﻿ 松弛模系矩阵分裂迭代法求解一类非线性互补问题
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 同济大学学报(自然科学版)  209, Vol. 47 Issue (2): 291-297.  DOI: 10.11908/J.ISSN.0253-374x.2019.02.019 0

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WANG Yan, YIN Junfeng, LI Rui. A Relaxation Modulus-based Matrix Splitting Iteration Method for a Class of Nonlinear Complementarity Problems[J]. Journal of Tongji University (Natural Science), 209, 47(2): 291-297. DOI: 10.11908/J.ISSN.0253-374x.2019.02.019

文章历史

1. 同济大学 数学科学学院，上海 200092;
2. 嘉兴学院 数理与信息工程学院，浙江 嘉兴 314001

A Relaxation Modulus-based Matrix Splitting Iteration Method for a Class of Nonlinear Complementarity Problems
WANG Yan 1, YIN Junfeng 1, LI Rui 1
1. School of Mathematical Sciences, Tongji University, Shanghai 200092, China;
2. College of Mathematics Physics and Information Engineering, Jiaxing University, Jiaxing 314001, China
Abstract: A relaxation modulus-based matrix splitting iteration method is proposed for solving a class of nonlinear complementarity problems. The convergence theory is established when the system matrix is H+-and the choice of relaxation parameters is given. Numerical examples show that the proposed methods are efficient and can accelerate the convergence performance of the modulus-based matrix splitting method with less iteration steps and CPU time.
Key words: matrix splitting    relaxation modulus-based iteration methods    nonlinear complementarity problems

 $\mathit{\boldsymbol{z}} \ge 0,\;\;\;\;\mathit{\boldsymbol{w}}: = \mathit{\boldsymbol{Mz}} + \mathit{\boldsymbol{q}} + f\left( \mathit{\boldsymbol{z}} \right) \ge 0,\;\;\;\;{\mathit{\boldsymbol{z}}^T}\mathit{\boldsymbol{w}} = 0$ (1)

1 松弛模系矩阵分裂迭代法

 $\begin{array}{*{20}{c}} {\left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }} + \mathit{\boldsymbol{F}}} \right){\mathit{\boldsymbol{x}}^{\left( k \right)}} = \mathit{\boldsymbol{G}}{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} + \left( {\mathit{\boldsymbol{ \boldsymbol{\varPsi} }} - \mathit{\boldsymbol{M}}} \right)\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| - }\\ {\gamma \left( {\mathit{\boldsymbol{q}} + f\left( {{\mathit{\boldsymbol{z}}^{\left( {k - 1} \right)}}} \right)} \right),} \end{array}$

 $\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)\mathit{\boldsymbol{x}} = {\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}\mathit{\boldsymbol{x}} + \left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right)\left| \mathit{\boldsymbol{x}} \right| - \left( {\mathit{\boldsymbol{q}} + f\left( \mathit{\boldsymbol{z}} \right)} \right)$ (2)

 $\left\{ \begin{array}{l} \left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{ \boldsymbol{F} }}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right){\mathit{\boldsymbol{x}}^{\left( {k - 1/2} \right)}} = {\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} + \left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right)\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;q - f\left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| + {\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right)} \right)\\ {\mathit{\boldsymbol{x}}^{\left( k \right)}} = \left( {I - \mathit{\boldsymbol{P}}} \right){\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} + \mathit{\boldsymbol{P}}{\mathit{\boldsymbol{x}}^{\left( {k - 1/2} \right)}} \end{array} \right.$ (3)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{x}}^{\left( k \right)}} = \left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{P}}} \right){\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} + \mathit{\boldsymbol{P}}{{\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)}^{ - 1}}\left[ {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} + } \right.}\\ {\left. {\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right)\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| - \mathit{\boldsymbol{q}} - f\left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| + {\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right)} \right)} \right]} \end{array}$ (4)

 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} = \frac{1}{\alpha }\left( {{\mathit{\boldsymbol{D}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}} - \mathit{\boldsymbol{\beta }}{\mathit{\boldsymbol{L}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}}} \right)\\ {\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} = \frac{1}{\alpha }\left( {\left( {1 - \alpha } \right){\mathit{\boldsymbol{D}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}} + \left( {\alpha - \beta } \right){\mathit{\boldsymbol{L}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}} + \alpha {\mathit{\boldsymbol{U}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}}} \right) \end{array} \right.$

α, β分别取α=β, α=β=1和α=1, β=0时，称为松弛模系逐步超松弛迭代格式(RMSOR)、松弛模系高斯赛德尔迭代格式(RMGS)和松弛模系雅克比迭代格式(RMJ).

