非奇异H-矩阵的迭代判别新条件

New Conditions for Iterative Discrimination of Nonsingular H-Matrices

摘要: 非奇异H-矩阵是一类应用广泛的特殊矩阵，在矩阵理论、控制论和神经网络等许多领域有重要的应用。本文利用矩阵的相关性质，通过对非占优行指标集进行细分和构造新迭代因子的方法，得到新正对角因子，提出一组具有迭代形式的非奇异H-矩阵判定的新条件，并相应给出了非奇异H-矩阵迭代判定算法，最后利用数值实例验证了算法的有效性。

Abstract: Nonsingular H-matrices are a kind of special matrices with wide applications in many fields such as matrix theory, control theory and neural networks. In this paper, by using the relevant properties of matrices and through the method of subdividing the set of non-dominant row indices and constructing new iterative factors, a new positive diagonal factor is obtained. A new set of conditions for determining nonsingular H-matrices in iterative form is presented, and the corresponding iterative discrimination algorithm for nonsingular H-matrices is given. The effectiveness of the algorithm is verified by numerical examples.

1. 引言

矩阵理论作为解决实际问题的有力工具，在众多研究领域中发挥着重要作用。非奇异H-矩阵作为矩阵理论中一类特殊且重要的矩阵，一直是学者们研究的重点对象。如何高效判定一个矩阵是否为非奇异H-矩阵成为学者研究的热点问题[1]-[10]。文献[1]-[5]基于非奇异H-矩阵的性质，利用细分和迭代的方法，得出一系列非奇异H-矩阵的判定条件及算法。本文在此基础上，通过对非占优行指标集进行精细化细分，同时构造新的迭代因子，进一步提出了一组非奇异H-矩阵迭代判定的新条件以及相应的判定算法，推广了文献[1]-[5]的结果。

本文采用如下记号和定义：

用$C^{n\times n}\left(R^{n\times n}\right)$ 表示$n\times n$ 阶复(实)矩阵的集合，设$A=\left(a_{ij}\right)\in C^{n\times n}$ ，$N=\left(1,2,\cdots ,n\right)$ ，$\alpha \in \left(0,1\right]$ ，记

$R_i=R_i\left(A\right)={\displaystyle \sum _{j\ne i}\left|a_{ij}\right|},i,j\in N,$

$N_1=\left\{i\in N:\left|a_{ii}\right|>R_i\left(A\right)\right\},N_2=\left\{i\in N:0<\left|a_{ii}\right|\le R_i\left(A\right)\right\}.$

${N^{\prime }}_2=\left\{i\in N_2:0<\left|a_{ii}\right|<R_i\left(A\right)\right\},{N^{″}}_2=\left\{i\in N_2:0<\left|a_{ii}\right|=R_i\left(A\right)\right\}.$

$N=N_1\cup N_2=N_1\cup {N^{\prime }}_2\cup {N^{″}}_2,Z=\left\{0,1,2,\cdots \right\},Z^{+}=\left\{1,2,3,\cdots \right\}.$

定义1 设$A=\left(a_{ij}\right)\in C^{n\times n}$ ，若对任意$i\in N$ ，有$\left|a_{ii}\right|\ge R_i$ ，则称$A$ 为对角占优矩阵。记所有对角占优矩阵的全体为$D_0$ 。若对任意$i\in N$ ，有$\left|a_{ii}\right|>R_i$ ，则称$A$ 是严格对角占优矩阵。记所有严格对角占优矩阵的全体为$D$ 。若存在正对角矩阵$X$ ，使得$AX\in D$ ，则称$A$ 为广义严格对角占优矩阵，也称为非奇异H-矩阵。记所有广义严格对角占优矩阵的全体为$D^{\text{*}}$ 。

定义2 设$A=\left(a_{ij}\right)\in C^{n\times n}$ 为不可约矩阵，若$\left|a_{ii}\right|\ge R_i$ ，且其中至少有一个严格不等式成立，则称$A$ 为不可约对角占优矩阵。

