# 非奇异H-矩阵的一组细分迭代实用判定方法A Set of Subdivision Iteration Practical Criteria for Nonsingular H-Matrix

DOI: 10.12677/AAM.2021.104138, PDF, HTML, XML, 下载: 10  浏览: 32  国家自然科学基金支持

Abstract: Nonsingular H-matrices have been widely used in many fields, such as computational mathematics, control theory, elastic mechanics, neural network system, etc. But it is very difficult to judge nonsingular H-matrix. In this paper, the criteria for nonsingular H-matrices are studied. By subdividing the matrix index set according to the requirements and constructing progressive diagonal matrix elements, a set of new subdividing iterative criteria for nonsingular H-matrices are obtained and proved. Finally, numerical examples show that the new decision condition is superior to the known results.

1. 引言

$\begin{array}{l}{N}_{1}=\left\{i\in N:0<|{a}_{ii}|<{\Lambda }_{i}\right\},\text{}{N}_{2}=\left\{i\in N:|{a}_{ii}|={\Lambda }_{i}\right\},\\ {N}_{3}=\left\{i\in N:|{a}_{ii}|>{\Lambda }_{i}\left(A\right)\right\},\text{}N={N}_{1}\oplus {N}_{2}\oplus {N}_{3}.\end{array}$

${N}_{1}$ 进一步划分 ${N}_{1}={N}_{{}_{1}}^{\left(1\right)}\cup {N}_{{}_{1}}^{\left(2\right)}\cup \cdots \cup {N}_{{}_{1}}^{\left(m\right)}$，其中m是任意正整数，取 $k\in {Z}^{+}$$k\le m$，且

${N}_{{}_{1}}^{\left(1\right)}=\left\{i\in N:0<|{a}_{ii}|<\frac{1}{m}{\Lambda }_{i}\left(A\right)\right\}$,

${N}_{{}_{1}}^{\left(k\right)}=\left\{i\in N:\frac{k-1}{m}{\Lambda }_{i}\left(A\right)<|{a}_{ii}|<\frac{k}{m}{\Lambda }_{i}\left(A\right)\right\}$,

2. 主要结果

${x}_{1i}^{\left(k\right)}=\frac{\frac{k}{m}\left({\Lambda }_{i}-\frac{k}{k+1}|{a}_{ii}|\right)}{{\Lambda }_{i}}\text{}\left(i\in {N}_{1}^{\left(k\right)},k=1,2,\cdots ,m\right)$,

${\delta }_{0,i}=\frac{{\Lambda }_{i}\left(A\right)}{|{a}_{ii}|}\text{}\left(\forall i\in {N}_{3}\right)$, ${\delta }_{l,i}=\frac{{P}_{l,i}}{|{a}_{ii}|}\text{}\left(\forall i\in {N}_{3},l\in {Z}^{\text{+}}\right)$,

${P}_{l,i}\left(A\right)=\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|{\delta }_{l-1,t}\left(\forall i\in {N}_{3},l\in {Z}^{\text{+}}\right)$,

${x}_{2i}=\frac{\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2},t\ne i}{\sum }|{a}_{it}|+\underset{t\in {N}_{3}}{\sum }|{a}_{it}|{\delta }_{l,t}}{|{a}_{ii}|}\text{}\left(\forall i\in {N}_{2},l\in {Z}^{+}\right)$,

${Q}_{l}=\underset{i\in {N}_{3}}{\mathrm{max}}\left(\frac{\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|{\delta }_{l,t}}{{P}_{l,i}\left(A\right)}\right)\text{}\left(l\in {Z}^{+}\right)$.

