├── .DS_Store ├── README.md ├── project1 ├── .DS_Store ├── Programe1.pdf ├── code │ ├── ex1.m │ ├── ex2.m │ ├── f.m │ ├── lagrange.m │ ├── newtonInterpol.m │ └── result │ │ ├── base0.png │ │ ├── base0f.png │ │ ├── f.png │ │ ├── lagche.png │ │ ├── lagchef.png │ │ ├── lagchef1.png │ │ ├── lagrunge.png │ │ ├── lagrunge2.png │ │ ├── linf.png │ │ ├── linrunge.png │ │ ├── newtonf.png │ │ ├── newtonf2.png │ │ ├── newtonrunge.png │ │ ├── newtonrunge2.png │ │ ├── runge.png │ │ ├── splinef.png │ │ ├── splinef1.png │ │ └── splinerunge.png └── report │ ├── Tsinghua2.png │ ├── main.aux │ ├── main.log │ ├── main.out │ ├── main.pdf │ ├── main.synctex.gz │ ├── main.tex │ └── main.toc ├── project2 ├── .DS_Store ├── code │ ├── .DS_Store │ ├── Gauss.m │ ├── GaussSeidel.m │ ├── Hilbert.m │ ├── Jacobi.m │ ├── SOR.m │ ├── bestOmega.m │ ├── ex1.m │ ├── ex2.m │ └── result │ │ ├── acond2.png │ │ ├── alogcond2.png │ │ ├── bestw.png │ │ ├── cond2.png │ │ ├── cond2fit.png │ │ ├── er10.png │ │ ├── er100.png │ │ ├── er50.png │ │ ├── gausserror.png │ │ ├── gsrho.png │ │ ├── gsrho2.png │ │ ├── jacobirho.png │ │ ├── jacobirho2.png │ │ ├── logcond2.png │ │ ├── sorrho.png │ │ └── sorrho2.png ├── progrem2.pdf └── report │ ├── .gitignore │ ├── Tsinghua2.png │ ├── main.pdf │ └── main.tex ├── 作业答案(参考) ├── Krylov子空间方法作业参考答案_906106982.pdf ├── eigth homework.pdf ├── second homework.pdf ├── time5.pdf ├── time6.pdf ├── 第一次作业_594105971.pdf ├── 第七次作业参考答案.pdf └── 第三次作业参考答案.pdf ├── 期中考题 └── 期中考试题目.pdf ├── 期末考题 └── 期末考试题目.pdf └── 课件 ├── FPIM-an.pdf ├── approximation2018ana.pdf ├── directmethod-an18.pdf ├── eigenvalue-an.pdf ├── interpolation-integration18a.pdf ├── introduction-2018a.pdf ├── krylov18.pdf └── lapre18a.pdf /.DS_Store: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/luoyt14/AdvancedNumericalAnalysis/HEAD/.DS_Store -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # 高等数值分析参考资料 2 | 3 | ## 大作业 4 | 5 | 1. 插值程序题project1 6 | 2. 病态线性方程组的求解问题project2 7 | 3. 使用Matlab实现 8 | 9 | ## 课件 10 | 11 | ## 2018年秋考题 12 | 13 | ## 作业参考答案(老师给的数值分析的答案，与高等数值分析作业基本一样) 14 | -------------------------------------------------------------------------------- /project1/.DS_Store: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/luoyt14/AdvancedNumericalAnalysis/HEAD/project1/.DS_Store -------------------------------------------------------------------------------- /project1/Programe1.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/luoyt14/AdvancedNumericalAnalysis/HEAD/project1/Programe1.pdf -------------------------------------------------------------------------------- /project1/code/ex1.m: 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<../code/result/lagrunge2.png> 1160 | 1161 | LaTeX Warning: `!h' float specifier changed to `!ht'. 1162 | 1163 | [3] 1164 | 1165 | LaTeX Warning: Text page 4 contains only floats. 