定义3 设$A=\left(a_{ij}\right)\in C^{n\times n}$ ，若$\left|a_{ii}\right|\ge R_i$ ，又对于满足$\left|a_{ii}\right|=R_i$ 的下标i，存在非零元素序列$a_{i{j}_1}{a}_{j_1{j}_2}\cdots a_{j_k{j}_{}}$ ，使得$\left|a_{jj}\right|>R_j$ ，则称$A$ 为具有非零元素链对角占优矩阵。

引理1 [2] 设$A=\left(a_{ij}\right)\in C^{n\times n}$ ，若$A$ 为不可约对角占优矩阵，则$A\in D^{*}$ 。

引理2 [3] 设$A=\left(a_{ij}\right)\in C^{n\times n}$ ，若$A$ 为具有非零元素链对角占优矩阵，则$A\in D^{*}$ 。

本研究总假设$\left|a_{ii}\right|\ne 0,R_i\ne 0$ ，规定当$t\in \varphi ,{\displaystyle \sum _{t\in \varphi }\cdot =0}$ 。

文献[1]给出如下主要结果：

定理1 设$A=\left(a_{ij}\right)\in C^{n\times n}$ ，若存在非负整数k，使得

$\left|a_{ii}\right|x_i>{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|{\delta }_{k+1,t}}+{\displaystyle \sum _{t\in N_2,t\ne i}\left|a_{it}\right|x_t},i\in N_2$

其中：

$r_0=1,r_1=\underset{i\in N_1}{\max }\frac{R_i}{\left|a_{ii}\right|},x_i=\frac{\left|a_{ii}\right|}{R_i}\left(\forall i\in N_2\right),$

${\delta }_{k+1,i}=\frac{1}{\left|a_{ii}\right|}\left[{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|r_k}+{\displaystyle \sum _{t\in N_2}\left|a_{it}\right|x_t}\right]\left(i\in N_1,k\in Z\right),$

$r_{k+1}=\underset{i\in N_1}{\max }{\delta }_{k+1,i}\left(k\in Z^{+}\right),$

则$A\in D^{*}$ 。

文献[2]给出如下主要结果：

定理2 设$A=\left(a_{ij}\right)\in C^{n\times n}$ ，若满足

$\left|a_{ii}\right|>\frac{R_i}{R_i-\left|a_{ii}\right|}\left[h{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|\frac{Q_t}{\left|a_{tt}\right|}}+{\displaystyle \sum _{t\in {N^{\prime }}_2,t\ne i}\left|a_{it}\right|\frac{R_t-\left|a_{tt}\right|}{R_t}}+s{\displaystyle \sum _{t\in {N^{″}}_2}\left|a_{it}\right|}\right],i\in {N^{″}}_2$

且对任意的$i\in {N^{″}}_2$ ，存在$t\in N_1\cup {N^{\prime }}_2$ ，使得$a_{it}\ne 0$ ，则$A\in D^{*}$ 。

其中：

$\omega =\underset{i\in N_1}{\max }\frac{{\displaystyle \sum _{t\in {N^{\prime }}_2}\left|a_{it}\right|+{\displaystyle \sum _{t\in {N^{″}}_2}\left|a_{it}\right|}}}{\left|a_{ii}\right|-{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}},m=\underset{i\in {N^{\prime }}_2}{\max }\frac{R_i-\left|a_{ii}\right|}{R_i},$

$s=\max \left\{\omega ,m\right\},Q_i=s{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}+{\displaystyle \sum _{t\in {N^{\prime }}_2}\left|a_{it}\right|+{\displaystyle \sum _{t\in {N^{″}}_2}\left|a_{it}\right|}},$

$h=\underset{i\in N_1}{\max }\frac{s\left({\displaystyle \sum _{t\in {N^{\prime }}_2}\left|a_{it}\right|+{\displaystyle \sum _{t\in {N^{″}}_2}\left|a_{it}\right|}}\right)}{Q_i-{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{Q_t}{\left|a_{tt}\right|}}.$

2. 主要结果

为了叙述方便，进一步引入以下记号：

$r=\underset{i\in N_1}{\max }\frac{{\displaystyle \sum _{t\in N_2}\left|a_{it}\right|}}{\left|a_{ii}\right|-{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}}\left(i\in N_1\right),$