2.1. 定理1

$A\in {M}_{n}\left(C\right)$，若存在 $l\in {Z}^{+}$，使得

$|{a}_{ii}|{x}_{1i}^{\left(k\right)}>\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)},t\ne i}{\sum }|{a}_{it}|{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|{\delta }_{l,t}\text{}\left(\forall i\in {N}_{1}^{\left(k\right)},k=1,2,\cdots ,m\right)$, (1)

$|{a}_{ii}|{x}_{2i}>\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2},t\ne i}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|{\delta }_{l,t}\text{}\left(\forall i\in {N}_{2}\right)$, (2)

$A\in \stackrel{˜}{D}$

$0<{x}_{1i}^{\left(k\right)}<1\text{}\left(\forall i\in {N}_{1}^{\left(k\right)},k=1,2,\cdots ,m\right)$,

${P}_{1,i}\left(A\right)=\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{0,t}<{\Lambda }_{i}\left(A\right)$,

$\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|\frac{{P}_{1,t}}{|{a}_{tt}|}<\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|\frac{{\Lambda }_{t}\left(A\right)}{|{a}_{tt}|}$,

${P}_{2,i}\left(A\right)<{P}_{1,i}\left(A\right)$。故对 $\forall i\in {N}_{3}$，有 ${P}_{2,i}\left(A\right)<{P}_{1,i}\left(A\right)<1$，假设当 $l=n-1$ 时， ${P}_{n,i}<{P}_{n-1,i}$，则当 $l=n$ 时，由

$\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|\frac{{P}_{n,t}}{|{a}_{tt}|}<\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|\frac{{P}_{n-1,t}}{|{a}_{tt}|}$,

$0<{\delta }_{l,i}<{\delta }_{l-1,i}<\cdots <{\delta }_{1,i}<\frac{{\Lambda }_{i}\left(A\right)}{|{a}_{ii}|}<1$.

$\frac{\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{2t}+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|{\delta }_{l,t}}{{P}_{l,i}\left(A\right)}\le \frac{\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{2t}+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|{\delta }_{l,t}}{\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|{\delta }_{l-1,t}}\le 1$.

$|{a}_{ii}|{x}_{1i}^{\left(k\right)}>\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)},t\ne i}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{2t}+\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\left({Q}_{l}{\delta }_{l,t}+\epsilon \right)\text{}\left(\forall i\in {N}_{{}_{1}}^{\left(k\right)},k=1,2,\cdots ,m\right)$, (3)

$|{a}_{ii}|{x}_{2i}>\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2},t\ne i}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{2t}+\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\left({Q}_{l}{\delta }_{l,t}+\epsilon \right)\text{}\left(\forall i\in {N}_{2}\right)$, (4)

${d}_{i}=\left\{\begin{array}{l}{x}_{1i}^{\left(k\right)},\text{}i\in {N}_{1}^{\left(k\right)}\hfill \\ {x}_{2i},\text{}i\in {N}_{2}\hfill \\ {Q}_{l}{\delta }_{l,i}+\epsilon ,\text{}i\in {N}_{3}\hfill \end{array}$

1) 对 $\forall i\in {N}_{1}^{\left(k\right)}\text{}\left(k=1,2,\cdots ,m\right)$，由(3)式得

$\begin{array}{c}{\Lambda }_{i}\left(B\right)=\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)},t\ne i}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\left({Q}_{l}{\delta }_{l,t}+\epsilon \right)\\ <|{a}_{ii}|{x}_{1i}^{\left(k\right)}=|{b}_{ii}|；\end{array}$

2) 对 $\forall i\in {N}_{2}$，由(4)式得

$\begin{array}{c}{\Lambda }_{i}\left(B\right)=\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2},t\ne i}{\sum }|{a}_{it}|{x}_{2t}+\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\left({Q}_{l}{\delta }_{l,t}+\epsilon \right)\\ <|{a}_{ii}|{x}_{2i}=|{b}_{ii}|；\end{array}$

3) 对 $\forall i\in {N}_{3}$$l\in {Z}^{+}$，有 $\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|<|{a}_{ii}|$，由

${Q}_{l}\ge \frac{\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{2t}+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|{\delta }_{l,t}}{{P}_{l,i}\left(A\right)}$,