1166 | 1167 | [4] 1168 | File: ../code/result/base0.png Graphic file (type bmp) 1169 | <../code/result/base0.png> 1170 | 1171 | 1172 | LaTeX Warning: `!h' float specifier changed to `!ht'. 1173 | 1174 | 1175 | Overfull \hbox (1.29395pt too wide) in paragraph at lines 155--156 1176 | []\TU/SimSun(0)/m/n/10.53937 其 次 我 们 使 用 \TU/lmr/m/n/10.53937 Chebyshev \TU/Sim 1177 | Sun(0)/m/n/10.53937 多 项 式 零 点 进 行 插 值，| 插 值 节 点 为 $\OML/cmm/m/it/10.53937 x[] 1178 | \OT1/cmr/m/n/10.53937 = \OML/cmm/m/it/10.53937 cos\OT1/cmr/m/n/10.53937 ([]\OML 1179 | /cmm/m/it/10.53937 ^^Y\OT1/cmr/m/n/10.53937 )(\OML/cmm/m/it/10.53937 i \OT1/cmr 1180 | /m/n/10.53937 = 0\OML/cmm/m/it/10.53937 ; \OT1/cmr/m/n/10.53937 1\OML/cmm/m/it/ 1181 | 10.53937 ; \OT1/cmr/m/n/10.53937 2\OML/cmm/m/it/10.53937 ; [] ; \OT1/cmr/m/n/1 1182 | 0.53937 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-------------------------------------------------------------------------------- /project1/report/main.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/luoyt14/AdvancedNumericalAnalysis/HEAD/project1/report/main.pdf -------------------------------------------------------------------------------- /project1/report/main.synctex.gz: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/luoyt14/AdvancedNumericalAnalysis/HEAD/project1/report/main.synctex.gz -------------------------------------------------------------------------------- /project1/report/main.tex: -------------------------------------------------------------------------------- 1 | \documentclass[a4paper]{article} 2 | 3 | \usepackage{fullpage} % Package to use full page 4 | \usepackage{parskip} % Package to tweak paragraph skipping 5 | \usepackage{tikz} % Package for drawing 6 | \usepackage{amsmath} 7 | \usepackage{hyperref} 8 | \usepackage{ctex} 9 | \usepackage{amssymb} 10 | \usepackage{amsthm} 11 | \usepackage{indentfirst} 12 | \setlength{\parindent}{2em} 13 | \usepackage{array} 14 | \usepackage{graphicx} 15 | \usepackage{float} 16 | \usepackage{subfigure} 17 | 18 | \renewcommand\thefigure{\thesection.\arabic{figure}} 19 | \makeatletter 20 | \@addtoreset{figure}{section} 21 | \makeatother 22 | 23 | \renewcommand\thesubfigure{\thesection.\arabic{figure}(\alph{subfigure})} 24 | \makeatletter 25 | \@addtoreset{figure}{section} 26 | \makeatother 27 | 28 | \makeatletter 29 | \renewcommand \theequation {% 30 | \ifnum \c@section>\z@ \@arabic\c@section.\fi \ifnum \c@subsection>\z@ 31 | \@arabic\c@subsection.\fi\ifnum \c@subsubsection>\z@ 32 | \@arabic\c@subsubsection.\fi\@arabic\c@equation} 33 | \@addtoreset{equation}{section} 34 | \@addtoreset{equation}{subsection} 35 | %\setcounter{section}{-1} 36 | \makeatother 37 | \newtheorem{theorem}{\hspace{2em}定理}[section] 38 | \hypersetup{ 39 | colorlinks=true, 40 | linkcolor=black 41 | } 42 | 43 | \title{高等数值分析} 44 | \author{罗雁天 \\ 45 | 2018310742} 46 | \date{\today} 47 | 48 | \begin{document} 49 | 50 | %\maketitle 51 | \newcommand{\HRule}{\rule{\linewidth}{0.5mm}} 