$N_2^{\left(1\right)}=\left\{i\in N_2:0<\left|a_{ii}\right|\le {\displaystyle \sum _{t\in N_1}\left|a_{it}\right|r}+{\displaystyle \sum _{t\in N_2,t\ne i}\left|a_{it}\right|}\right\},$

$N_2^{\left(2\right)}=\left\{i\in N_2:\left|a_{ii}\right|>{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|r}+{\displaystyle \sum _{t\in N_2,t\ne i}\left|a_{it}\right|}\right\},$

$m_i=\frac{{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|r+{\displaystyle \sum _{t\in N_2,t\ne i}\left|a_{it}\right|}}}{R_i+\left|a_{ii}\right|}\left(i\in N_2\right),\lambda =\underset{i\in N_2}{\max }\left\{r,m_i\right\},$

$f_{1,i}={\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|r}+{\displaystyle \sum _{t\in N_2}\left|a_{it}\right|}\left(i\in N_1\right),$

$f_{l+1,i}={\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{l,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\left(i\in N_1,l\in Z^{+}\right),$

${\eta }_{l+1}=\underset{i\in N_2^{\left(2\right)}}{\max }\left\{\frac{{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|}+{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|}}{\left|a_{ii}\right|}\right\}\left(l\in Z\right),$

$P_{l+1,i}={\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|}+{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|{\eta }_{l+1}}\left(l\in Z\right),$

$h_l=\underset{i\in N_2^{\left(2\right)}}{\max }\left\{\frac{{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+\lambda {\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|}}{P_{l+1,i}^{}-{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|}\frac{P_{l+1,t}}{\left|a_{tt}\right|}}\right\}\left(l\in Z\right).$

2.1. 定理3

设$A=\left(a_{ij}\right)\in C^{n\times n}$ ，若对$i\in N_2^{\left(1\right)}\left(l\in Z\right)$ ，有

$\left|a_{ii}\right|m_i>{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)},t\ne i}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\frac{P_{l+1,t}}{\left|a_{tt}\right|},$ (1)

且对于$i\in N_1$ ，${\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|\ne 0}$ ，${\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\ne 0}$ ，则$A$ 是非奇异H-矩阵。

证明

由$m_i,r$ 的定义可知，有$0\le m_i<1,0<r<1$ 。由数学归纳法可知：

$0\le \frac{f_{l+1,i}}{\left|a_{ii}^{}\right|}<\cdots <\frac{f_{1,i}}{\left|a_{ii}^{}\right|}\le r<1,i\in N_1,l\in Z^{+}.$ (2)

又根据$N_2$ 细分的条件，知$0<{\eta }_{l+1}<1$ 。

对于$i\in N_2^{\left(2\right)}$ ，有

$\left|a_{ii}\right|{\eta }_{l+1}\ge {\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|}+{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|},$

可即有$0<\frac{P_{l+1,i}}{\left|a_{ii}\right|}<{\eta }_{l+1}<1$ 。同时，由$P_{l+1,i}$ 的表达式可知，当$i\in N_2^{\left(2\right)}$ 时，

$0<\frac{{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+\lambda {\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|}}{P_{l+1,i}-{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|}\frac{P_{l+1,t}}{\left|a_{tt}\right|}}<\frac{P_{l+1,i}-{\eta }_{l+1}{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|}}{P_{l+1,i}-{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|}\frac{P_{l+1,t}}{\left|a_{tt}\right|}}<1,$

故

$0<h_l<1,i\in N_2^{\left(2\right)},l\in Z.$

根据式(1)有

$\left|a_{ii}\right|m_i>{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)},t\ne i}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\frac{P_{l+1,t}}{\left|a_{tt}\right|},$

取充分小的正数$\epsilon$ ，使$\epsilon$ 满足

$0<\frac{f_{l+1,t}}{\left|a_{tt}\right|}\left(1+\epsilon \right)<1\left(i\in N_1\right),0<h_l\frac{P_{l+1,i}}{\left|a_{ii}\right|}+\epsilon <1\left(i\in N_2^{\left(2\right)}\right),$