$\mathit{\boldsymbol{ \boldsymbol{\varOmega} = }}\frac{1}{\mathit{\gamma }}\mathit{\boldsymbol{I, \boldsymbol{\varGamma} = }}\frac{1}{\mathit{\gamma }}\mathit{\boldsymbol{ \boldsymbol{\varPsi} }}$, P=I，算法2即为模系矩阵分裂迭代方法.

 $\left\{ \begin{array}{l} \left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right){\mathit{\boldsymbol{x}}^{\left( {k - 1/2} \right)}} = {\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} + \left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right)\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;q - f\left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| + {\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right)} \right)\\ {\mathit{\boldsymbol{x}}^{\left( k \right)}} = \left( {1 - \varepsilon } \right){\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} + \varepsilon {\mathit{\boldsymbol{x}}^{\left( {k - 1/2} \right)}} \end{array} \right.$ (5)

2 收敛性分析

H+-矩阵在数学物理问题、控制论、电力系统理论、经济数学以及弹性力学等众多领域中都有广泛应用，例如经济价值模型、反网络分析模型、经济学中的投入产出增长模型和概率统计中的Markov链等问题.

MRn×n为一个实n×n矩阵，它的比较矩阵〈M〉=(〈mij)定义为

 ${\left\langle m \right\rangle _{ij}} = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} \left| {{m_{ij}}} \right|,\\ - \left| {{m_{ij}}} \right|, \end{array}&\begin{array}{l} i = j,\\ i \ne j, \end{array} \end{array}} \right.\;\;\;\;i,j = 1,2, \cdots ,n$

 ${\left\| {{\mathit{\boldsymbol{M}}^{ - 1}}\mathit{\boldsymbol{N}}} \right\|_\infty } \le \mathop {\max }\limits_{1 \le i \le n} \frac{{{{\left( {\left| \mathit{\boldsymbol{N}} \right|\mathit{\boldsymbol{e}}} \right)}_i}}}{{{{\left( {\left\langle \mathit{\boldsymbol{M}} \right\rangle \mathit{\boldsymbol{e}}} \right)}_i}}}$

 ${\left[ {\left( {\left\langle {\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right\rangle + \left\langle {\mathit{\boldsymbol{F \boldsymbol{\varOmega} }}} \right\rangle - \left| {\mathit{\boldsymbol{G \boldsymbol{\varOmega} }}} \right|} \right)\mathit{\boldsymbol{D}}\mathit{e}} \right]_\mathit{i}} > 0,i = 1,2, \cdots ,n$

 $0 \le \frac{{{\rm{d}}{f_i}}}{{{\rm{d}}{z_i}}} \le {{\bar j}_i},\;\;\;i = 1,2, \cdots ,n$

 $\begin{array}{l} {\mathit{\boldsymbol{x}}_ * } = \left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{P}}} \right){\mathit{\boldsymbol{x}}_ * } + \mathit{\boldsymbol{P}}{\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)^{ - 1}}\left[ {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}{\mathit{\boldsymbol{x}}_ * } + } \right.\\ \;\;\;\left. {\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right)\left| {{\mathit{\boldsymbol{x}}_ * }} \right| - \mathit{\boldsymbol{q}} - f\left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| {{\mathit{\boldsymbol{x}}_ * }} \right| + {\mathit{\boldsymbol{x}}_ * }} \right)} \right)} \right] \end{array}$ (6)