$\begin{array}{c}{Q}_{l}{P}_{l,i}\left(A\right)\ge \underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|{\delta }_{l,t}\\ \ge \underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|{\delta }_{l,t}\end{array}$ (5)

$\begin{array}{c}{\Lambda }_{i}\left(B\right)=\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|\left({Q}_{l}{\delta }_{l,t}+\epsilon \right)\\ =\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|{\delta }_{l,t}+\epsilon \underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|\\ \le {Q}_{l}{P}_{l,i}\left(A\right)+\epsilon \underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|\\ <{Q}_{l}{\delta }_{l,i}|{a}_{ii}|+\epsilon |{a}_{ii}|=|{b}_{ii}|\end{array}$

2.1.1. 推论1

$A\in {M}_{n}\left(C\right)$，若存在 $l\in {Z}^{+}$，使得

$|{a}_{ii}|\frac{{\Lambda }_{i}\left(A\right)-\frac{1}{2}|{a}_{ii}|}{{\Lambda }_{i}\left(A\right)}>\underset{t\in {N}_{1},t\ne i}{\sum }|{a}_{it}|\frac{{\Lambda }_{t}\left(A\right)-\frac{1}{2}|{a}_{tt}|}{{\Lambda }_{t}\left(A\right)}+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\text{}\left(\forall i\in {N}_{1}\right)$,

$|{a}_{ii}|{x}_{2i}>\underset{t\in {N}_{1}}{\sum }|{a}_{it}|\frac{{\Lambda }_{t}\left(A\right)-\frac{1}{2}|{a}_{tt}|}{{\Lambda }_{t}\left(A\right)}+\underset{t\in {N}_{2},t\ne i}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|{\delta }_{l,t}\text{}\left(\forall i\in {N}_{2}\right)$,

$A\in \stackrel{˜}{D}$

${N}_{{}_{1}}^{\left(1\right)}=\left\{i\in N:0<|{a}_{ii}|<\frac{1}{2}{\Lambda }_{i}\left(A\right)\right\}$, ${N}_{{}_{1}}^{\left(2\right)}=\left\{i\in N:\frac{1}{2}{\Lambda }_{i}\left(A\right)<|{a}_{ii}|<{\Lambda }_{i}\left(A\right)\right\}$,

2.1.2. 推论2

$A\in {M}_{n}\left(C\right)$，若存在 $l\in {Z}^{+}$，使得

$\begin{array}{c}|{a}_{ii}|\frac{\frac{1}{2}\left({\Lambda }_{i}-\frac{1}{2}|{a}_{ii}|\right)}{{\Lambda }_{i}}>\underset{t\in {N}_{1}^{\left(1\right)},t\ne i}{\sum }|{a}_{it}|\frac{\frac{1}{2}\left({\Lambda }_{t}\left(A\right)-\frac{1}{2}|{a}_{ii}|\right)}{{\Lambda }_{t}\left(A\right)}+\underset{t\in {N}_{1}^{\left(2\right)}}{\sum }|{a}_{it}|\frac{{\Lambda }_{t}\left(A\right)-\frac{2}{3}|{a}_{ii}|}{{\Lambda }_{t}\left(A\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\text{}\left(\forall i\in {N}_{1}^{\left(1\right)}\right)，\end{array}$

$\begin{array}{c}|{a}_{ii}|\frac{{\Lambda }_{i}\left(A\right)-\frac{2}{3}|{a}_{ii}|}{{\Lambda }_{i}\left(A\right)}>\underset{t\in {N}_{1}^{\left(1\right)}}{\sum }|{a}_{it}|\frac{\frac{1}{2}\left({\Lambda }_{t}\left(A\right)-\frac{1}{2}|{a}_{ii}|\right)}{{\Lambda }_{t}\left(A\right)}+\underset{t\in {N}_{1}^{\left(2\right)},t\ne i}{\sum }|{a}_{it}|\frac{{\Lambda }_{t}\left(A\right)-\frac{2}{3}|{a}_{ii}|}{{\Lambda }_{t}\left(A\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\text{}\left(\forall i\in {N}_{1}^{\left(2\right)}\right)，\end{array}$