52 | \begin{titlepage} 53 | \begin{center} 54 | % Upper part of the page 55 | \includegraphics[width=0.4\textwidth]{Tsinghua2.png}\\[1cm] 56 | \textsc{\Large \texttt{高等数值分析}}\\[1cm] 57 | % Title 58 | \HRule \\[1cm] 59 | {\Huge \bfseries 插值程序题}\\[0.4cm] 60 | \HRule \\[3.5cm] 61 | % Author and supervisor 62 | \begin{minipage}{0.4\textwidth} 63 | \begin{center} 64 | \Large 65 | \begin{tabular}{cc} 66 | \texttt{作者：} & 罗雁天 \\[0.5cm] 67 | \texttt{学号：} & 2018310742 \\[0.5cm] 68 | \texttt{日期：} & \today 69 | \end{tabular} 70 | \end{center} 71 | \end{minipage} 72 | \vfill 73 | \end{center} 74 | \end{titlepage} 75 | 76 | \tableofcontents 77 | \newpage 78 | 79 | \section{题目描述} 80 | 用多项式函数逼近一般的函数是数值计算的一类基本问题，一来多项式函数形式简单，计算只需有限次加、减、乘、除就可完成，且多项式的导数和原函数还是多项式函数，在不考虑舍入误差时可以在计算机上准确表达和运算；二来，多项式插值来逼近一般的函数还可以用于积分、微分等的运算。 81 | 82 | 在本次大作业中，我们将利用Newton插值、Lagrange插值、分段线性插值、三次自然样条插值4种插值方式对Runge函数($R(x)=1/(1+25x^2)$)和自定义分段函数$f(x)$(式\ref{eq1})在区间$[-1,1]$上进行插值，并对结果进行分析。 83 | 84 | \begin{equation} 85 | \label{eq1} 86 | f(x)=\left\{ 87 | \begin{array}{cc} 88 | \sin \pi x & -1\le x< 0, \\ 89 | \cos \pi x & 0\le x< 1/2,\\ 90 | 0 & 1/2\le x \le 1 91 | \end{array} 92 | \right. 93 | \end{equation} 94 | 95 | \section{Runge函数的插值分析} 96 | 我们首先绘制出Runge函数的真实图像如图\ref{fig:1}所示，其中采样间隔为0.01，一共201个采样点，以此作为我们的标准图像，之后使用4种插值逼近Runge函数的图像与此图像进行对比。 97 | 98 | \begin{figure}[!h] 99 | \centering 100 | \includegraphics[width=0.8\textwidth]{../code/result/runge} 101 | \caption{\label{fig:1}Runge函数真实图像} 102 | \end{figure} 103 | 104 | \subsection{Newton插值} 105 | 使用等距节点$x_i=-1+ih,h=0.1,0\le i\le 20$，对Runge函数进行20次Newton插值。同样选取采样间隔为0.01，一共201个采样点上绘制出20次等距节点Newton插值Runge函数的图像如图\ref{fig:2}所示。可以看出在边界的地方出现了Gibbs现象，与原图像相差极大，由于Newton插值与Lagrange插值等价，因此，我们将此部分的误差分析统一放置到Lagrange插值部分进行分析。从图\ref{fig:2}中我们不能清楚的看出中间部分的插值效果，因此我们将中间局部的插值图像和原图像绘制到一张图上如图\ref{fig:3}所示，从中可以看出，在中间部分插值的结果和原图像是非常相近的。综上，等距节点的Newton插值在非边界部分的逼近效果较好，在边界部分的误差很大，所以等距节点的Newton插值可以求区间内部的值，在边界附近的值误差很大。 106 | \begin{figure}[!h] 107 | \centering 108 | \includegraphics[width=0.85\textwidth]{../code/result/newtonrunge} 109 | \caption{\label{fig:2}20次Newton等距节点插值Runge函数图像} 110 | \end{figure} 111 | 112 | \begin{figure}[!h] 113 | \centering 114 | \includegraphics[width=0.85\textwidth]{../code/result/newtonrunge2} 115 | \caption{\label{fig:3}20次Newton等距节点插值Runge函数局部图像与原图像对比} 116 | \end{figure} 117 | 118 | \subsection{Lagrange插值} 119 | 为讨论Newton插值的误差，首先我们使用等距节点$x_i=-1+ih,h=0.1,0\le i\le 20$，对Runge函数进行20次Lagrange插值。同样选取采样间隔为0.01，一共201个采样点上绘制出20次等距节点Lagrange插值Runge函数的图像如图\ref{fig:4}所示。为了能够看清区间内部的插值效果，我们绘制出局部图与原图的对比图如图\ref{fig:5}所示，从中可以看出，在中间部分插值的结果和原图像是非常相近的。从图\ref{fig:4}中可以看出在边界的地方出现了与Newton插值一样的Gibbs现象，与原图像相差极大。 