一组关于非奇异H-矩阵的细分迭代判别新条件

A Set of New Criteria for the Iterative Discrimination of Subdivision of Nonsingular H-Matrices

摘要: 本文根据非奇异H-矩阵与α-对角占优矩阵之间的关系，通过细分矩阵的下标区间，以及构造出新的迭代系数，得出了一组关于非奇异H-矩阵的细分迭代判别新条件，该条件改进了近期的某些结果，最后给出的几个数值算例说明了其有效性。

Abstract: In this paper, we produced a set of new conditions for subdivided and iterative criteria of nonsingular H-matrices by the method of subdivided region and selected iterative coefficient, based on the nonsingular H-matrix and α-diagonally dominant matrix the relationship between diagonally dominant matrices. These conditions improved some recent results. Finally, several numerical examples were given to illustrate their validity.

1. 引言

非奇异H-矩阵是一种很重要的矩阵类。它在数学物理、统计学、神经网络等各个领域中都有着重要的地位。对于判定线性方程组(尤其是超大型方程组)是否具有稳定解时，往往需要先通过判定其系数矩阵是否为非奇异H-矩阵来体现。如：当一个线性方程组的系数矩阵为非奇异H-矩阵，那么该方程组对Jacobi，Gauss-seidel，SOR，SSOR，AOR等经典算法均是收敛的，即该方程组具有稳定解。因此，非奇异H-矩阵判定问题一直是研究的热点，近年来国内外许多学者给出了一些实用的判定条件。

范迎松、徐仲、陆全等人在文献 [1] 中先使用了细分矩阵的下标区间的办法，并由新构造的递进系数提出了关于非奇异H-矩阵新的判别准则；尹军茹等人在文献 [2] 中使用细分矩阵下标区间的方法，构造出新的迭代系数得到了不一样的判别准则。在此基础上，山瑞平等人在文献 [3] 中，根据广义严格对角占优矩阵与非奇异H-矩阵之间蕴含的关系，细分了下标区间，并构造出不同的正对角矩阵，从而得到更好的判定条件。此外，庹清、刘长太等学者近年来在此领域也得出了一些很好的结果(见文献 [4] [5] [6] [7])。

本文中，用 $M_n\left(C\right)\left(M_n\left(R\right)\right)$ 来表示n阶复(实)矩阵的集合。设 $A=\left(a_{ij}\right)\in M_n\left(C\right)$， 记 $R_i=R_i\left(A\right)={\displaystyle \underset{j\ne i}{\sum }\left|a_{ij}\right|},C_i=C_i\left(A\right)={\displaystyle \underset{j\ne i}{\sum }\left|a_{ji}\right|},\forall i,j\in N,N=1,2,\cdots ,n$。为使所讨论的矩阵为非零矩阵且所论内容有意义，以下假设：在全文中， $\left|a_{ii}\right|\ne 0,R_i\ne 0,C_i\ne 0\left(\forall i\in N\right)$，且规定 ${\displaystyle \underset{t\in \varnothing }{\sum }·}=0$。

定义1：设 $A=\left(a_{ij}\right)\in M_n\left(C\right)$，若 $\left|a_{ii}\right|\ge R_i\left(A\right),\forall i\in N$，则称A为对角占优矩阵，记为 $A\in D^0$；若 $\left|a_{ii}\right|>R_i\left(A\right),\forall i\in N$，则称A为严格对角占优矩阵，记为 $A\in D$；若存在正对角阵X，使得X右乘到矩阵A后的乘积矩阵为严格对角占优矩阵，则称A为广义严格对角占优矩阵，记为 $A\in \tilde{D}$。