$\begin{array}{c}|{a}_{ii}|{x}_{2i}>\underset{t\in {N}_{1}^{\left(1\right)}}{\sum }|{a}_{it}|\frac{\frac{1}{2}\left({\Lambda }_{t}\left(A\right)-\frac{1}{2}|{a}_{ii}|\right)}{{\Lambda }_{t}\left(A\right)}+\underset{t\in {N}_{1}^{\left(2\right)}}{\sum }|{a}_{it}|\frac{{\Lambda }_{t}\left(A\right)-\frac{2}{3}|{a}_{ii}|}{{\Lambda }_{t}\left(A\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{t\in {N}_{2},t\ne i}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\text{}\left(\forall i\in {N}_{2}\right)，\end{array}$

$A\in \stackrel{˜}{D}$

2.2. 定理2

$A\in {M}_{n}\left(C\right)$$A$ 不可约，若存在 $l\in {Z}^{+}$，使得

$|{a}_{ii}|{x}_{1i}^{\left(k\right)}\ge \underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)},t\ne i}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\text{}\left(\forall i\in {N}_{{}_{1}}^{\left(k\right)},k=1,2,\cdots ,m\right)$,(6)

$|{a}_{ii}|{x}_{2i}\ge \underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2},t\ne i}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|{\delta }_{l,t}\text{}\left(\forall i\in {N}_{2}\right)$, (7)

${d}_{i}=\left\{\begin{array}{l}{x}_{1i}^{\left(k\right)},\text{}i\in {N}_{1}^{\left(k\right)}\hfill \\ {x}_{2i},\text{}i\in {N}_{2}\hfill \\ {Q}_{l}{\delta }_{l,i},\text{}i\in {N}_{3}\hfill \end{array}$

1) 对 $\forall i\in {N}_{{}_{1}}^{\left(k\right)}\left(k=1,2,\cdots ,m\right)$，由(6)式得

${\Lambda }_{i}\left(B\right)=\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)},t\ne i}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\le |{a}_{ii}|{x}_{1i}^{\left(k\right)}=|{b}_{ii}|$ ;

2) 对 $\forall i\in {N}_{2}$，由(7)式得

${\Lambda }_{i}\left(B\right)=\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2},t\ne i}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\le |{a}_{ii}|{x}_{2i}=|{b}_{ii}|$ ;

3) 对 $\forall i\in {N}_{3}$$l\in {Z}^{+}$，由(5)式得

${\Lambda }_{i}\left(B\right)=\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|{\delta }_{l,t}\le {Q}_{l}{\delta }_{l,i}|{a}_{ii}|\text{=}|{b}_{ii}|$.

2.3. 定理3

${J}_{1}={U}_{k=1}^{m}{J}_{1}^{\left(k\right)}$，其中

${J}_{1}^{\left(k\right)}=\left\{i\in {N}_{1}^{\left(k\right)}:|{a}_{ii}|{x}_{1i}^{\left(k\right)}>\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)},t\ne i}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\right\}\text{}\left(k=1,2,\cdots ,m\right)$,

${J}_{2}=\left\{i\in {N}_{2}:|{a}_{ii}|{x}_{2i}>\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2},t\ne i}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\right\}$,

${J}_{3}=\left\{i\in {N}_{3}:{Q}_{l}|{a}_{ii}|{\delta }_{l,i}>\underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3},t\ne i}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\right\}$.