120 | 121 | 122 | \begin{figure}[!h] 123 | \centering 124 | \includegraphics[width=0.85\textwidth]{../code/result/lagrunge} 125 | \caption{\label{fig:4}20次Lagrange等距节点插值Runge函数图像} 126 | \end{figure} 127 | 128 | \begin{figure}[!h] 129 | \centering 130 | \includegraphics[width=0.85\textwidth]{../code/result/lagrunge2} 131 | \caption{\label{fig:5}20次Lagrange等距节点插值Runge函数局部图像与原图像对比} 132 | \end{figure} 133 | 134 | 分析其原因，由于插值函数不可避免存在误差，设$\hat{f_i}=f_i+\epsilon$是扰动后的值，而$\hat{L}_n(x)$是以$\hat{f_0},\hat{f_1},\cdots,\hat{f_n}$为插值数据的多项式，则: 135 | 136 | \begin{equation} 137 | f(x)-\hat{L}_n(x)=f(x)-L_n(x)+[L_n(x)-\hat{L}_n(x)] 138 | \end{equation} 139 | 而 140 | \begin{equation} 141 | L_n(x)-\hat{L}_n(x)=\sum_{j=0}^{n}\epsilon_jl_j(x) 142 | \end{equation} 143 | 假定$n=2m+1,\epsilon_m\ne 0$,其他$\epsilon_j=0$，则 144 | \begin{equation} 145 | |f(x)-L_n(x)|-|f(x)-\hat{L}_n(x)|=\epsilon_ml_m(x) 146 | \end{equation} 147 | 由此可以看出，若$l_m(x)$在某些点$x^*$很大，那么$\epsilon_ml_m(x^*)$也变得非常大。这就意味着即使是函数值的微小扰动也将带来插值函数的巨大变化，误差会过分放大。因此我们绘制出在$x=0$点处的节点基函数如图\ref{fig:6}所示，从此图中可以看出，在边界点处的值很大，由此误差被过分放大，出现了等距节点Lagrange插值和Newton插值中的Gibbs现象。 148 | 149 | \begin{figure}[!h] 150 | \centering 151 | \includegraphics[width=0.85\textwidth]{../code/result/base0} 152 | \caption{\label{fig:6}x=0处的节点基函数} 153 | \end{figure} 154 | 155 | 其次我们使用Chebyshev多项式零点进行插值，插值节点为$x_i=cos(\frac{2i+1}{42}\pi)(i=0,1,2,\cdots,20)$.绘制出插值之后的图像如图\ref{fig:7}所示，从图中可以看出，使用Chebyshev多项式的零点作为插值节点插值得到的图像比使用等距节点插值得到的图像效果好很多，在边界点虽然也有波动，但是幅度已经很小了。 156 | 157 | \begin{figure}[!h] 158 | \centering 159 | \includegraphics[width=0.85\textwidth]{../code/result/lagche} 160 | \caption{\label{fig:7}Chebyshev多项式零点Lagrange插值Runge函数图像} 161 | \end{figure} 162 | 163 | \subsection{分段线性插值} 164 | 使用等距节点$x_i=-1+ih,h=0.1,0\le i\le 20$，对Runge函数进行分段线性插值。同样选取采样间隔为0.01，一共201个采样点上绘制出分段线性插值Runge函数的图像如图\ref{fig:8}所示。从图中可以看出，分段线性插值对原图像的逼近效果是很好的，唯一的缺点便是在插值节点处不光滑，导致图上有毛刺出现，因此在求数值微分的时候使用分段线性插值可能会效果不佳。 165 | 166 | \begin{figure}[!h] 167 | \centering 168 | \includegraphics[width=0.85\textwidth]{../code/result/linrunge} 169 | \caption{\label{fig:8}分段线性插值Runge函数图像} 170 | \end{figure} 171 | 172 | \subsection{三次自然样条插值} 173 | 使用等距节点$x_i=-1+ih,h=0.1,0\le i\le 20$，对Runge函数进行三次自然样条插值。同样选取采样间隔为0.01，一共201个采样点上绘制出三次自然样条插值Runge函数的图像如图\ref{fig:9}所示。从图中可以看出，三次自然样条插值得到的图像基本与原图像一致，并且足够光滑。结合课上的知识，三次自然样条函数插值满足插值节点的一阶、二阶导数都连续，因此光滑性比分段线性插值要好。 174 | 175 | \begin{figure}[!h] 176 | \centering 177 | \includegraphics[width=0.85\textwidth]{../code/result/splinerunge} 178 | \caption{\label{fig:9}三次自然样条插值Runge函数图像} 179 | \end{figure} 180 | 181 | \subsection{结论} 182 | 综合以上讨论，对于Runge函数而言，采用等距节点的Lagrange(或者Newton)插值在边界点会出现Gibbs现象使得误差极大，不相容，采用Chebyshev多项式零点的Lagrange(或者Newton)插值在边界点虽然会出现波动，但是误差很小，是相容的；采用分段线性插值满足相容性并且误差较之前方法更小，但是其在插值节点不光滑，导数不存在；三次自然样条函数插值，满足相容性，误差最小，并且具有一定的光滑性，在插值节点二阶可导，是对于Runge函数而言最优的插值方式。 183 | 184 | \section{不连续函数f(x)的插值分析} 185 | 在本节将对不连续函数$f(x)$(式\ref{eq1})利用Newton插值、Lagrange插值、分段线性插值、三次自然样条插值4种插值方式进行分析。 186 | 187 | 类似的，我们首先做出函数$f(x)$的图像，采样间隔为0.01，共201个采样节点，如图\ref{fig:10}所示。以此作为我们的标准图像，之后使用4种插值逼近Runge函数的图像与此图像进行对比。 