On comparison results for $K$-nonnegative double splittings of different $K$-monotone matrices

1.   Introduction

Let $\mathbb{R}^n$ be the set of $n$-dimensional vectors, $K$ be a proper cone in $\mathbb{R}^n$. Let $\pi(K)$ denote the set of $n\times n$ matrices which leave the proper cone $K\subseteq \mathbb{R}^n$ invariant, then $\pi(K)$ is closed under multiplication and is a proper cone in $\mathbb{R}^{n\times n}$ ^[1]. It should be noted that both the nonnegative cone $\mathbb{R}^n_+$ and the ice cream cone $\{x\in\mathbb{R}^n|(x_2^2+x_3^2+\cdots+x_n^2)^{\frac{1}{2}} < x_1\}$ are particular proper cones.

In this note, we consider the linear system

where $A\in\mathbb{R}^{n\times n}$ is a nonsingular matrix and $A^{-1}\in\pi(K)$, $b\in \mathbb{R}^n$ is given and $x\in\mathbb{R}^n$ is unknown. Wo$\acute{\rm z}$niki in ^[18] introduced the double splitting of $A$ as

where $P$ is nonsingular, and the approximate solution $x_{i+1}$ of Eq (1.1) is generated by three successive iterations:

It follows from ^[6] that the iterative scheme (1.3) can be rewritten in the following equivalent form

where $I$ denotes the identity matrix with compatible size.

The iterative scheme (1.4) converges to the unique solution $x_{\star} = A^{-1}b$ of (1.1) if and only if the spectral radius of the iteration matrix

is less than one, i.e., $\rho(W) < 1$. Hou ^[7] gave convergence and comparison theorems for $K$-double splittings of an $K$-monotone matrix. Wang ^[19] presented convergence and the comparison results for $K$-nonnegative double splittings of an $K$-monotone matrix.

The smaller $\rho(W)$, the faster convergence of the iterative scheme (1.4). One approach for improving the convergence rate of the corresponding iterative method is the preconditioning techniques ^[2]. More precisely, we may solve the preconditioned linear systems

instead of (1.1), here $Q$, called the preconditioner, is nonsingular. When there are two or more preconditioners for the linear system (1.1), which one is the most efficient one is worth studying. The comparison theorem between the spectral radii of iteration matrices is a useful tool for judging the efficiency of preconditioner ^[5]. Therefore, in this note we will present the comparison results between the spectral radii of the corresponding iteration matrices arising from $K$-nonnegative splittings of different preconditioned matrices. For this purpose, we assume that the preconditioned matrices $QA$ with different preconditioners $Q$ satisfy $(QA)^{-1}\in\pi(K)$. The obtained results of this paper are the generalizations of the corresponding results in ^[7,16,19].

The rest of this article is organized as follows. In Section 2, some definitions and results are reviewed. In Section 3, the main comparison results for $K$-nonnegative double splittings of different preconditioned matrices are given. Finally, in Section 4, conclusions are {drawn}.

2.   Preliminaries

In this section, some definitions and lemmas, which will be used throughout the paper, will be given.

Definition 2.1. A vector $x$ in $\mathbb{R}^n$ is called $K$-nonnegative ($K$-positive) if $x$ belongs to $K$ ($x$ belongs to ${\rm int}K$, the the interior of $K$) and is denoted by $x\geq_K 0$ ($x > _K 0$). If $x, \; y\in\mathbb{R}^n$ satisfying $x-y\geq_K 0$ ($x-y > _K 0$), we denote $x\geq_{K} y$ ($x > _{K}y$).

Definition 2.2. An $n\times n$ real matrix $A$ is called $K$-nonnegative ($K$-positive) if $AK\subseteq K$ (respectively, $A(K-\{0\})\subseteq {\rm int}K$) and is denoted as $A\geq_K 0$ ($A > _K 0$). Similarly, for $n\times n$ real matrices $A$ and $B$ we denote $A-B\geq_K 0$ ($A-B > _K 0$) by $A\geq_K B$ ($A > _K B$).

Clearly, $A$ is $K$-nonnegative is equivalent to $A\in\pi(K)$ ^[7]. Basic properties of the $K$-nonnegative matrix are given in ^[1,7,8]. It should be remarked that the properties of $K$-nonnegative matrices are very similar to the theory of nonnegative matrices, see for example ^[1,3,4,8].

Based on the definition of $K$-nonnegative matrix, we can give the definition of $K$-monotone matrix.

Definition 2.3. Let $A\in\mathbb{R}^{n\times n}$ be nonsingular, $A$ is called $K$-monotone if $A^{-1}\geq_K 0$, i.e., $A^{-1}\in\pi(K)$.