定义2：设 $A=\left(a_{ij}\right)\in M_n\left(C\right)$， $\alpha \in \left(0,1\right]$，若 $\left|a_{ii}\right|\ge \alpha R_i\left(A\right)+\left(1-\alpha \right)C_i\left(A\right),\forall i\in N$，则称A为α-对角占优矩阵，记为 $A\in D^0\left(\alpha \right)$；若 $\left|a_{ii}\right|>\alpha R_i\left(A\right)+\left(1-\alpha \right)C_i\left(A\right),\forall i\in N$，则称A为严格α-对角占优矩阵，记为 $A\in D\left(\alpha \right)$；若存在正对角阵X，使得X右乘到矩阵A后的乘积矩阵为严格α-对角占优矩阵，则称A为广义严格α-对角占优矩阵，记为 $A\in \tilde{D}\left(\alpha \right)$。

定义3：设为不可约矩阵，，若 $\left|a_{ii}\right|\ge \alpha R_i\left(A\right)+\left(1-\alpha \right)C_i\left(A\right)$，则称A为不可约α-对角占优矩阵。

定义4：设 $A=\left(a_{ij}\right)\in M_n\left(C\right)$ 为α-对角占优阵， $\alpha \in \left(0,1\right]$，若对满足的每个下标i，存在非零元素链 $a_{i{i}_1}{a}_{i_1{i}_2}\cdots a_{i_{r}k}$，使得 $k\in \left\{i\in N:\left|a_{ij}\right|=R_i^{\alpha }\left(A\right)C_i^{\left(1-\alpha \right)}\left(A\right)\right\}\ne \varnothing$，则称A为具非零元素链α-对角占优矩阵。

引理1 [8] 设 $A=\left(a_{ij}\right)\in M_n\left(C\right)$， $\alpha \in \left(0,1\right]$， $A\in \tilde{D}$ 当且仅当 $A\in \tilde{D}\left(\alpha \right)$。

引理2 [8] 设 $A=\left(a_{ij}\right)\in M_n\left(C\right)$ 为不可约α-对角占优矩阵，且至少有一个对角占优行，则 $A\in \tilde{D}$。

引理3 [9] 设 $A=\left(a_{ij}\right)\in M_n\left(C\right)$ 为具有非零元素链α-对角占优矩阵，则 $A\in \tilde{D}$。

以下是下标集的符号解释：

$N_1=\left\{i\in N:0<\left|a_{ii}\right|<\alpha R_i+\left(1-\alpha \right)C_i\right\}$，，

$N_3=\left\{i\in N:\left|a_{ii}\right|>\alpha R_i\left(A\right)+\left(1-\alpha \right)C_i\left(A\right)\right\}$，。

若 $N_3=\varnothing$，则 $A\notin \tilde{D}$；若 $N_1\cup N_2=\varnothing$，显然 $A\in \tilde{D}$。为使所论内容有意义，以下假设： $N_1\oplus N_2\ne \varnothing$， $N_3\ne \varnothing$。

文献 [3] 给出的主要结果如下：

首先，划分下标区间：令 $N_1=N_1^{\left(1\right)}\cup N_1^{\left(2\right)}\cup \cdots \cup N_1^{\left(m\right)}$， $N_1^{\left(1\right)}=\left\{i\in N:0<\left|a_{ii}\right|<\frac{1}{m}\left[\alpha R_i+\left(1-\alpha \right)C_i\right]\right\}$， $N_1^{\left(k\right)}=\left\{i\in N:\frac{k-1}{m}\left[\alpha R_i+\left(1-\alpha \right)C_i\right]\le \left|a_{ii}\right|\le \frac{k}{m}\left[\alpha R_i+\left(1-\alpha \right)C_i\right]\right\},k=2,3,\cdots ,m$。

记，

$\forall i\in N_1^{\left(k\right)},k=1,2,\cdots ,m$ 时， $x_{1i}^{\left(k\right)}=\frac{k\left|a_{ii}\right|}{m\left[\alpha R_i+\left(1-\alpha \right)C_i\right]}$；

$\forall i\in N_2,p\ge 1$ 时， $x_{2i}=\frac{1}{pm}$；

$\forall i\in N_3$ 时， $r_0=1$，

$r_1=\underset{i\in N_3}{\max }\left\{\frac{\alpha R_i+\left(1-\alpha \right)C_i}{\left|a_{ii}\right|}\right\}$，