188 | 189 | \begin{figure}[!h] 190 | \centering 191 | \includegraphics[width=0.85\textwidth]{../code/result/f} 192 | \caption{\label{fig:10}f(x)函数图像} 193 | \end{figure} 194 | 195 | \subsection{Newton插值} 196 | 使用等距节点$x_i=-1+ih,h=0.1,0\le i\le 20$，对Runge函数进行20次Newton插值。同样选取采样间隔为0.01，一共201个采样点上绘制出20次等距节点Newton插值Runge函数的图像如图\ref{2fig:1}所示。可以看出在边界的地方出现了Gibbs现象，与原图像相差极大，误差我们已经在之前分析过了，在此不再赘述。 197 | 198 | \begin{figure}[!h] 199 | \centering 200 | \includegraphics[width=0.85\textwidth]{../code/result/newtonf} 201 | \caption{\label{2fig:1}f(x)函数图像} 202 | \end{figure} 203 | 204 | \subsection{Lagrange插值} 205 | 由于等距节点的Lagrange插值结果与Newton插值的结果一致，因此在此不再进行等距节点的Lagrange插值。 206 | 207 | 我们使用Chebyshev多项式零点进行插值，插值节点为$x_i=cos(\frac{2i+1}{42}\pi)(i=0,1,2,\cdots,20)$.插值图像如图\ref{2fig:2}所示，从中可以看出，插值图像与原图像还是有一定差距的，我们计算采样点的误差，发现在x=0处的相对误差高达30.7676，因此可以发现函数的不连续点对Chebyshev多项式零点Lagrange插值有一定的影响。 208 | 209 | \begin{figure}[!h] 210 | \centering 211 | \includegraphics[width=0.85\textwidth]{../code/result/lagchef} 212 | \caption{\label{2fig:2}Chebyshev多项式零点插值f(x)图像} 213 | \end{figure} 214 | 215 | 介于此，我们考虑将原始图像的两部分分开进行Chebyshev多项式零点插值，对于左半部分，我们需要使用坐标变换\ref{eq2}将区间放缩到$[-1,1]$之间，对于右半部分，我们需要使用坐标变换\ref{eq3}将区间放缩到$[-1,1]$之间，不妨左侧我们使用10个插值节点，右侧使用11个插值节点，则左侧插值节点为$x_i=\frac{1}{2}(\cos(\frac{2i+1}{20}\pi)-1),i=0,1,\cdots,9$，右侧插值节点为$x_i=\frac{1}{2}(\cos(\frac{2i+1}{22}\pi)+1),i=0,1,\cdots,10$。插值之后的图像与原图像对比如图\ref{2fig:3}所示，由此可以看出，将间断点绕开做分段的Chebyshev多项式零点的Lagrange插值相比于直接做Chebyshev多项式零点的Lagrange插值的效果要好很多，间断点左侧插值效果很好，基本和原图像完全重合，间断点右侧虽然没有与原图像完全重合，但是误差也很小，最大相对误差为0.4526。综上，对于有间断点的函数进行Chebyshev多项式零点插值，可以考虑以间断点进行分段，然后分段插值，效果会好一些。 216 | \begin{equation} 217 | \label{eq2} 218 | x=\frac{t-1}{2},\quad t\in[-1,1] 219 | \end{equation} 220 | \begin{equation} 221 | \label{eq3} 222 | x=\frac{t+1}{2},\quad t\in[-1,1] 223 | \end{equation} 224 | 225 | \begin{figure}[!h] 226 | \centering 227 | \includegraphics[width=0.85\textwidth]{../code/result/lagchef1} 228 | \caption{\label{2fig:3}分段Chebyshev多项式零点插值f(x)图像} 229 | \end{figure} 230 | 231 | \subsection{分段线性插值} 232 | 使用等距节点$x_i=-1+ih,h=0.1,0\le i\le 20$，对分段函数f(x)进行分段线性插值。同样选取采样间隔为0.01，一共201个采样点上绘制出分段线性插值f(x)的图像如图\ref{2fig:4}所示。从图中可以看出，除了间断点附近的逼近效果不好之外，其余点的插值效果都很好。因此，函数间断点的存在也会影响到分段线性插值的一致收敛性，使得在间断点附近出现较大的误差。 233 | \begin{figure}[!h] 234 | \centering 235 | \includegraphics[width=0.85\textwidth]{../code/result/linf} 236 | \caption{\label{2fig:4}分段线性插值f(x)图像} 237 | \end{figure} 238 | 239 | \subsection{三次自然样条插值} 240 | 使用等距节点$x_i=-1+ih,h=0.1,0\le i\le 20$，对分段函数f(x)进行三次自然样条插值。同样选取采样间隔为0.01，一共201个采样点上绘制出三次自然样条插值f(x)的图像如图\ref{2fig:5}所示。从图中可以看出，类似于分段线性插值，除了间断点附近的逼近效果不好之外，其余点的插值效果都很好。因此，函数间断点的存在也会影响到分段线性插值的一致收敛性，使得在间断点附近出现较大的误差。 241 | 242 | \begin{figure}[!h] 243 | \centering 244 | \includegraphics[width=0.85\textwidth]{../code/result/splinef} 245 | \caption{\label{2fig:5}三次自然样条插值f(x)图像} 246 | \end{figure} 247 | 248 | 对此，我们将原函数分成左右两部分连续函数，分别采用三次自然样条函数插值，如图\ref{2fig:6}所示，从此图中可以看出，以间断点为界，将原函数分段分别进行三次样条插值的效果很好的减小了间断点附近的误差。 