Hou and Li ^[8] proposed the definition of the $K$-nonnegative single splittings, and Wang ^[19] introduced the $K$-nonnegative double splittings as:

Definition 2.4. Let $A$ be a nonsingular matrix. Then,

(i). the single splitting $A = M-N$ is a $K$-nonnegative single splitting if $M^{-1}N\geq_K 0$ ^[8];

(ii). the double splitting $A = P-R-S$ is a $K$-nonnegative double splitting if $P^{-1}R\geq_K 0$ and $P^{-1}S\geq_K 0$ ^[19].

It should be remarked that the $K$-nonnegative single splitting is termed as $K$-weak splitting in ^[9] or $K$-weak splitting of the first type in ^[8]. Moreover, rewriting (1.2) as

then it is a single splitting of $A$. If the double splitting (1.2) is $K$-nonnegative, then the single splitting (2.1) is also $K$-nonnegative.

3.   Main result

In this section, we will present the comparison results for the $K$-nonnegative double splittings of different $K$-monotone matrices. Assume that there are two preconditioners $Q_1$ and $Q_2$ for the linear system (1.1), then we have two preconditioned linear systems with coefficient matrices $A_1 = Q_1A$ and $A_2 = Q_2A$, respectively. Moreover, we further assume that the preconditioned matrices $A_1$ and $A_2$ are $K$-monotone matrices. Some excellent comparison results for $K$-nonnegative single splittings of $A_1$ and $A_2$ are given in ^[9,11].

Let

be $K$-nonnegative double splittings of $A_{1}$ and $A_{2}$, respectively. Define the corresponding iteration matrices

For $i = 1, \; 2$, if we split

as

with

then $W_i = \mathbb{M}^{-1}_i\mathbb{N}_i$. Therefore, we can get comparison results for the double splitting (3.1) by investigating the splitting (3.2).

Theorem 3.1. Let $A_1$ and $A_2$ be $K$-monotone matrices, the splittings (3.1) be $K$-nonnegative and convergent. Suppose $A_2^{-1}\geq_K A_1^{-1}$, $S_2\geq_K S_1$ and $R_2+S_2\geq_K R_1+S_1$, then

Proof. Firstly, it is easy to see that the splittings $\mathbb{A}_{i} = \mathbb{M}_i-\mathbb{N}_i$ are $K$-nonnegative as the splittings (3.1) are $K$-nonnegative. Secondly, note that

and

hold under the assumptions. Hence, it follows from [9,Theorem 3.13] that $\rho(W_1)\leq \rho(W_2)$.

The conditions $S_2\geq_K S_1$ and $R_2\geq_K R_1$ imply $S_2\geq_K S_1$ and $R_2+S_2\geq_K R_1+S_1$, so from Theorem 3.1, we have the following corollary.

Corollary 3.2. Let $A_1$ and $A_2$ be $K$-monotone matrices, the splittings (3.1) be $K$-nonnegative and convergent. Suppose $A_2^{-1}\geq_K A_1^{-1}$, $S_2\geq_K S_1$ and $R_2\geq_K R_1$, then

Remark 3.3. When we pay our attention to the particular proper cone $K = \mathbb{R}^n_+$, then Theorem 3.1 and Corollary 3.2 are Theorem 3.12 and Corollary 3.13 in ^[16], respectively.

The following example shows that the conditions that the splittings (3.1) are convergent cannot be dropped in Theorems 3.1.

Example 3.4. Let $K = \{x\in\mathbb{R}^3|(x_2^2+x_3^2)^{\frac{1}{2}} < x_1\}$. Assume that

Then we have

If $A_{1}$ and $A_{2}$ are splitted as

respectively, here

and

It is easy to see that

and

i.e., $A_1$ and $A_2$ are $K$-monotone matrices, although $A_1$ and $A_2$ are not monotone matrices. By calculating, we have

and

Here the splittings (3.3) are not nonnegative double splittings, but $K$-nonnegative double splittings. It is easy to verify that $A_2^{-1}\geq_K A_1^{-1}$, $S_2\geq_K S_1$ and $R_2+S_2\geq_K R_1+S_1$, but $\rho(W_1) > 1$ and $\rho(W_2) > 1$. In fact, we have

In particular, if we restrict our discussion on the different $K$-nonnegative double splittings of one $K$-monotone matrix $A$, then from Theorem 3.1, Corollary 3.2 and [9,Theorem 3.5], the following corollaries are obtained.