 $\begin{array}{l} f\left( {{\mathit{\boldsymbol{z}}^{\left( k \right)}}} \right) - f\left( {{\mathit{\boldsymbol{z}}_ * }} \right) = f\left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| {{\mathit{\boldsymbol{x}}^{\left( k \right)}}} \right| + {\mathit{\boldsymbol{x}}^{\left( k \right)}}} \right)} \right) - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;f\left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| {{\mathit{\boldsymbol{x}}_ * }} \right| + {\mathit{\boldsymbol{x}}_ * }} \right)} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{J}}^{\left( k \right)}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| {{\mathit{\boldsymbol{x}}^{\left( k \right)}}} \right| - \left| {{\mathit{\boldsymbol{x}}_ * }} \right| + {\mathit{\boldsymbol{x}}^{\left( k \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right) \end{array}$

 $\begin{array}{l} {\mathit{\boldsymbol{x}}^{\left( k \right)}} - {\mathit{\boldsymbol{x}}_ * } = \left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{P}}} \right)\left( {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\boldsymbol{P}}{\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)^{ - 1}}\left[ {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}\left( {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right) + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right)\left( {\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| - \left| {{\mathit{\boldsymbol{x}}_ * }} \right|} \right) - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {f\left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| + {\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right)} \right) - } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {f\left( {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| {{\mathit{\boldsymbol{x}}_ * }} \right| + {\mathit{\boldsymbol{x}}_ * }} \right)} \right)} \right)} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{P}}} \right)\left( {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\boldsymbol{P}}{\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)^{ - 1}}\left[ {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}\left( {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right) + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right)\left( {\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| - \left| {{\mathit{\boldsymbol{x}}_ * }} \right|} \right) - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {{\mathit{\boldsymbol{J}}^{\left( {k - 1} \right)}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\left( {\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| - \left| {{\mathit{\boldsymbol{x}}_ * }} \right| + {\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right)} \right] \end{array}$ (7)

 $\begin{array}{l} \left| {{\mathit{\boldsymbol{x}}^{\left( k \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right| = \\ \;\;\left| {\mathit{\boldsymbol{P}}{{\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)}^{ - 1}}\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)\left( {{\mathit{\boldsymbol{P}}^{ - 1}} - \mathit{\boldsymbol{I}}} \right)\left( {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right) + } \right.\\ \;\;\mathit{\boldsymbol{P}}{\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)^{ - 1}}\left[ {\left( {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right)\left( {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right) + } \right.\\ \;\;\left. {\left. {\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{M \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right)\left( {\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}}} \right| - \left| {{\mathit{\boldsymbol{x}}_ * }} \right|} \right)} \right]} \right| \le \\ \;\;\mathit{\boldsymbol{P}}\left| {{{\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)}^{ - 1}}} \right|\left[ {\left| {\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)\left( {{\mathit{\boldsymbol{P}}^{ - 1}} - \mathit{\boldsymbol{I}}} \right) + {\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} - } \right.} \right.\\ \;\;\left. {\left. {\mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right| + \left| {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{M \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right|} \right]\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right| \le \\ \;\;\mathit{\boldsymbol{P}}{\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + \left\langle {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right\rangle } \right)^{ - 1}}\left[ {\left| {\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)\left( {{\mathit{\boldsymbol{P}}^{ - 1}} - \mathit{\boldsymbol{I}}} \right) + {\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} - } \right.} \right.\\ \;\;\left. {\left. {\mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right| + \left| {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{M \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right|} \right]\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right| = \\ \;\;\mathit{\boldsymbol{\hat L}}\left| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right| \end{array}$

 $\mathit{\boldsymbol{\tilde F}} = \mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + \left\langle {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right\rangle$ (8)
 $\begin{array}{l} \mathit{\boldsymbol{\tilde G}} = \left| {\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right)\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{P}}} \right) + {\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}\mathit{\boldsymbol{P}}} \right| + \\ \;\;\;\;\;\;\left| {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - \mathit{\boldsymbol{M \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right|\mathit{\boldsymbol{P}} \end{array}$ (9)