$|{a}_{ii}|{x}_{1i}^{\left(k\right)}\ge \underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)},t\ne i}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2}}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\text{}\left(\forall i\in {N}_{{}_{1}}^{\left(k\right)},k=1,2,\cdots ,m\right)$, (8)

$|{a}_{ii}|{x}_{2i}\ge \underset{k=1}{\overset{m}{\sum }}\left(\underset{t\in {N}_{1}^{\left(k\right)}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{x}_{1t}^{\left(k\right)}\right)+\underset{t\in {N}_{2},t\ne i}{\sum }|{a}_{it}|{x}_{2t}+{Q}_{l}\underset{t\in {N}_{3}}{\sum }|{a}_{it}|\text{\hspace{0.17em}}{\delta }_{l,t}\text{}\left(\forall i\in {N}_{2}\right)$, (9)

3. 数值实例

$A=\left(\begin{array}{cccccc}4& 1& 1& 0& 0& 2\\ 2.5& 5.5& 0& 1& 2& 0\\ 1.5& 0& 3& 1.5& 0& 2\\ 1& 0& 1& 3& 0& 5\\ 1& 1& 0& 2& 25& 6\\ 0& 0& 0.5& 0& 4.5& 15\end{array}\right)$

$|{a}_{33}|{x}_{3}=1.2<2.357144+\left(\underset{i\in {N}_{3}}{\mathrm{max}}{M}_{i}\right)|{a}_{36}|{x}_{6}$ ;

$|{a}_{33}|{x}_{3}=1.2<2.357144+|{a}_{36}|{\delta }_{l,6}$ ;

$|{a}_{33}|\frac{{\Lambda }_{3}-|{a}_{33}|}{{\Lambda }_{3}}=1.2<2.357144+\frac{h{P}_{6}}{|{a}_{66}|}|{a}_{36}|$ ;

$|{a}_{33}|{x}_{13}=1.2<2.357144+|{a}_{36}|{\delta }_{2,6}$ ;

$|{a}_{11}|{x}_{21}=1.880381>1.51488=|{a}_{12}|{x}_{22}+|{a}_{13}|{x}_{13}+{Q}_{2}|{a}_{16}|{\delta }_{2,6}$,

$|{a}_{22}|{x}_{22}=3.640229>2.261694=|{a}_{21}|{x}_{21}+|{a}_{24}|{x}_{14}+{Q}_{2}|{a}_{25}|{\delta }_{2,5}$,

$|{a}_{33}|{x}_{13}=2.1>2.036735=|{a}_{31}|{x}_{21}+|{a}_{34}|{x}_{14}+{Q}_{2}|{a}_{36}|{\delta }_{2,6}$,

$|{a}_{44}|{x}_{14}=2.357143>1.552647=|{a}_{41}|{x}_{2,2}+|{a}_{43}|{x}_{13}+{Q}_{2}|{a}_{46}|{\delta }_{2,6}$ ;

$X=diag\left(0.470095,0.661859,0.7,0.785714,0.150371,0.076511\right)$,

$AX\in D$，则矩阵A是非奇异H-矩阵。

$A=\left(\begin{array}{cccccc}5& 2& 1& 1& 0& 1\\ 1& 4& 1& 0& 2& 0\\ 0.5& 0& 2& 1.5& 1& 1\\ 0& 0.5& 0& 2.5& 0& 5\\ 1& 1& 0& 2& 25& 5\\ 1& 0& 1& 0& 6& 20\end{array}\right)$

$|{a}_{33}|{x}_{3}=1<1.318182+\left(\underset{i\in {N}_{3}}{\mathrm{max}}{M}_{i}\right)\left(|{a}_{35}|{x}_{5}+|{a}_{36}|{x}_{6}\right)$

$|{a}_{33}|{x}_{3}=1<1.318182+|{a}_{35}|{\delta }_{l,5}+|{a}_{36}|{\delta }_{l,6}$ ;

$|{a}_{33}|\frac{{\Lambda }_{3}-|{a}_{33}|}{{\Lambda }_{3}}=1<1.181818+\frac{h{P}_{5}}{|{a}_{55}|}|{a}_{35}|+\frac{h{P}_{6}}{|{a}_{66}|}|{a}_{36}|$ ;