249 | 250 | \begin{figure}[!h] 251 | \centering 252 | \includegraphics[width=0.85\textwidth]{../code/result/splinef1} 253 | \caption{\label{2fig:6}以间断点为界，左右两部分分别进行三次自然样条插值f(x)的图像} 254 | \end{figure} 255 | 256 | \subsection{结论} 257 | 对于有间断点的函数进行插值，等距节点的Lagrange法和Newton法的误差较大，不太适合进行此种函数的插值；直接进行Chebyshev多项式零点的Lagrange插值在间断点附近的误差较大，其他地方误差较小，以间断点为界，分段进行Chebyshev多项式零点的Lagrange插值可以有效地减小间断点附近的误差；分段线性插值和三次自然样条插值插值在距离间断点较远的地方误差很小，在间断点附近误差较大，为了减小间断点附近的误差，可以以间断点为界，分段进行三次自然样条插值来减小误差，但是采用此种方法，会使得在间断点不光滑，打破了三次自然样条插值满足一定光滑性的特点，因此，在误差和光滑性之间还需取舍，使得效果最佳。 258 | 259 | \section{实验总结} 260 | 本次大作业使用Newton插值、Lagrange插值、分段线性插值、三次自然样条插值4种插值方式对连续函数和有间断点的函数分别进行插值，对比了4种插值方式的特点以及连续函数和有间断点函数插值的区别，复习了巩固了上课讲过的函数插值的知识，对插值有了更加深刻的理解。 261 | \end{document} -------------------------------------------------------------------------------- /project1/report/main.toc: -------------------------------------------------------------------------------- 1 | \contentsline {section}{\numberline {1}题目描述}{2}{section.1} 2 | \contentsline {section}{\numberline {2}Runge函数的插值分析}{2}{section.2} 3 | \contentsline {subsection}{\numberline {2.1}Newton插值}{3}{subsection.2.1} 4 | \contentsline {subsection}{\numberline {2.2}Lagrange插值}{3}{subsection.2.2} 5 | \contentsline {subsection}{\numberline {2.3}分段线性插值}{5}{subsection.2.3} 6 | \contentsline {subsection}{\numberline {2.4}三次自然样条插值}{7}{subsection.2.4} 7 | \contentsline {subsection}{\numberline {2.5}结论}{7}{subsection.2.5} 8 | \contentsline {section}{\numberline {3}不连续函数f(x)的插值分析}{8}{section.3} 9 | \contentsline {subsection}{\numberline {3.1}Newton插值}{8}{subsection.3.1} 10 | \contentsline {subsection}{\numberline {3.2}Lagrange插值}{8}{subsection.3.2} 11 | \contentsline {subsection}{\numberline {3.3}分段线性插值}{10}{subsection.3.3} 12 | \contentsline {subsection}{\numberline {3.4}三次自然样条插值}{12}{subsection.3.4} 13 | \contentsline {subsection}{\numberline {3.5}结论}{12}{subsection.3.5} 14 | \contentsline {section}{\numberline {4}实验总结}{13}{section.4} 15 | -------------------------------------------------------------------------------- /project2/.DS_Store: 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\renewcommand\thefigure{\thesection.\arabic{figure}} 19 | \makeatletter 20 | \@addtoreset{figure}{section} 21 | \makeatother 22 | 23 | \renewcommand\thesubfigure{\thesection.\arabic{figure}(\alph{subfigure})} 24 | \makeatletter 25 | \@addtoreset{figure}{section} 26 | \makeatother 27 | 28 | \makeatletter 29 | \renewcommand \theequation {% 30 | \ifnum \c@section>\z@ \@arabic\c@section.\fi \ifnum \c@subsection>\z@ 31 | \@arabic\c@subsection.\fi\ifnum \c@subsubsection>\z@ 32 | \@arabic\c@subsubsection.