 $\left\{ \begin{array}{l} \left\{ {\left[ {2\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + 2{\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}} - \left( {2\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + \left| {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| + \left| {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| - } \right.} \right.} \right.\\ \;\;\;\;\;\;\;\;\;\;{\left. {\left. {\left. {\left\langle {\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right\rangle } \right)\mathit{\boldsymbol{P}}} \right]\mathit{\boldsymbol{De}}} \right\}_\mathit{\boldsymbol{i}}} > 0,\\ \;\;\;\;\;\;\;\;\;\;当\;{\left( {\mathit{\boldsymbol{Pe}}} \right)_i} \ge 1\;时\\ \left\{ {\left[ {\left( {\left\langle {\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right\rangle + \left| {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| - \left| {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right|} \right)\mathit{\boldsymbol{P}} - } \right.} \right.\\ \;\;\;\;\;\;\;\;\;\;{\left. {\left. {2\left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|} \right]\mathit{\boldsymbol{De}}} \right\}_\mathit{\boldsymbol{i}}} > 0,\\ \;\;\;\;\;\;\;\;\;\;当\;0 < {\left( {\mathit{\boldsymbol{Pe}}} \right)_i} < 1\;时 \end{array} \right.$ (10)

 $\begin{array}{l} {\left\| {{\mathit{\boldsymbol{D}}^{ - 1}}{\mathit{\boldsymbol{P}}^{ - 1}}\mathit{\boldsymbol{\hat LPD}}} \right\|_\infty } = {\left\| {{\mathit{\boldsymbol{D}}^{ - 1}}{\mathit{\boldsymbol{P}}^{ - 1}}\mathit{\boldsymbol{P}}{{\mathit{\boldsymbol{\tilde F}}}^{ - 1}}\mathit{\boldsymbol{\tilde GG}}{\mathit{\boldsymbol{P}}^{ - 1}}\mathit{\boldsymbol{PD}}} \right\|_\infty } = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left\| {{\mathit{\boldsymbol{D}}^{ - 1}}{{\mathit{\boldsymbol{\tilde F}}}^{ - 1}}\mathit{\boldsymbol{\tilde GD}}} \right\|_\infty } = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left\| {{{\left( {\mathit{\boldsymbol{\tilde FD}}} \right)}^{ - 1}}\mathit{\boldsymbol{\tilde GD}}} \right\|_\infty } \end{array}$

 ${\left\| {{\mathit{\boldsymbol{D}}^{ - 1}}{\mathit{\boldsymbol{P}}^{ - 1}}\mathit{\boldsymbol{LPD}}} \right\|_\infty } = {\left\| {{{\left( {\mathit{\boldsymbol{\tilde FD}}} \right)}^{ - 1}}\mathit{\boldsymbol{\tilde GD}}} \right\|_\infty } \le \mathop {\max }\limits_{1 \le i \le n} \frac{{{{\left( {\mathit{\boldsymbol{\tilde GDe}}} \right)}_i}}}{{{{\left( {\mathit{\boldsymbol{\tilde FDe}}} \right)}_i}}}$

 $\begin{array}{l} {\mathit{\boldsymbol{D}}_{\mathit{\boldsymbol{\tilde G}}}} \le \left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right)\left| {\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{I}}} \right| + \left| {{\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right|\mathit{\boldsymbol{P}} + \\ \;\;\;\;\;\;\;\;\left| {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{D}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right|\mathit{\boldsymbol{P}} \end{array}$

${\mathit{\boldsymbol{B}}_{\mathit{\boldsymbol{\tilde G}}}}$的对角元满足

 $\begin{array}{l} {\mathit{\boldsymbol{B}}_{\mathit{\boldsymbol{\tilde G}}}} = - \left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{P}}} \right) + {\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}\mathit{\boldsymbol{P}}} \right| - \left| {{\mathit{\boldsymbol{B}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}}} \right|\mathit{\boldsymbol{P}} \ge \\ \;\;\;\;\;\;\;\; - \left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\left| {\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{I}}} \right| - \left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\mathit{\boldsymbol{P}} - \left| {{\mathit{\boldsymbol{B}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}}} \right|\mathit{\boldsymbol{P}} \end{array}$