同时满足

$\epsilon <\frac{\left|a_{ii}\right|m_i-{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}-{\displaystyle \sum _{t\in N_2^{\left(1\right)},t\ne i}\left|a_{it}\right|m_t}-h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\frac{P_{l+1,t}}{\left|a_{tt}\right|}}{{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}},$ (3)

构造正对角矩阵$X=\text{diag}\left(x_1,x_2,\cdots ,x_n\right)$ ，并记$B=AX=\left(b_{ij}\right)$ ，其中

$x_i=\{\begin{array}{ll}\frac{f_{l+1,i}}{\left|a_{ii}\right|}\left(1+\epsilon \right),\hfill & i\in N_1,\hfill \\ m_i,\hfill & i\in N_2^{\left(1\right)},\hfill \\ h_l\frac{P_{l+1,i}}{\left|a_{ii}\right|}+\epsilon ,\hfill & i\in N_2^{\left(2\right)}.\hfill \end{array}$

1) 对任意的$i\in N_1$ ，由式(2)可得：

$\left|a_{ii}\right|>R_i>{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|r+}{\displaystyle \sum _{t\in N_2}\left|a_{it}\right|}\ge {\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|}+{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|},$

并且$l=0$ 时，有

$\left|a_{ii}\right|\frac{f_{1,i}}{\left|a_{ii}\right|}={\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|r+}{\displaystyle \sum _{t\in N_2}\left|a_{it}\right|}>{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+h_0{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\frac{P_{1,t}}{\left|a_{tt}\right|}},$

因此

$\begin{array}{l}\left|b_{ii}\right|-R_i\left(B\right)\\ =\left|a_{ii}\right|\frac{f_{1,i}}{\left|a_{ii}\right|}\left(1+\epsilon \right)-\left[{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{1,t}}{\left|a_{tt}\right|}\left(1+\epsilon \right)+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\left(h_0\frac{P_{1,t}}{\left|a_{tt}\right|}+\epsilon \right)}\right]\\ =\left|a_{ii}\right|\frac{f_{1,i}}{\left|a_{ii}\right|}-\left[{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+h_0{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\frac{P_{1,t}}{\left|a_{tt}\right|}}\right]\\ +\epsilon \left[\left|a_{ii}\right|-{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{1,t}}{\left|a_{tt}\right|}-{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\right]\\ >0.\end{array}$

并且$l\ge 1$ 时，有

$\begin{array}{l}\left|a_{ii}\right|\frac{f_{l+1,i}}{\left|a_{ii}\right|}={\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{l,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\\ >{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\frac{P_{l+1,t}}{\left|a_{tt}\right|}},\end{array}$

因此

$\begin{array}{l}\left|b_{ii}\right|-R_i\left(B\right)\\ =\left|a_{ii}\right|\frac{f_{l+1,i}}{\left|a_{ii}\right|}\left(1+\epsilon \right)-\left[{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}\left(1+\epsilon \right)+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\left(h_l\frac{P_{l+1,t}}{\left|a_{tt}\right|}+\epsilon \right)}\right]\\ =\left|a_{ii}\right|\frac{f_{l+1,i}}{\left|a_{ii}\right|}-\left[{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\frac{P_{l+1,t}}{\left|a_{tt}\right|}}\right]\\ +\epsilon \left[\left|a_{ii}\right|-{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}-{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\right]\\ >0.\end{array}$

2) 对任意的$i\in N_2^{\left(1\right)}$ ，由式(1)、式(3)可知：

$\begin{array}{l}\left|b_{ii}\right|-R_i\left(B\right)\\ =\left|a_{ii}\right|m_i-\left[{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}\left(1+\epsilon \right)+{\displaystyle \sum _{t\in N_2^{\left(1\right)},t\ne i}\left|a_{it}\right|m_t}+{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\left(h_l\frac{P_{l+1,t}}{\left|a_{tt}\right|}+\epsilon \right)}\right]\\ =\left|a_{ii}\right|m_i-\left[{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)},t\ne i}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\frac{P_{l+1,t}}{\left|a_{tt}\right|}}\right]-\epsilon \left[{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\right]\\ >0.\end{array}$