$r_{l+1}=\underset{i\in N_3}{\max }\left\{\frac{\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1i}^{\left(k\right)}}\right)}+\alpha {\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2i}}+r_l{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|}\right]+\left(1-\alpha \right)C_i}{\left|a_{ii}\right|}\right\},l\in Z^{+}$，

，

${\delta }_{l+1,i}=\frac{M_{l+1,i}\left(A\right)}{\left|a_{ii}\right|},l\in Z$，

$W_{l+1,i}\left(A\right)=\frac{\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}\right]+\left(1-\alpha \right)C_i}{M_{l+1,i}\left(A\right)-\alpha {\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\delta }_{l+1,i}}},h_l=\underset{i\in N_3}{\max }{W}_{l+1,i}\left(A\right),l\in Z$。

定理. 设 $A\in M_n\left(C\right)$，，若存在 $l_0\in Z,p\ge 1$，使满足

$\left|a_{ii}\right|x_{1i}^{\left(k\right)}>\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)},t\ne i}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+h_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\delta }_{l_0+1,t}}\right]+\left(1-\alpha \right)C_i{x}_{1i}^{\left(k\right)},\forall i\in N_1^{\left(k\right)};k=1,2,\cdots ,m$，

$\left|a_{ii}\right|x_{2i}>\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2,t\ne i}{\sum }\left|a_{it}\right|x_{2t}}+h_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\delta }_{l_0+1,t}}\right]+\left(1-\alpha \right)C_i{x}_{2i},\forall i\in N_2$，

则A是非奇异H-矩阵。

2. 主要结果

设 $A=\left(a_{ij}\right)\in M_n\left(C\right)$， $\alpha \in \left(0,1\right]$，将不占优行的下标区间划分为m个区间，即 $N_1=N_1^{\left(1\right)}\cup N_1^{\left(2\right)}\cup \cdots \cup N_1^{\left(m\right)}$，其中m是任意正整数。

具体分法即，

$N_1^{\left(1\right)}=\left\{i\in N:0<\left|a_{ii}\right|<\frac{1}{m}\left[\alpha R_i+\left(1-\alpha \right)C_i\right]\right\}$，

$N_1^{\left(k\right)}=\left\{i\in N:\frac{k-1}{m}\left[\alpha R_i+\left(1-\alpha \right)C_i\right]\le \left|a_{ii}\right|<\frac{k}{m}\left[\alpha R_i+\left(1-\alpha \right)C_i\right]\right\},k=2,3,\cdots ,m$，

根据划分规则易知： $N_1^{\left(k\right)}$ 或为空集。

为叙述方便，引入以下符号：

$\forall i\in N_1^{\left(k\right)}$， $k=1,2,\cdots ,m$，；

$\forall i\in N_2$， $p\ge 1$， $x_{2i}=\frac{1}{pm}$；

$\forall i\in N_3$， $k=1,2,\cdots ,m$， $l=1,2,\cdots$，

${\Omega }_i^{\left(0\right)}=r_0=1$， $r_l=\underset{i\in N_3}{\max }\left(\frac{\alpha {\displaystyle \underset{t\in N_1}{\sum }\left|a_{it}\right|}+\alpha {\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|}+\left(1-\alpha \right)C_i}{\left|a_{ii}\right|-\alpha {\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l-1\right)}}}\right)$，

$P_{l,i}=\alpha \left({\displaystyle \underset{t\in N_1}{\sum }\left|a_{it}\right|}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|}+r_l{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l-1\right)}}\right)+\left(1-\alpha \right)C_i$，

${\Omega }_i^{\left(l\right)}=\frac{P_{l,i}}{\left|a_{ii}\right|}$，

，

$H_l=\underset{i\in N_3}{\max }{M}_{l,i}$，

由上易知， $P_{l+1,i}\le P_{l,i}$， ${\Omega }_i^{\left(l\right)}\le {\Omega }_i^{\left(l-1\right)}\le \cdots \le {\Omega }_i^{\left(1\right)}\le {\Omega }_i^{\left(0\right)}=1$， $r_l\le r_{l-1}\le \cdots \le r_1<r_0=1$。