 $\begin{array}{l} {\left( {\mathit{\boldsymbol{\tilde FDe}}} \right)_i} - {\left( {\mathit{\boldsymbol{\tilde GDe}}} \right)_i} = {\left[ {\left( {\mathit{\boldsymbol{\tilde F}} - {\mathit{\boldsymbol{D}}_{\mathit{\boldsymbol{\tilde G}}}} + {\mathit{\boldsymbol{B}}_{\mathit{\boldsymbol{\tilde G}}}}} \right)\mathit{\boldsymbol{De}}} \right]_i} \ge \\ \left\{ {\left[ {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + \left\langle {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right\rangle - \left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right)\left| {\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{I}}} \right| - } \right.} \right.\\ \left| {{\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right|\mathit{\boldsymbol{P}} - \left| {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - {\mathit{\boldsymbol{D}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right|\mathit{\boldsymbol{P}} - \\ {\left. {\left. {\left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\left| {\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{I}}} \right| - \left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\mathit{\boldsymbol{P}} - \left| {{\mathit{\boldsymbol{B}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}}} \right|\mathit{\boldsymbol{P}}} \right]\mathit{\boldsymbol{De}}} \right\}_i}. \end{array}$ (11)

 $\begin{array}{l} {\left( {\mathit{\boldsymbol{\tilde FDe}}} \right)_i} - {\left( {\mathit{\boldsymbol{\tilde GDe}}} \right)_i} \ge \left\{ {\left[ {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + \left\langle {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right\rangle - } \right.} \right.\\ \;\;\;\;\;\;\;\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right)\left( {\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{I}}} \right) - \left| {{\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right|\mathit{\boldsymbol{P}} - \\ \;\;\;\;\;\;\;\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - {\mathit{\boldsymbol{D}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right)\mathit{\boldsymbol{P}} - \left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\left( {\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{I}}} \right) - \\ \;\;\;\;\;\;\;{\left. {\left. {\left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\mathit{\boldsymbol{P}} - \left| {{\mathit{\boldsymbol{B}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}}} \right|\mathit{\boldsymbol{P}}} \right]\mathit{\boldsymbol{De}}} \right\}_i} \ge \left\{ {\left[ {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + \left\langle {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right\rangle - } \right.} \right.\\ \;\;\;\;\;\;\;\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right)\left( {\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{I}}} \right) - \left| {{\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\mathit{\boldsymbol{P}} - \left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - {\mathit{\boldsymbol{D}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}}} \right)\mathit{\boldsymbol{P}} - \\ \;\;\;\;\;\;\;{\left. {\left. {\left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\left( {\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{I}}} \right) - \left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\mathit{\boldsymbol{P}} - \left| {{\mathit{\boldsymbol{B}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}}} \right|\mathit{\boldsymbol{P}}} \right]\mathit{\boldsymbol{De}}} \right\}_i} = \\ \;\;\;\;\;\;\;\left\{ {\left[ {2\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + 2{\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}} - \left( {2\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + \left| {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| + \left| {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| - } \right.} \right.} \right.\\ \;\;\;\;\;\;\;{\left. {\left. {\left. {\left\langle {\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right\rangle } \right)\mathit{\boldsymbol{P}}} \right]\mathit{\boldsymbol{De}}} \right\}_i} > 0 \end{array}$ (12)

 $\begin{array}{l} {\left( {\mathit{\boldsymbol{\tilde FDe}}} \right)_i} - {\left( {\mathit{\boldsymbol{\tilde GDe}}} \right)_i} \ge \left\{ {\left[ {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + \left\langle {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right\rangle - } \right.} \right.\\ \;\;\;\;\;\;\;\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right)\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{P}}} \right) - \left| {{\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right|\mathit{\boldsymbol{P}} - \\ \;\;\;\;\;\;\;\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - {\mathit{\boldsymbol{D}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}} - \mathit{\boldsymbol{J}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\left( {k - 1} \right)}} \right)\mathit{\boldsymbol{P}} - \left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{P}}} \right) - \\ \;\;\;\;\;\;\;{\left. {\left. {\left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\mathit{\boldsymbol{P}} - \left| {{\mathit{\boldsymbol{B}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}}} \right|\mathit{\boldsymbol{P}}} \right]\mathit{\boldsymbol{De}}} \right\}_i} \ge \left\{ {\left[ {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + \left\langle {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right\rangle - } \right.} \right.\\ \;\;\;\;\;\;\;\left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + {\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right)\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{P}}} \right) - \left| {{\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\mathit{\boldsymbol{P}} - \left( {\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} - {\mathit{\boldsymbol{D}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}}} \right)\mathit{\boldsymbol{P}} - \\ \;\;\;\;\;\;\;{\left. {\left. {\left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\left( {\mathit{\boldsymbol{I}} - \mathit{\boldsymbol{P}}} \right) - \left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\mathit{\boldsymbol{P}} - \left| {{\mathit{\boldsymbol{B}}_{\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}}}} \right|\mathit{\boldsymbol{P}}} \right]\mathit{\boldsymbol{De}}} \right\}_i} = \\ \;\;\;\;\;\;\;\left\{ {\left[ {\left( {\left\langle {\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right\rangle + \left| {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| - \left| {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right|} \right)\mathit{\boldsymbol{P}} - } \right.} \right.\\ \;\;\;\;\;\;\;{\left. {\left. {2\left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|} \right]\mathit{\boldsymbol{De}}} \right\}_i} > 0. \end{array}$ (13)