3) 对任意的$i\in N_2^{\left(2\right)}\left(l\in Z\right)$ ，由h的表达式可得：

$\begin{array}{l}{h}_l{P}_{l+1,i}\ge {\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+\lambda {\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|\frac{P_{l+1,t}}{\left|a_{tt}\right|}},\\ \left|a_{ii}\right|>{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|r+}{\displaystyle \sum _{t\in N_2,t\ne i}\left|a_{it}\right|}\ge {\displaystyle \sum _{t\in N_1}\left|a_{it}\right|\frac{f_{l+1,t}}{\left|a_{tt}\right|}+}{\displaystyle \sum _{t\in N_2,t\ne i}\left|a_{it}\right|},\end{array}$

从而

$\begin{array}{l}\left|b_{ii}\right|-R_i\left(B\right)\\ =\left|a_{ii}\right|\left(h_l\frac{P_{l+1,i}}{\left|a_{ii}\right|}+\epsilon \right)-\left[{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}\left(1+\epsilon \right)+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|\left(h_l\frac{P_{l+1,t}}{\left|a_{tt}\right|}+\epsilon \right)}\right]\\ =\left|a_{ii}\right|h_l\frac{P_{l+1,i}}{\left|a_{ii}\right|}-\left[{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|\frac{P_{l+1,t}}{\left|a_{tt}\right|}}\right]\\ +\epsilon \left[\left|a_{ii}\right|-{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}-{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|}\right]\\ >0.\end{array}$

综上所述，我们有$\left|b_{ii}\right|>R_i\left(B\right)\left(i\in N\right)$ ，所以$A\in D^{*}$ ，证毕。

2.2. 定理4

设$A=\left(a_{ij}\right)\in C^{n\times n}$ 且不可约，若对$i\in N_2^{\left(1\right)}\left(l\in Z\right)$ ，有

$\left|a_{ii}\right|m_i\ge {\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)},t\ne i}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\frac{P_{l+1,t}}{\left|a_{tt}\right|},$ (4)

且上式至少有一个严格不等式成立，则$A$ 是非奇异H-矩阵。

证明

构造正对角矩阵$X=\text{diag}\left(x_1,x_2,\cdots ,x_n\right)$ ，并记$B=AX=\left(b_{ij}\right)$ ，其中

$x_i=\{\begin{array}{ll}\frac{f_{l+1,i}}{\left|a_{ii}\right|},\hfill & i\in N_1,\hfill \\ m_i,\hfill & i\in N_2^{\left(1\right)},\hfill \\ h_l\frac{P_{l+1,i}}{\left|a_{ii}\right|},\hfill & i\in N_2^{\left(2\right)}.\hfill \end{array}$

1) 对任意的$i\in N_1$ ，并且$l=0$ 时，有

$\left|a_{ii}\right|\frac{f_{1,i}}{\left|a_{ii}\right|}={\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|r+}{\displaystyle \sum _{t\in N_2}\left|a_{it}\right|}>{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+h_0{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\frac{P_{1,t}}{\left|a_{tt}\right|}},$

因此

$\begin{array}{l}\left|b_{ii}\right|-R_i\left(B\right)\\ =\left|a_{ii}\right|\frac{f_{1,i}}{\left|a_{ii}\right|}-\left[{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+h_0{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\frac{P_{1,t}}{\left|a_{tt}\right|}}\right]\\ >0.\end{array}$

并且$l\ge 1$ 时，有

$\begin{array}{c}\left|a_{ii}\right|\frac{f_{l+1,i}}{\left|a_{ii}\right|}={\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{l,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\\ >{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\frac{P_{l+1,t}}{\left|a_{tt}\right|}},\end{array}$

因此

$\begin{array}{l}\left|b_{ii}\right|-R_i\left(B\right)\\ =\left|a_{ii}\right|\frac{f_{l+1,i}}{\left|a_{ii}\right|}-\left[{\displaystyle \sum _{t\in N_1,t\ne i}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\frac{P_{l+1,t}}{\left|a_{tt}\right|}}\right]\\ \ge 0.\end{array}$

