% Copyright (C) 2018 Liu Yifan, Zhao Tianyu 
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\begin{document}
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        %   求解多右端线性方程组的一种改进共轭梯度算法研究\\
        %   %{\bfseries \zihao{2} !如果不多加这一行字复制进Word会乱码,导致不能查重!}
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	\begin{abstract}
	\zihao{-4}{
	目前在众多学科领域都需要求解多右端项线性方程组,本文
	..
	
	..
	
	..
	到校正矩阵Frobenius范数的最小值。最后通过Julia程序对其进行对比数值试验。
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	{\bfseries  关键词: 关键词1,关键词2} }
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	\begin{abstract}
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	At present, it is necessary to solve multi-right-end linear equations in many disciplines. This paper 
	..
	
	..
	algorithms can converge to the minimum of the Frobenius norm value.
	\\
	\\
	{\bf Keywords: 
KW1,KW2}     }
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	\zihao{3} \bfseries  论文题目
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	\section*{结论}	
	
	
	\addcontentsline{toc}{section}{结论}
	本文从最优化理论出发
    \begin{enumerate}
    \item 针对基于Krylov子空间的迭代算法——共轭梯度算法,
    \item 在阐述清楚共轭
    \item 最后讨论针...进而给出两者的等价性证明。
    \end{enumerate}
	
	
	
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	\addcontentsline{toc}{section}{参\ 考\ 文\ 献}
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	\section*{\centerline{致\hspace*{3em}谢}}
	\addcontentsline{toc}{section}{致\hspace*{1em}谢}
	%\zihao{-4}{
	四年的大学时光流逝的飞快,非常荣幸能够有机会在大学的最后时刻完成这篇致谢,向一直帮助我的师长、家人和朋友们表达我最衷心的感谢。回顾大学四年的学习与生活,我收获了许多,也成长了许多。
	
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	\section*{\centerline{附\hspace*{3em}录}}
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