$|{a}_{44}|{x}_{14}=0.113638<0.25+|{a}_{46}|{\delta }_{2,6}$ ;

$|{a}_{11}|{x}_{21}=3.244364>2.200322=|{a}_{12}|{x}_{22}+|{a}_{13}|{x}_{13}+|{a}_{14}|{x}_{14}+{Q}_{1}|{a}_{16}|{\delta }_{1,6}$,

$|{a}_{22}|{x}_{22}=2.048485>1.561094=|{a}_{21}|{x}_{21}+|{a}_{23}|{x}_{13}+{Q}_{1}|{a}_{25}|{\delta }_{1,5}$,

$|{a}_{33}|{x}_{13}=1.333333>1.149808=|{a}_{31}|{x}_{21}+|{a}_{34}|{x}_{14}+{Q}_{1}|{a}_{35}|{\delta }_{1,5}+{Q}_{1}|{a}_{36}|{\delta }_{1,6}$,

$|{a}_{44}|{x}_{14}=0.965909>0.871309=|{a}_{42}|{x}_{21}+{Q}_{1}|{a}_{46}|{\delta }_{1,6}$ ;

$X=diag\left(0.648873,0.512121,0.666667,0.386364,0.122777,0.123049\right)$,

$AX\in D$，则矩阵A是非奇异H-矩阵。

$A=\left(\begin{array}{ccccccc}5& 1& 0.5& 0& 1& 1.5& 1\\ 1& 6& 0& 1& 1& 1& 2\\ 1& 2& 10& 3.5& 4.4& 1& 1.5\\ 0.5& 1.5& 3& 9& 2& 2& 2.5\\ 1& 2& 0& 1& 9& 4& 2\\ 1& 1& 0& 2& 2& 24& 2\\ 1& 0& 1& 1& 5& 4& 25\end{array}\right)$

$|{a}_{55}|{x}_{5}=0.9<3.217391+\left(\underset{i\in {N}_{3}}{\mathrm{max}}{M}_{i}\right)\left(|{a}_{56}|{x}_{6}+|{a}_{57}|{x}_{7}\right)$ ;

$|{a}_{55}|{x}_{5}=0.9<3.217391+|{a}_{56}|{\delta }_{l,6}+|{a}_{57}|{\delta }_{l,7}$ ;

$|{a}_{33}|\frac{{\Lambda }_{3}-|{a}_{33}|}{{\Lambda }_{3}}=2.53731<4.200869+\frac{h{P}_{6}}{|{a}_{66}|}|{a}_{36}|+\frac{h{P}_{7}}{|{a}_{77}|}|{a}_{37}|$ ;

$|{a}_{55}|{x}_{15}=0.9<1.217391+|{a}_{56}|{\delta }_{2,6}+|{a}_{57}|{\delta }_{2,7}$ ;

$|{a}_{11}|{x}_{21}=1.95517>1.259377=|{a}_{12}|{x}_{22}+|{a}_{13}|{x}_{13}+|{a}_{15}|{x}_{15}+{Q}_{2}|{a}_{16}|{\delta }_{2,6}+{Q}_{2}|{a}_{17}|{\delta }_{2,7}$,

$|{a}_{22}|{x}_{22}=2.236338>1.5441=|{a}_{21}|{x}_{22}+|{a}_{24}|{x}_{14}+|{a}_{25}|{x}_{15}+{Q}_{2}|{a}_{26}|{\delta }_{2,6}+{Q}_{2}|{a}_{27}|{\delta }_{2,7}$,

$|{a}_{33}|{x}_{13}=4.402985>4.356896=|{a}_{31}|{x}_{21}+|{a}_{32}|{x}_{22}+|{a}_{34}|{x}_{14}+|{a}_{35}|{x}_{15}+{Q}_{2}|{a}_{36}|{\delta }_{2,6}+{Q}_{2}|{a}_{37}|{\delta }_{2,7}$,