2) 对任意的$i\in N_2^{\left(1\right)}$ ，由式(1)可知：

$\begin{array}{l}\left|b_{ii}\right|-R_i\left(B\right)\\ =\left|a_{ii}\right|m_i-\left[{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)},t\ne i}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|\frac{P_{l+1,t}}{\left|a_{tt}\right|}}\right]\\ \ge 0.\end{array}$

3) 对任意的$i\in N_2^{\left(2\right)}\left(l\in Z\right)$ ，由h的表达式可得：

$h_l{P}_{l+1,i}\ge {\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+\lambda {\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|\frac{P_{l+1,t}}{\left|a_{tt}\right|}},$

故有

$\begin{array}{l}\left|b_{ii}\right|-R_i\left(B\right)\\ =\left|a_{ii}\right|h_l\frac{P_{l+1,i}}{\left|a_{ii}\right|}-\left[{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)},t\ne i}\left|a_{it}\right|\frac{P_{l+1,t}}{\left|a_{tt}\right|}}\right]\\ \ge 0.\end{array}$

综上所述，我们有$\left|b_{ii}\right|>R_i\left(B\right)\left(i\in N\right)$ ，且由假设知至少有一个严格不等式成立。由矩阵$A$ 不可约知矩阵$B$ 不可约，则$B$ 为不可约对角占优矩阵。由引理1知$A\in D^{*}$ ，证毕。

2.3. 定理5

设$A=\left(a_{ij}\right)\in C^{n\times n}$ ，若对$i\in N_2^{\left(1\right)}\left(l\in Z\right)$ ，有

$\left|a_{ii}\right|m_i\ge {\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)},t\ne i}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\frac{P_{l+1,t}}{\left|a_{tt}\right|},$ (5)

$J\left(A\right)=\left\{i\in N_2^{\left(1\right)}:\left|a_{ii}\right|m_i>{\displaystyle \sum _{t\in N_1}\left|a_{it}\right|}\frac{f_{l+1,t}}{\left|a_{tt}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)},t\ne i}\left|a_{it}\right|m_t}+h_l{\displaystyle \sum _{t\in N_2^{\left(2\right)}}\left|a_{it}\right|}\frac{P_{l+1,t}}{\left|a_{tt}\right|}\right\}\ne \varphi ,$

且对$\forall i\in N_2^{\left(1\right)}-J\left(A\right)$ ，存在非零元素链$a_{i{i}_1}{a}_{i_1{i}_2}\cdots a_{i_r{k}_{}}$ ，其中$i\ne i_1,i_1\ne i_2,\cdots ,i_r\ne k$ ，使得$k\in J\left(A\right)$ ，则$A\in D^{*}$ 。

3. 算法

由定理3的判定条件，得到了一组判定非奇异H-矩阵的迭代新算法，且对矩阵$A$ 总假定：

$R_i\ne 0,C_i\ne 0\left(i\in N\right)$

3.1. 算法1

输入：矩阵$A=\left(a_{ij}\right)\in C^{n\times n}$ 。

输出：$X=X^{\left(1\right)}{X}^{\left(2\right)}\cdots X^{\left(y\right)}$ ，如果$A\in D^{*}$ 。

步骤1 如果$N_1\left(A\right)=\varphi$ 或存在$i\in N$ 使$a_{ii}=0$ ，则$A\notin D^{*}$ ，停止；否则，

步骤2 设$y=1,A^{\left(0\right)}=A,X^{\left(0\right)}=I$ 。

步骤3 计算$A^{\left(y\right)}=A^{\left(y-1\right)}{X}^{\left(y-1\right)}=\left(a_{ij}^{\left(y\right)}\right)$ 。