2.1. 定理1

设 $A\in M_n\left(C\right)$， $\alpha \in \left(0,1\right]$，若存在 $l_0\in Z,p\ge 1$，使满足：

$\begin{array}{l}\left|a_{ii}\right|x_{1i}^{\left(k\right)}>\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)},t\ne i}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]\\ +\left(1-\alpha \right)C_i{x}_{1i}^{\left(k\right)},\forall i\in N_1^{\left(k\right)};k=1,2,\cdots ,m\end{array}$ (1)

(2)

则A是非奇异H-矩阵。

证明

由于 $x_{1i}^{\left(k\right)}=\frac{k\left|a_{ii}\right|}{m\left[\alpha R_i+\left(1-\alpha \right)C_i\right]}$，得到 $0<x_{1i}^{\left(k\right)}\le \frac{\left|a_{ii}\right|}{\alpha R_i+\left(1-\alpha \right)C_i}<1,\forall i\in N_1^{\left(k\right)},k=1,2,\cdots ,m$，从而 $0<x_{1i}^{\left(k\right)}<1,\forall i\in N_1^{\left(k\right)}$。

由于 $x_{2i}=\frac{1}{pm},p\ge 1$，m是任意正整数，所以 $0<x_{2i}\le 1,\forall i\in N_2$。

根据定义得到，

$\begin{array}{c}{M}_{l,i}=\frac{\alpha {\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+\alpha {\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+\left(1-\alpha \right)C_i}{P_{l,i}-\alpha r_l{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l\right)}}}\\ \le \frac{\alpha {\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+\alpha {\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+\left(1-\alpha \right)C_i}{P_{l,i}-\alpha r_l{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l-1\right)}}}\\ \le \frac{\alpha {\displaystyle \underset{t\in N_1}{\sum }\left|a_{it}\right|}+\alpha {\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|}+\left(1-\alpha \right)C_i}{P_{l,i}-\alpha r_l{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l-1\right)}}}=1\end{array}$

又因为 $H_l=\underset{i\in N_3}{\max }{M}_{l,i}$，故 $H_l\le 1$， ${\Omega }_i^{\left(l\right)}\le 1$。

根据(1)式和(2)式，存在 $1_0\in Z$，可取充分小的正数 $\epsilon >0$，使 $\epsilon$ 同时满足：

$\begin{array}{l}\epsilon \alpha {\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|}<\left|a_{ii}\right|x_{1i}^{\left(k\right)}-\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]\\ -\left(1-\alpha \right)C_i{x}_{1i}^{\left(k\right)},\forall i\in N_1^{\left(k\right)};k=1,2,\cdots ,m\end{array}$ (3)

$\epsilon \alpha {\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|}<\left|a_{ii}\right|x_{2t}-\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^k,t\ne i}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]-\left(1-\alpha \right)C_i{x}_{2i},\forall i\in N_2$ (4)

再根据 $M_{l,i}$， $P_{l,i}$ 的定义，以及， ${\Omega }_t^{\left(l\right)}=\frac{P_{l,i}}{\left|a_{ii}\right|}$，对于任意 $i\in N_3$，有：

$H_{l_0}{P}_{l_0,i}\ge \alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^k}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{r}_{l_0}{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]+\left(1-\alpha \right)C_i$，

又因为 ，所以得到：

$\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{r}_{l_0}{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]+\left(1-\alpha \right)C_i-H_{l_0}{\Omega }_t^{\left(l_0\right)}\left|a_{ii}\right|\le 0$，

即 $H_{l_0}{\Omega }_t^{\left(l_0\right)}\left|a_{ii}\right|-\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{r}_{l_0}{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]+\left(1-\alpha \right)C_i\ge 0$，

当 $r_{l_0}=r_0=1$ 时，易知上述等式依旧成立。

因为 $\left|a_{ii}\right|>\alpha R_i+\left(1-\alpha \right)C_i\ge \alpha {\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|}+\left(1-\alpha \right)C_i$，所以 $\left|a_{ii}\right|-\alpha {\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|}+\left(1-\alpha \right)C_i\ge 0$，