 ${a_i} < \varepsilon < {b_i}$

 $\begin{array}{*{20}{c}} {{a_i} = \frac{{{{\left( {2\left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\mathit{\boldsymbol{De}}} \right)}_i}}}{{{{\left[ {\left( {\left\langle {\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right\rangle + \left| {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| - \left| {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right|} \right)\mathit{\boldsymbol{De}}} \right]}_i}}}}\\ {{b_i} = \frac{{{{\left[ {\left( {2\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + 2{\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right)\mathit{\boldsymbol{De}}} \right]}_i}}}{{{{\left[ {\left( {2\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + \left| {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| + \left| {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| - \left\langle {\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right\rangle } \right)\mathit{\boldsymbol{De}}} \right]}_i}}}}\\ {i = 1,2, \cdots ,n} \end{array}$ (14)

 $\begin{array}{l} {\left[ {\left( {2\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + 2{\mathit{\boldsymbol{D}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right)\mathit{\boldsymbol{De}}} \right]_i} - \left[ {\left( {2\mathit{\boldsymbol{ \boldsymbol{\varGamma} }} + \left| {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| + \left| {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| - } \right.} \right.\\ \;\;\;\;\;\;\;{\left. {\left. {\left\langle {\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right\rangle } \right)\mathit{\boldsymbol{De}}} \right]_i} = \left[ {\left( {\left\langle {\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right\rangle + \left| {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| - } \right.} \right.\\ \;\;\;\;\;\;\;{\left. {\left. {\left| {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right|} \right)\mathit{\boldsymbol{De}}} \right]_i} > 0 \end{array}$

 $\begin{array}{l} {\left[ {\left( {\left\langle {\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right\rangle + \left| {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| - \left| {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right|} \right)\mathit{\boldsymbol{De}}} \right]_i} - 2{\left( {\left| {{\mathit{\boldsymbol{B}}_{{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}}}} \right|\mathit{\boldsymbol{De}}} \right)_i} = \\ \;\;\;\;\;\;\;\;\;{\left[ {\left( {\left\langle {\mathit{\boldsymbol{M \boldsymbol{\varOmega} }}} \right\rangle + \left| {{\mathit{\boldsymbol{F}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right| - \left| {{\mathit{\boldsymbol{G}}_\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}} \right|} \right)\mathit{\boldsymbol{De}}} \right]_i} > 0 \end{array}$

 $\varepsilon = 1 + \frac{{{{\left( {{\mathit{\boldsymbol{x}}^{\left( {k - 1/2} \right)}} - {\mathit{\boldsymbol{x}}_ * }} \right)}^{\rm{T}}}\left( {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}^{\left( {k - 1/2} \right)}}} \right)}}{{\left\| {{\mathit{\boldsymbol{x}}^{\left( {k - 1} \right)}} - {\mathit{\boldsymbol{x}}^{\left( {k - 1/2} \right)}}} \right\|_2^2}}$