$|{a}_{44}|{x}_{14}=3.717391>3.34477=|{a}_{41}|{x}_{21}+|{a}_{42}|{x}_{22}+|{a}_{43}|{x}_{13}+|{a}_{45}|{x}_{15}+{Q}_{2}|{a}_{46}|{\delta }_{2,6}+{Q}_{2}|{a}_{47}|{\delta }_{2,7}$,

$|{a}_{55}|{x}_{15}=2.925>2.366521=|{a}_{51}|{x}_{21}+|{a}_{52}|{x}_{22}+|{a}_{54}|{x}_{14}+{Q}_{2}|{a}_{56}|{\delta }_{2,6}+{Q}_{2}|{a}_{57}|{\delta }_{2,7}$ ;

$X=diag\left(0.391034,0.372722,0.440299,0.413043,0.325,0.133992,0.140515\right)$,

$AX\in D$，则矩阵A是非奇异H-矩阵。

4. 结论

1) 通过例1表明，本文的迭代判定方法比文献 [1] 定理1和文献 [3] 定理1的非迭代的判定方法好，且迭代次数少于文献 [2] 和文献 [4] 定理中的迭代判定定理。

2) 通过例2表明，本文推论2和文献 [3] 定理1将非占优行指标集分为两个区间时，本文推论2的判定方法要优于文献 [3] 的定理1，且迭代一次就能判定，而文献 [2] 和文献 [4] 迭代一次时不能判定。

3) 通过例3表明，本文细分非占优行指标集区间后判定范围比文献 [3] 定理的判定范围更广，并且在m相同的情况下，本文定理1的迭代次数要少于文献 [2] 和文献 [4] 的判定定理。

NOTES

*通讯作者。

 [1] 王健, 徐仲, 陆全. 判定广义严格对角占优矩阵的一组新条件[J]. 计算数学, 2011, 33(3): 225-232. [2] 王健, 徐仲, 陆全. 广义严格对角占优矩阵判定的新迭代准则[J]. 应用数学学报, 2010, 33(6): 961-966. [3] 韩涛, 陆全, 徐仲, 杜永恩. 一组非奇异H-矩阵的新判据[J]. 工程数学学报, 2011, 28(4): 498-504. [4] 范迎松, 陆全, 徐仲, 高慧敏. 非奇异H-矩阵的一组细分迭代判别准则[J]. 工程数学学报, 2012, 29(6): 877-882. [5] 干泰彬, 黄廷祝. 非奇异H-矩阵的实用充分条件[J]. 计算数学, 2004, 26(1): 109-116. [6] 谢清明. 关于H-矩阵的实用判定的注记[J]. 应用数学学报, 2006, 29(6): 1080-1084. [7] Varga, R.S. (1976) On Recurring Theorems on Diagonal Dominance. Linear Algebra and Its Applications, 13, 1-9. https://doi.org/10.1016/0024-3795(76)90037-9 [8] 朱海, 王健, 廖貅武, 徐仲. 非奇H-矩阵的实用判别准则[J]. 数学的实践与认识, 2014, 44(7): 280-285. [9] 陈茜, 庹清. 非奇异H-矩阵的一组新判定法[J]. 工程数学学报, 2020, 37(3): 325-334. [10] 刘长太. 广义严格对角占优矩阵的简捷判据[J]. 数学的实践与认识, 2018, 48(22): 217-221. [11] 庹清, 朱砾, 刘建州. 一类非奇异H-矩阵判定的新条件[J]. 计算数学, 2008, 30(2): 177-182. [12] Gan, T.-B., Huang, T.-Z., Evans, D.J., et al. (2005) Sufficient Conditions for H-Matrices. International Journal of Computer Mathematics, 82, 247-258. https://doi.org/10.1080/00207160412331291053