综上，取 $\epsilon >0$ 且充分小，得：

$\begin{array}{l}{H}_{l_0}{\Omega }_t^{\left(l_0\right)}\left|a_{ii}\right|-\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{r}_{l_0}{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]+\left(1-\alpha \right)C_i\\ +\epsilon \left[\left|a_{ii}\right|-\alpha {\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|}+\left(1-\alpha \right)C_i\right]>0\end{array}$ (5)

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

，

1) $\forall i\in N_1^{\left(k\right)}$，由(3)式得，

$\begin{array}{l}\alpha R_i\left(B\right)+\left(1-\alpha \right)C_i\left(B\right)\\ =\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)},t\ne i}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|\left(H_{l_0}{\Omega }_t^{\left(l_0\right)}+\epsilon \right)}\right]+\left(1-\alpha \right)C_i\left(A\right)x_{1i}^{\left(k\right)}\\ =\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)},t\ne i}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]+\left(1-\alpha \right)C_i\left(A\right)x_{1i}^{\left(k\right)}+\epsilon \alpha {\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|}\\ <\left|a_{ii}\right|x_{1i}^{\left(k\right)}=\left|b_{ii}\right|\end{array}$

2) $\forall i\in N_2$，由(4)式得，

$\begin{array}{l}\alpha R_i\left(B\right)+\left(1-\alpha \right)C_i\left(B\right)\\ =\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2,t\ne i}{\sum }\left|a_{it}\right|x_{2t}}+{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|\left(H_{l_0}{\Omega }_t^{\left(l_0\right)}+\epsilon \right)}\right]+\left(1-\alpha \right)C_i\left(A\right)x_{2i}\\ =\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2,t\ne i}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]+\left(1-\alpha \right)C_i\left(A\right)x_{2i}+\epsilon \alpha {\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|}\\ <\left|a_{ii}\right|x_{2i}=\left|b_{ii}\right|\end{array}$

3) $\forall i\in N_3$，因为 $H_l\le 1$， ${\Omega }_i^{\left(l\right)}\le 1$，所以 $H_{l_0}{\Omega }_t^{\left(l_0\right)}\le 1$，再由(5)式得，

$\begin{array}{l}\left|b_{ii}\right|-\alpha R_i\left(B\right)-\left(1-\alpha \right)C_i\left(B\right)\\ =\left(H_{l_0}{\Omega }_t^{(l_0)}+\epsilon \right)\left|a_{ii}\right|-\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|\left(H_{l_0}{\Omega }_t^{\left(l_0\right)}+\epsilon \right)}\right]\\ -\left(1-\alpha \right)C_i\left(A\right)\left(H_{l_0}{\Omega }_t^{\left(l_0\right)}+\epsilon \right)\\ =\epsilon \left[\left|a_{ii}\right|-\alpha {\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|}-\left(1-\alpha \right)C_i\left(A\right)\right]+H_{l_0}{\Omega }_t^{\left(l_0\right)}\left|a_{ii}\right|\\ -\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]-\left(1-\alpha \right)C_i\left(A\right)H_{l_0}{\Omega }_t^{(l0)}\end{array}$

$\begin{array}{l}>\epsilon \left[\left|a_{ii}\right|-\alpha {\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|}-\left(1-\alpha \right)C_i\left(A\right)\right]+H_{l_0}{\Omega }_t^{\left(l_0\right)}\left|a_{ii}\right|\\ -\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]-\left(1-\alpha \right)C_i\left(A\right)\\ >0\end{array}$

综上所述， $\forall i\in N$， $\left|b_{ii}\right|>\alpha R_i\left(B\right)+\left(1-\alpha \right)C_i\left(B\right)$，总是能够成立，即满足 $B=AX\in D\left(\alpha \right)$，所以 $A\in \tilde{D}\left(\alpha \right)$，根据引理4，则 $A\in \tilde{D}$，证毕。