\fi\@arabic\c@equation} 33 | \@addtoreset{equation}{section} 34 | \@addtoreset{equation}{subsection} 35 | %\setcounter{section}{-1} 36 | \makeatother 37 | \newtheorem{theorem}{\hspace{2em}定理}[section] 38 | \hypersetup{ 39 | colorlinks=true, 40 | linkcolor=black 41 | } 42 | 43 | \title{高等数值分析} 44 | \author{罗雁天 \\ 45 | 2018310742} 46 | \date{\today} 47 | 48 | \begin{document} 49 | 50 | %\maketitle 51 | \newcommand{\HRule}{\rule{\linewidth}{0.5mm}} 52 | \begin{titlepage} 53 | \begin{center} 54 | % Upper part of the page 55 | \includegraphics[width=0.4\textwidth]{Tsinghua2.png}\\[1cm] 56 | \textsc{\Large \texttt{高等数值分析}}\\[1cm] 57 | % Title 58 | \HRule \\[1cm] 59 | {\Huge \bfseries 病态线性方程组的求解}\\[0.4cm] 60 | \HRule \\[3.5cm] 61 | % Author and supervisor 62 | \begin{minipage}{0.4\textwidth} 63 | \begin{center} 64 | \Large 65 | \begin{tabular}{cc} 66 | \texttt{作者：} & 罗雁天 \\[0.5cm] 67 | \texttt{学号：} & 2018310742 \\[0.5cm] 68 | \texttt{日期：} & \today 69 | \end{tabular} 70 | \end{center} 71 | \end{minipage} 72 | \vfill 73 | \end{center} 74 | \end{titlepage} 75 | 76 | \tableofcontents 77 | \newpage 78 | 79 | \section{题目描述} 80 | 理论分析表明，数值求解病态线性方程组很困难。考虑求解如下的线性方程组，$Hx=b$，其中$H$是Hilbert矩阵，$H=(h_{ij}),h_{ij}=\frac{1}{i+j-1},i,j=1,2,\cdots,n$。本次大作业从条件数、高斯消去法、Jacobi迭代法、Gauss-Seidel迭代法、SOR迭代法等角度分析上述病态线性方程组并进行对比。 81 | 82 | \section{Hilbert矩阵2-条件数和阶数的关系} 83 | \subsection{使用Matlab自带的cond()函数进行计算} 84 | 由于Matlab自带了求2-条件数的函数$cond()$，因此我们首先采用此种方式讨论Hilbert矩阵2-条件数和阶数的关系。 85 | 86 | 我们首先计算了几个低阶的条件数如表\ref{tab:table1}所示。从表中我们可以看出，随着矩阵阶数$n$的增长，2-条件数增加幅度很快，因此我们采用对数坐标绘制2-条件数和矩阵阶数n的关系曲线。 87 | 88 | \begin{table}[htbp] 89 | \centering 90 | \caption{Hilbert矩阵2-条件数与阶数的关系表格} 91 | \label{tab:table1} 92 | \begin{tabular}{|c|c|c|c|c|c|} 93 | \hline 94 | 阶数n & 1 & 2 & 3 & 4 & 5 \\ 95 | \hline 96 | 2-条件数 & 1.0000 & 19.2815 & 524.0568 & 15513.7387 & 476607.2502\\ 97 | \hline 98 | \end{tabular} 99 | \end{table} 100 | 101 | 取矩阵的阶数从$1\to100$，在对数坐标下绘制2-条件数和矩阵阶数n的关系曲线如图\ref{fig:1}所示。从图中我们可以看出，当阶数较低(大约$1\to 13$)时，对数化2-条件数大约与阶数呈现线性关系，当阶数变高时，对数化的2-条件数波动起来，不再增加，根据我们对Hilbert矩阵病态性的知识，图\ref{fig:1}中阶数较大时的曲线显然不正确，由此可以说明Matlab自带的cond()函数在矩阵阶数较高时计算的条件数误差较大。因此我们考虑另一种方法计算矩阵的条件数。 102 | 103 | \begin{figure}[!h] 104 | \centering 105 | \includegraphics[width=0.8\textwidth]{../code/result/logcond2} 106 | \caption{\label{fig:1}使用cond()函数计算的2-条件数和矩阵阶数n在对数坐标下的曲线} 107 | \end{figure} 108 | 109 | \subsection{使用2-条件数的定义进行计算} 110 | 根据2-条件数的定义$cond_2(H)=||H||_2||H^{-1}||_2$，Matlab中有专门针对Hilbert矩阵逆矩阵的函数$invhilb()$，因此我们可以采用定义法来计算Hilbert矩阵的2-条件数。同样在对数坐标下，绘制出此种方法计算出的2-条件数和矩阵阶数的关系图如图\ref{fig:2}所示，从此图中可以看出，随着矩阵阶数的增加，对数化的2-条件数近似与阶数呈现线性关系，符合我们对Hilbert矩阵病态性的理解。 111 | 112 | 我们将对数化的2-条件数和矩阵阶数进行线性回归，得到拟合公式为: $cond2=10^{1.5257n-2.0758}$，相关系数$r\approx 1$，拟合之后图像如图\ref{fig:3}所示。 113 | 114 | \begin{figure}[!h] 115 | \centering 116 | \includegraphics[width=0.8\textwidth]{../code/result/alogcond2} 117 | \caption{\label{fig:2}使用定义计算的2-条件数和矩阵阶数n在对数坐标下的曲线} 118 | \end{figure} 119 | 120 | \begin{figure}[!h] 121 | \centering 122 | \includegraphics[width=0.8\textwidth]{../code/result/cond2fit} 123 | \caption{\label{fig:3}使用定义计算的2-条件数和矩阵阶数n在对数坐标下的曲线} 124 | \end{figure} 125 | 126 | \section{Gauss消去法} 127 | 用Gauss消去法将Hilbert矩阵消成上三角矩阵，然后求解结果。