Corollary 3.5. Let $A$ be $K$-monotone matrix, the splittings $A = P_{1}-R_1-S_1 = P_{2}-R_2-S_2$ be $K$-nonnegative and convergent. Suppose $S_2\geq_K S_1$ and $R_2+S_2\geq_K R_1+S_1$, then

Corollary 3.6. Let $A$ be $K$-monotone matrix, the splittings $A = P_{1}-R_1-S_1 = P_{2}-R_2-S_2$ be $K$-nonnegative and convergent. Suppose $S_2\geq_K S_1$ and $R_2\geq_K R_1$, then

Corollary 3.5 and 3.6 are just Theorem 2 and Corollary 1 in ^[19], respectively. In summary, Theorem 3.1 extends some results in ^[16] and ^[19].

In what follows, taking the inverses of $P_1$ and $P_2$ into account, we will derive another comparison theorem. Note that for $i = 1, \; 2$,

Hence, $\mathbb{M}_1^{-1}\geq_K\mathbb{M}_2^{-1}$ if $P_1^{-1}\geq_K P_2^{-1}$ and $P_1^{-1}S_1\leq_K P_2^{-1}S_2$. If we assume $A_1\geq_K A_2$ additionally, then the conditions of [9,Theorem 3.15] are satisfied. Therefore, from [9,Theorem 3.15], we have the following comparison result.

Theorem 3.7. Let $A_1$ and $A_2$ be $K$-monotone matrices, the splittings (3.1) be $K$-nonnegative and convergent. Suppose $A_1\geq_K A_2$, $P_1^{-1}\geq_K P_2^{-1}$ and $P_1^{-1}S_1\leq_K P_2^{-1}S_2$, then

If we turn our attention to the particular proper cone $K = \mathbb{R}^n_+$, then following conclusion is a direct consequence of Theorem 3.7.

Corollary 3.8. Let $A_1$ and $A_2$ be monotone matrices, the splittings (3.1) be nonnegative and convergent. Suppose $A_1\geq A_2$, $P_1^{-1}\geq P_2^{-1}$ and $P_1^{-1}S_1\leq P_2^{-1}S_2$, then

Moreover, if we consider the different nonnegative double splittings of one matrix $A$, then from Theorem 3.7, Corollary 3.8, the following corollaries are obtained.

Corollary 3.9. Let $A$ be $K$-monotone matrix, the splittings $A = P_{1}-R_1-S_1 = P_{2}-R_2-S_2$ be $K$-nonnegative and convergent. Suppose $P_1^{-1}\geq_K P_2^{-1}$ and $P_1^{-1}S_1\leq_K P_2^{-1}S_2$, then

Remark 3.10. Corollary 3.9 extends [7,Theorem 3.1 (ii)] and [7,Theorem 3.3 (ii)].

Corollary 3.11. Let $A$ be monotone matrices, the splittings $A = P_{1}-R_1-S_1 = P_{2}-R_2-S_2$ be nonnegative and convergent. Suppose $P_1^{-1}\geq P_2^{-1}$ and $P_1^{-1}S_1\leq P_2^{-1}S_2$, then

Remark 3.12. We assume in Corollary 3.11 that the splittings $A = P_{1}-R_1-S_1 = P_{2}-R_2-S_2$ be nonnegative and convergent, while it assumed that $A = P_{1}-R_1-S_1$ be regular, $A = P_{2}-R_2-S_2$ be nonegative and both be convergent in [16,Theorem 3.9]. So Corollary 3.11 is a new comparison results for different nonnegative double splittings of one monotone matrix, which has weaker conditions than Theorem 3.9 in ^[16].

The regular splitting $A = P_{1}-R_1-S_1$ is nonnegative, but not vice versa. The following example shows that the inequality $\rho(W_1)\leq \rho(W_2)$ holds for nonnegative splittings $A = P_{1}-R_1-S_1 = P_{2}-R_2-S_2$ of {one monotone matrix $A$ } instead of the regular splitting $A = P_{1}-R_1-S_1$ and the nonnegative splitting $A = P_2-R_2-S_2$.

Example 3.13. Let the monotone matrix

be splitted as $A = P_1-R_1-S_1 = P_2-R_2-S_2$ with

and

Some calculations yield

It should be noted that both splittings $A = P_1-R_1-S_1 = P_2-R_2-S_2$ are nonnegative splittings, not regular splittings. It is easy to see that

It follows from Corollary 3.11 that $\rho(W_1)\leq \rho(W_2)$. In fact, we have

4.   Conclusions

In this paper, we established the comparison results for two $K$-nonnegative double splittings of different $K$-monotone matrices, the obtained results generalized the corresponding results in ^[7,16,19].

Acknowledgments

This work was supported by the Natural Science Foundation of Northwest Normal University (No. NWNU-LKQN-17-5), the Postgraduate Research Funding Project of Northwest Normal University (No. 2020KYZZ001118) and the National Natural Science Foundation of China (No. 61967014).

Conflict of interest

The authors declare no conflict of interest.