注1 本文定理1对文献 [3] 的定理1有所改进，由于 $r_1\le 1$,故做每步迭代时都有所进步(即判定条件比文献 [3] 的弱)。

我们可以在定理1中取 $\alpha =1,p=1$，得出推论1，如下所示。

2.1.1. 推论1

设 $A\in M_n\left(C\right)$，若存在 $l_0\in Z$，使满足

$\left|a_{ii}\right|x_{1i}>{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)},t\ne i}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{2t}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}},\forall i\in N_1;k=1,2,\cdots ,m$

$\left|a_{ii}\right|x_{2i}^{\left(k\right)}>{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2,t\ne i}{\sum }\left|a_{2t}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}},\forall i\in N_2^{(k)}$

则A是非奇异H-矩阵。

注2 由于 $r_l\le 1$，故做每步迭代时都有所进步(即判定条件比文献 [1] 的定理1条件弱)，也就是说本文推论1推广了文献 [1] 的定理1。

还可以在定理1中取 $\alpha =1,p=1,m=1$，得出推论2。

2.1.2. 推论2

设 $A\in M_n\left(C\right)$，若存在 $l_0\in Z$，使满足

$\left|a_{ii}\right|\frac{\left|a_{ii}\right|}{R_i}>{\displaystyle \underset{t\in N_1\cup N_2,t\ne i}{\sum }\left|a_{it}\right|\frac{\left|a_{ii}\right|}{R_i}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}},\forall i\in N_1;k=1,2,\cdots ,m$

则A是非奇异H-矩阵。

同理可以推出下面两个定理：

2.2. 定理2

设 $A\in M_n\left(C\right)$， $\alpha \in \left(0,1\right]$，矩阵A不可约，若存在 $l_0\in Z,p\ge 1$，使满足

$\begin{array}{l}\left|a_{ii}\right|x_{1i}^{\left(k\right)}\ge \alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)},t\ne i}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]\\ +\left(1-\alpha \right)C_i{x}_{1i}^{\left(k\right)},\forall i\in N_1^{\left(k\right)};k=1,2,\cdots ,m\end{array}$ (6)

$\left|a_{ii}\right|x_{2i}\ge \alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2,t\ne i}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]+\left(1-\alpha \right)C_i{x}_{2i},\forall i\in N_2$ (7)

且(6)或(7)中至少有一个严格不等式成立，则A是非奇异H-矩阵。

2.3. 定理3

记，

$K_1=\left\{i\in N_1^{\left(k\right)}:\left|a_{ii}\right|x_{1i}^{(k)}>\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)},t\ne i}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]+\left(1-\alpha \right)C_i{x}_{1i}^{\left(k\right)}\right\};k=1,2,\cdots ,m$

$K_1={\displaystyle \underset{k=1}{\overset{m}{\cup }}{K}_1^{\left(k\right)}}$，

$K_2=\left\{i\in N_2:\left|a_{ii}\right|x_{2i}>\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2,t\ne i}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]+\left(1-\alpha \right)C_i{x}_{2i}\right\}$，

$K_3=\left\{i\in N_3:\left|a_{ii}\right|H_{l_0}{\Omega }_i^{\left(l_0\right)}>\alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3,t\ne i}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]+\left(1-\alpha \right)C_i{H}_{l_0}{\Omega }_i^{\left(l_0\right)}\right\}$

设 $A\in M_n\left(C\right)$， $\alpha \in \left(0,1\right]$，若存在 $l_0\in Z,p\ge 1$，使满足

$\begin{array}{l}\left|a_{ii}\right|x_{1i}^{\left(k\right)}\ge \alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)},t\ne i}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]\\ +\left(1-\alpha \right)C_i{x}_{1i}^{\left(k\right)},\forall i\in N_1^{\left(k\right)};k=1,2,\cdots ,m\end{array}$ (8)