 ${\varepsilon ^{\left( k \right)}} = 1 + \frac{{{{\left( {{\mathit{\boldsymbol{x}}^{\left( {k - 3/2} \right)}} - {\mathit{\boldsymbol{x}}^{\left( {k - 1/2} \right)}}} \right)}^{\rm{T}}}\left( {{\mathit{\boldsymbol{x}}^{\left( {k - 2} \right)}} - {\mathit{\boldsymbol{x}}^{\left( {k - 3/2} \right)}}} \right)}}{{\left\| {{\mathit{\boldsymbol{x}}^{\left( {k - 2} \right)}} - {\mathit{\boldsymbol{x}}^{\left( {k - 3/2} \right)}}} \right\|_2^2}}$ (15)

 $\varepsilon = \left\{ \begin{array}{l} 1,\;\;\;\;\;\;\;\;当\;k = 1\;时\\ {\varepsilon ^{\left( k \right)}},\;\;\;\;\;当\;k > 1\;时\;{\varepsilon ^{\left( k \right)}} \in \left( {a,b} \right)\\ a,\;\;\;\;\;\;\;\;当\;k > 1\;时\;{\varepsilon ^{\left( k \right)}} \le a\\ b,\;\;\;\;\;\;\;\;当\;k > 1\;时\;{\varepsilon ^{\left( k \right)}} \ge b \end{array} \right.$ (16)

3 数值实验

 $\mathit{\boldsymbol{E}}\left( {{\mathit{\boldsymbol{z}}^{\left( k \right)}}} \right): = \frac{{{{\left\| {\min \left( {\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{z}}^{\left( k \right)}} + \mathit{\boldsymbol{q}} + \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{z}}^{\left( k \right)}}} \right),{\mathit{\boldsymbol{z}}^{\left( k \right)}}} \right)} \right\|}_2}}}{{{{\left\| {\min \left( {\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{z}}^{\left( 0 \right)}} + \mathit{\boldsymbol{q}} + \mathit{\boldsymbol{f}}\left( {{\mathit{\boldsymbol{z}}^{\left( 0 \right)}}} \right),{\mathit{\boldsymbol{z}}^{\left( 0 \right)}}} \right)} \right\|}_2}}} \le {10^{ - 6}}$

 $\mathit{\boldsymbol{\hat M}} = \left( {\begin{array}{*{20}{c}} \mathit{\boldsymbol{S}}&{ - \mathit{\boldsymbol{I}}}&0& \cdots &0\\ { - \mathit{\boldsymbol{I}}}&\mathit{\boldsymbol{S}}&{}&{}& \vdots \\ 0&{}&{}&{}&0\\ \vdots &{}&{}&S&{ - I}\\ 0& \cdots &0&{ - I}&S \end{array}} \right) \in {{\bf{R}}^{n \times n}}$

 $q = {\left( { - 1,1, \cdots ,{{\left( { - 1} \right)}^n}} \right)^{\rm{T}}} \in {{\bf{R}}^n},$
 $\mathit{\boldsymbol{f}}\left( \mathit{\boldsymbol{z}} \right) = {\left( {\sqrt {\mathit{\boldsymbol{z}}_1^2 + 0.01} , \cdots ,\sqrt {\mathit{\boldsymbol{z}}_\mathit{\boldsymbol{n}}^2 + 0.01} } \right)^{\rm{T}}} \in {{\bf{R}}^n}.$

 $\begin{array}{*{20}{c}} {\left( {0,1.200\;0} \right),\left( {0.333\;3,1.200\;0} \right),}\\ {\left( {0.428\;6,1.133\;3} \right)} \end{array}$ (17)

 $\left( {0,1.333\;3} \right),\left( {0,1.333\;3} \right),\left( {0,1.259\;3} \right)$ (18)

 $\mathit{\boldsymbol{M}} = \left( {\begin{array}{*{20}{c}} \mathit{\boldsymbol{S}}&{ - \mathit{\boldsymbol{I}}}&{ - \mathit{\boldsymbol{I}}}& \cdots &{}&0\\ {}&\mathit{\boldsymbol{S}}&{ - \mathit{\boldsymbol{I}}}&{ - \mathit{\boldsymbol{I}}}&{}&{}\\ \vdots &{}&{}&{}&{}& \vdots \\ {}&{}&{}&\mathit{\boldsymbol{S}}&{ - \mathit{\boldsymbol{I}}}&{ - \mathit{\boldsymbol{I}}}\\ {}&{}&{}&{}&\mathit{\boldsymbol{S}}&{ - \mathit{\boldsymbol{I}}}\\ 0&{}& \cdots &{}&{}&\mathit{\boldsymbol{S}} \end{array}} \right) \in {{\bf{R}}^{n \times n}}$