步骤4 如果$N_1\left(A^{\left(y\right)}\right)=\varphi$ ，则$A\notin D^{*}$ ，停止；否则，

步骤5 如果$N_2\left(A^{\left(y\right)}\right)=\varphi$ ，则$A\in D^{*}$ ，停止；否则，

步骤6 计算$r^{\left(y\right)},N_2^{\left(1\right)}\left(A^{\left(y\right)}\right),N_2^{\left(2\right)}\left(A^{\left(y\right)}\right),m_i^{\left(y\right)},{\lambda }^{\left(y\right)},f_{1,i}^{\left(y\right)},f_{l+1,i}^{\left(y\right)},{\eta }_{l+1}^{\left(y\right)},P_{l+1,i}^{\left(y\right)},h_l^{\left(y\right)}$ 。

步骤7 如果对任意$i\in N_2^{\left(1\right)}\left(A^{\left(y\right)}\right)$ ，满足

$\left|a_{ii}^{\left(y\right)}\right|m_i^{\left(y\right)}>{\displaystyle \sum _{t\in N_1\left(A^{\left(y\right)}\right)}\left|a_{it}^{\left(y\right)}\right|}\frac{f_{l+1,t}^{\left(y\right)}}{\left|a_{tt}^{\left(y\right)}\right|}+{\displaystyle \sum _{t\in N_2^{\left(1\right)}\left(A^{\left(y\right)}\right),t\ne i}\left|a_{it}^{\left(y\right)}\right|m_t^{\left(y\right)}}+h_l^{\left(y\right)}{\displaystyle \sum _{t\in N_2^{\left(2\right)}\left(A^{\left(y\right)}\right)}\left|a_{it}^{\left(y\right)}\right|}\frac{P_{l+1,t}^{\left(y\right)}}{\left|a_{tt}^{\left(y\right)}\right|},$

且对于$i\in N_1\left(A^{\left(y\right)}\right)$ ，${\displaystyle \sum _{t\in N_2^{\left(1\right)}\left(A^{\left(y\right)}\right)}\left|a_{it}^{\left(y\right)}\right|\ne 0}$ ，${\displaystyle \sum _{t\in N_2^{\left(2\right)}\left(A^{\left(y\right)}\right)}\left|a_{it}^{\left(y\right)}\right|\ne 0}$ ，则$A\in D^{*}$ 。停止；否则，

步骤8 设$x=\left(x_{ij}\right)$ ，其中

$x_i=\{\begin{array}{ll}\frac{f_{l+1,i}^{\left(y\right)}}{\left|a_{ii}^{\left(y\right)}\right|},\hfill & i\in N_1\left(A^{\left(y\right)}\right),\hfill \\ m_i^{\left(y\right)},\hfill & i\in N_2^{\left(1\right)}\left(A^{\left(y\right)}\right),\hfill \\ h_l^{\left(y\right)}\frac{P_{l+1,i}^{\left(y\right)}}{\left|a_{ii}^{\left(y\right)}\right|},\hfill & i\in N_2^{\left(2\right)}\left(A^{\left(y\right)}\right),\hfill \end{array}$

步骤9 $X^{\left(y\right)}=\text{diag}\left(x\right),y=y+1$ ，返回步骤3。

对该算法，有如下结论。

3.2. 定理6

如果算法经过有限步迭代后终止，且产生一个正对角阵，则$A\in D^{*}$ 。

证明 假设算法经过$y$ 次迭代后停止，得到一个严格对角占优矩阵$A^{\left(y\right)}$ ，由算法的过程可知$A^{\left(y\right)}=A^{\left(y-1\right)}{X}^{\left(y-1\right)}=\cdots =A{X}^{\left(1\right)}{X}^{\left(2\right)}\cdots X^{\left(y-1\right)}$ ，即存在正对角矩阵$X=X^{\left(1\right)}{X}^{\left(2\right)}\cdots X^{\left(y-1\right)}$ ，使得$AX\in D$ ，从而$A\in D^{*}$ 。

4. 数值算例

例1 设矩阵

$A=\left(\begin{array}{cccccc}7& 0& 3& 1& 0& 2\\ 0& 3& -1& 2& 0& 1\\ 1& 0& \frac{77}{2}& -825& 2& 1\\ 2& 1& 0& \frac{165}{2}& 0& 1\\ 2& 1& 2& 2& \frac{143}{2}& -715\\ 0& 1& 0& 1& 0& \frac{143}{2}\end{array}\right),$

