(𝜔−1)(𝜎−1)≥0. (23) Then, the double splitting defined by (22) is regular.

Let 𝑊_𝐽=𝑃_𝐽⁻¹𝑅_𝐽𝑃_𝐽⁻¹𝑆_𝐽𝐼0,𝑇_𝐽=𝑃_𝐽⁻¹𝑅_𝐽+𝑆_𝐽. (24) Then, we have the following result.

Theorem 11. Under the conditions of Lemma 10, then (1)𝜌(𝑇_𝐽)≤𝜌(𝑊_𝐽)<1 for 𝜌(𝑊_𝐽)<1; (2)𝜌(𝑇_𝐽)≥𝜌(𝑊_𝐽)>1 for 𝜌(𝑊_𝐽)>1.

Proof. From Theorem 8, it is easy to see that Theorem 11 holds.

Let 𝑃_𝐺=1𝐷𝜔𝜎𝐼−𝜎𝐷⁻¹𝑈,𝑆_𝐺=(𝜔−1)(𝜎−1)𝑅𝜔𝜎𝐷,_𝐺=1()𝐷𝜔𝜎𝜔𝜎𝐿−𝜔−1𝐼−𝜎𝐷⁻¹𝑈−(𝜎−1)𝐷. (25) Then, the iterative method (5) corresponding to the double splitting 𝐴=𝑃_𝐺−𝑅_𝐺−𝑆_𝐺 (26) is called the Gauss-Seidel double SOR method [1, 15].

Let 𝑊_𝐺=𝑃_𝐺⁻¹𝑅_𝐺𝑃_𝐺⁻¹𝑆_𝐺𝐼0,𝑇_𝐺=𝑃_𝐺⁻¹𝑅_𝐺+𝑆_𝐺. (27) 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(𝑇_𝐺)≤𝜌(𝑊_𝐺)<1 for 𝜌(𝑊_𝐺)<1; (2)𝜌(𝑇_𝐺)≥𝜌(𝑊_𝐺)>1 for 𝜌(𝑊_𝐺)>1.

From Theorems 8, 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.

Acknowledgment

This research was supported by NSFC Tianyuan Mathematics Youth Fund (11026040, 11226337), Science & Technology Development Plan of Henan Province (no. 122300410316) and by Natural Science Foundations of Henan Province (no. 13A110022).