 $\begin{array}{l} \mathit{\boldsymbol{q}} = {\left( { - 1,1, \cdots ,{{\left( { - 1} \right)}^n}} \right)^{\rm{T}}} \in {{\bf{R}}^n},f\left( \mathit{\boldsymbol{z}} \right) = \\ \;\;\;\;\;{\left( {{z_1} - \sin {z_1}, \cdots ,{z_n} - \sin {z_n}} \right)^{\rm{T}}} \in {{\bf{R}}^n}. \end{array}$

 $\left( {0,1.333\;3} \right),\left( {0,1.333\;3} \right),\left( {0,1.259\;3} \right)$ (18)
 $\left( {0.748\;9,1.062\;5} \right)$ (19)

 $\left( {0,1.294\;8} \right),\left( {0,1.294\;8} \right),\left( {0,1.199\;1} \right)$ (20)

 图 1 例1两种算法的残差比较 Fig.1 Residual comparison of two algorithms for Example 1
 图 2 例2两种算法的残差比较 Fig.2 Residual comparison of two algorithms for Example 2
4 结论

 [1] ORTEGA J M, RHEINBOLDT W C. Iterative solution of nonlinear equations in several variables[M]. New York: Academic Press, 1970 [2] BAI Z Z. Parallel nonlinear AOR method and its convergence[J]. Computers and Mahtematics with Applications, 1996, 31(2): 21 DOI:10.1016/0898-1221(95)00190-5 [3] FERRIS M C, PANG J S. Engineering and economic applications of complementarity problems[J]. SIAM Review, 1997, 39(4): 669 DOI:10.1137/S0036144595285963 [4] COTTLE R W, PANG J S, STONE R E. The linear complementarity problem[M]. San Diego: Academic Press, 2009 [5] BAI Z Z. Modulus-based matrix splitting iteration methods for linear complementarity problems[J]. Numerical Linear Algebra with Applications, 2010, 17(6): 917 DOI:10.1002/nla.v17.6 [6] ZHANG L L. Two-step modulus-based matrix splitting iteration method for linear complementarity problems[J]. Numerical Algorithms, 2011, 57(1): 83 DOI:10.1007/s11075-010-9416-7 [7] ZHENG N, YIN J F. Accelerated modulus-based matrix splitting iteration methods for linear complementarity problems[J]. Numerical Algorithms, 2013, 64(2): 245 DOI:10.1007/s11075-012-9664-9 [8] ZHENG H, LI W, VONG S. A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems[J]. Numerical Algorithms, 2017, 74(1): 137 DOI:10.1007/s11075-016-0142-7 [9] BAI Z Z, ZHANG L L. Modulus-based synchronous multisplitting iteration methods for linear complementarity problems[J]. Numerical Linear Algebra with Applications, 2013, 20(3): 425 DOI:10.1002/nla.1835 [10] BAI Z Z, ZHANG L L. Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems[J]. Numerical Algorithms, 2013, 62(1): 59 DOI:10.1007/s11075-012-9566-x [11] ZHANG L L. Two-step modulus-based synchronous multisplitting iteration methods for linear complementarity problems[J]. Journal of Computational Mathematics, 2015, 33(1): 100 DOI:10.4208/jcm [12] XIA Z C, LI C L. Modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problem[J]. Applied Mathematics and Computation, 2015, 271: 34 DOI:10.1016/j.amc.2015.08.108 [13] LI R, YIN J F. Accelerated modulus-based matrix splitting iteration methods for a class of restricted nonlinear complementarity problems[J]. Numerical Algorithms, 2017, 75(2): 339 DOI:10.1007/s11075-016-0243-3 [14] HU J G. Estimates of ‖B-1C‖∞ and their applications[J]. Mathematica Numerica Sinica, 1982, 4: 272 [15] BERMAN A, PLEMMONS R J. Nonnegative matrix in the mathematical science[M]. Philadelphia: SIAM Publisher, 1994