在判定矩阵$A$ 是否为非奇异H-矩阵时，利用文献[1]中的记号，有$N_1=\left\{1,4,6\right\}$ ，$N_2=\left\{2,3,5\right\}$ ，则当$k=1$ 时，

$\left|a_{22}\right|x_2=1.7864<\left|a_{21}\right|{\delta }_{2,1}+\left|a_{23}\right|x_3+\left|a_{24}\right|{\delta }_{2,4}+\left|a_{25}\right|x_5+\left|a_{26}\right|{\delta }_{2,6}=33.8552,$

故无法用文献[1]中的定理1来判定。利用文献[2]中的记号有${N^{″}}_2=\varphi$ ，当外迭代次数为4时，此文献[2]中的定理4可以判定矩阵$A$ 是非奇异H-矩阵。同理利用文献[4]中的记号，可验证当外迭代次数为10时，可判定矩阵$A$ 是非奇异H-矩阵。利用文献[5]中的记号，可验证当外迭代次数为2时，可判定矩阵$A$ 是非奇异H-矩阵。

而利用本文定理3的条件有$N_1=\left\{1,4,6\right\}$ ，$N_2=\left\{2,3,5\right\}$ ，$N_2^{\left(1\right)}=\left\{2,3,5\right\}$ ，$N_2^{\left(2\right)}=\varphi$ ，当$l=1$ 时，有$r=0.75$ ，$m_2=0.4643$ ，$m_3=0.7165$ ，$m_5=0.6834$ ，$f_{2,1}=0.3197$ ，$f_{2,4}=0.0241$ ，$f_{2,6}=0.0070$ 。

即有

$\left|a_{22}\right|m_2=1.3929>\left|a_{21}\right|f_{2,1}+\left|a_{23}\right|m_3+\left|a_{24}\right|f_{2,4}+\left|a_{25}\right|m_5+\left|a_{26}\right|f_{2,6}=0.7717,$

$\left|a_{33}\right|m_2=27.5853>\left|a_{31}\right|f_{2,1}+\left|a_{32}\right|m_2+\left|a_{34}\right|f_{2,4}+\left|a_{35}\right|m_5+\left|a_{36}\right|f_{2,6}=21.576,$

$\left|a_{55}\right|m_2=48.8631>\left|a_{51}\right|f_{2,1}+\left|a_{52}\right|m_2+\left|a_{53}\right|m_3+\left|a_{54}\right|f_{2,4}+\left|a_{56}\right|f_{2,6}=7.5899,$

事实上，取$X=\text{diag}\left(0.31970.46430.71650.02410.68340.0070\right)$ 。

则有

$AX=\left(\begin{array}{cccccc}2.2379& 0& 2.1495& 0.0241& 0& 0.0140\\ 0& 1.3929& -0.7165& 0.0482& 0& 0.0070\\ 0.3197& 0& 27.5853& -19.8825& 1.3668& 0.0070\\ 0.6394& 0.4643& 0& 1.9883& 0& 0.0070\\ 0.6394& 0.4643& 1.4330& 0.0482& 48.8631& -5.0050\\ 0& 0.4643& 0& 0.0241& 0& 0.5005\end{array}\right),$

易验证$AX\in D^{*}$ ，则矩阵$A$ 为非奇异H-矩阵。

5. 结论

通过数值算例表明，在引入区间细分策略和递进式迭代因子构造方法后，本文提出的迭代判定条件较文献[1]中的定理1具有更高的判定精度，对比文献[2]-[5]，本文判定条件的适用范围也更为广泛。研究表明，该非奇异H-矩阵的迭代判定新条件，一方面拓展了现有判定理论框架，另一方面有效提升了非奇异H-矩阵的判定效率。

致 谢

感谢陈茜老师对本项目的悉心指导和帮助。

基金项目

湖南省教育厅科研项目(23C0342)和湖南省教育厅优秀青年项目(24B0743)。

NOTES

^*通讯作者。