$\left|a_{ii}\right|x_{2i}\ge \alpha \left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}\left({\displaystyle \underset{t\in N_1^{\left(k\right)}}{\sum }\left|a_{it}\right|x_{1t}^{\left(k\right)}}\right)}+{\displaystyle \underset{t\in N_2,t\ne i}{\sum }\left|a_{it}\right|x_{2t}}+H_{l_0}{\displaystyle \underset{t\in N_3}{\sum }\left|a_{it}\right|{\Omega }_t^{\left(l_0\right)}}\right]+\left(1-\alpha \right)C_i{x}_{2i},\forall i\in N_2$ (9)

且(8)或(9)中至少有一个严格不等式成立，若对，存在非零元素链 $a_{i{i}_1}{a}_{i_1{i}_2}\cdots a_{i_{r}k}$，使得 $k\in {\displaystyle \underset{k=1}{\overset{3}{\cup }}{K}_i}$，则A是非奇异H-矩阵。

3. 数值实例

例设

$A=\left(\begin{array}{ccccc}2& 0& 0& 1& 1\\ 0& 3.2& 1& 2& 3\\ 1& 1& 18& 4& 5\\ 1& 1& 3& 25& 4\\ 0& 1.5& 2& 3& 20\end{array}\right)$。

易知 $N_1=\left\{2\right\},N_2=\left\{1\right\},N_3=\left\{3,4,5\right\}$，当取 $\alpha =\frac{1}{2},p=\frac{3}{2},l_0=1$ 时，容易算得 $x_{1i}^{\left(k\right)}=x_{12}=\frac{64}{95}=0.6737$ (保留四位小数，后同)， $x_{2i}=x_{21}=\frac{2}{3}=0.6666$。

继而能够得出：

${\delta }_{2,3}=0.3258,{\delta }_{2,4}=0.2951,{\delta }_{2,5}=0.4028,h_1=0.9613$ (文献 [1] 结果)；

${\Omega }_3^{\left(l\right)}=0.3258,{\Omega }_4^{\left(l\right)}=0.298,{\Omega }_5^{\left(l\right)}=0.4143,H_1=0.8975$ (本文结果)，

当 时，根据

显然A不满足文献 [1] 中定理1的条件，不能判定其为非奇异H-矩阵。

但是，由本文得出的结果，有

$\begin{array}{l}2.1558=3.2\times 0.6737=\left|a_{22}\right|x_{12}>\alpha \left[\left|a_{21}\right|x_{21}+H_1\left(\left|a_{23}\right|{\Omega }_3^{\left(l\right)}+\left|a_{24}\right|{\Omega }_4^{\left(l\right)}+\left|a_{25}\right|{\Omega }_5^{\left(l\right)}\right)\right]+\left(1-\alpha \right)C_2{x}_{21}\\ =\frac{1}{2}\times \left[0+0.8795\times \left(0.3258+2\times 0.298+3\times 0.4143\right)\right]+\frac{1}{2}\times 3.5\times 0.6737=2.1309\end{array}$

显然矩阵A满足本文定理1的条件，可知矩阵A是非奇异H-矩阵。

其中取正对角矩阵：

$X=\left(\begin{array}{ccccc}0.6737& 0& 0& 0& 0\\ 0& 0.\stackrel{\dot{}}{6}& 0& 0& 0\\ 0& 0& 0.2865& 0& 0\\ 0& 0& 0& 0.2621& 0\\ 0& 0& 0& 0& 0.3643\end{array}\right)$，

所以 $B=AX=\left(\begin{array}{ccccc}1.3474& 0& 0& 0.2621& 0.3643\\ 0& 2.1331& 0.2865& 0.5243& 1.0929\\ 0.6737& 0.\stackrel{\dot{}}{6}& 5.157& 1.0484& 1.8215\\ 0.6737& 0.\stackrel{\dot{}}{6}& 0.8595& 6.5525& 1.4572\\ 0& 0.\stackrel{\dot{}}{9}& 0.573& 0.7863& 7.286\end{array}\right)$，B是一个严格α-对角占优矩阵。

致谢

感谢陈茜等同学对本文提供的建议和帮助。

基金项目

国家自然科学基金(11461027)和湖南省教育厅科研基金(16A173)，吉首大学研究生创新基金(JGY201932)。

NOTES

^*通讯作者。