我们将阶数$n=2,5,10,20,50,100$的误差列表如表\ref{tab:table2}所示，从表中我们可以看出，随着矩阵阶数的增加，Gauss消去法的误差上升较快，当阶数为13时，误差就已经达到了3.0655，相对误差已经很大了，因此Gauss消去法不适和高阶Hilbert矩阵求解。我们绘制出$n=1\to100$时的Gauss消去法求解的相对误差曲线如图\ref{fig:4}所示。 128 | 129 | \begin{table}[htbp] 130 | \centering 131 | \caption{Gauss消去法相对误差与阶数关系表格} 132 | \label{tab:table2} 133 | \begin{tabular}{|p{4cm}<{\centering}|p{6cm}<{\centering}|} 134 | \hline 135 | 阶数n & Gauss消去法的相对误差 \\ 136 | \hline 137 | 2 & 5.66104886700368e-16 \\ 138 | \hline 139 | 5 & 1.55303820484067e-12 \\ 140 | \hline 141 | 10 & 0.000223773106799740 \\ 142 | \hline 143 | 20 & 23.5417423737487 \\ 144 | \hline 145 | 50 & 240.055736859534 \\ 146 | \hline 147 | 100 & 78.1201736046372 \\ 148 | \hline 149 | \end{tabular} 150 | \end{table} 151 | 152 | \begin{figure}[!h] 153 | \centering 154 | \includegraphics[width=0.8\textwidth]{../code/result/gausserror} 155 | \caption{\label{fig:4}Gauss消去法相对误差与阶数关系曲线图} 156 | \end{figure} 157 | 158 | \section{Jacobi矩阵迭代法} 159 | 设$H=D-L-U$，其中$D=diag(h_{11},h_{22},\cdots,h_{nn})$表示Hilbert矩阵$H$的对角线，$L$表示$H$的左下角元素的相反数，是一个下三角矩阵，$U$表示$H$的右上角元素的相反数，是一个上三角矩阵。因此，线性方程组$Hx=b$可以转换为$x=B_Jx+f$，其中$B_J=D^{-1}(L+U),f=D^{-1}b$，由此得到Jacobi迭代格式$x^{(k+1)}=B_Jx^{(k)}+f$。由于此种迭代格式只有当矩阵$B_J$的谱半径小于1时，迭代才是收敛的，因此，我们首先绘制出Jacobi迭代矩阵$B_J$的谱半径与阶数$n$的曲线图如图\ref{fig:5}所示，从图\subref{fig:b}中可以看出，当阶数$n>2$时，Jacobi迭代矩阵的谱半径就已经超过1了，因此当$n>2$时，Jacobi迭代法不收敛。当$n=2$时，我们设置当两次迭代的变化小于1e-6时，停止迭代，此时，Jicobi迭代相对误差为3.18555931744235e-07，误差比Gauss消去法在n=2时的误差还要大，因此，Jacobi迭代法不适合Hilbert矩阵线性方程组的求解。 160 | 161 | \begin{figure}[!h] \centering 162 | \subfigure[Jacobi迭代矩阵谱半径图] { \label{fig:a} 163 | \includegraphics[width=0.48\columnwidth]{../code/result/jacobirho} 164 | } 165 | \subfigure[Jacobi迭代矩阵谱半径与1比较图] { \label{fig:b} 166 | \includegraphics[width=0.48\columnwidth]{../code/result/jacobirho2} 167 | } 168 | \caption{Jacobi迭代矩阵谱半径图} 169 | \label{fig:5} 170 | \end{figure} 171 | 172 | \section{Gauss-Seidel矩阵迭代法} 173 | 设$H=D-L-U$，其中$D=diag(h_{11},h_{22},\cdots,h_{nn})$表示Hilbert矩阵$H$的对角线，$L$表示$H$的左下角元素的相反数，是一个下三角矩阵，$U$表示$H$的右上角元素的相反数，是一个上三角矩阵。因此，线性方程组$Hx=b$可以转换为$x=B_Gx+f$，其中$B_G=(D+L)^{-1}U,f=(D+L)^{-1}b$。与Jacobi矩阵迭代法类似，只有当矩阵$B_G$的谱半径小于1时，迭代才是收敛的，因此我们绘制出Gauss-Seidel迭代矩阵$B_G$的谱半径示意图如图\ref{fig:6}所示，从图中我们可以看出，当阶数$n\ge 13$时，谱半径已经近似$=1$了，因此此时Gauss-Seidel迭代法收敛速度很慢了。由此可知，Gauss-Seidel迭代法比Jacobi迭代法支持的Hilbert矩阵阶数高一点，但是仍然不能够用于高阶Hilbert矩阵线性方程组的求解。 174 | 175 | \begin{figure}[!h] 176 | \centering 177 | \includegraphics[width=0.8\textwidth]{../code/result/gsrho} 178 | \caption{\label{fig:6}Gauss-Seidel迭代矩阵谱半径与阶数n的关系图} 179 | \end{figure} 180 | 181 | \section{SOR迭代法} 182 | 设$H=D-L-U$，其中$D=diag(h_{11},h_{22},\cdots,h_{nn})$表示Hilbert矩阵$H$的对角线，$L$表示$H$的左下角元素的相反数，是一个下三角矩阵，$U$表示$H$的右上角元素的相反数，是一个上三角矩阵。因此，线性方程组$Hx=b$可以转换为$x=L_wx+f$，其中$L_w=(D-wL)^{-1}((1-w)D+wU),f=w*(D-wL)^{-1}b$。显然当$w=1$时，SOR迭代法即为Gauss-Seidel迭代法，并且根据上课学到的知识，只有当$028$时，谱半径$\rho$接近1，因此此时SOR方法收敛很慢。由此可知，SOR迭代法比Jacobi迭代法以及Gauss-Seidel迭代法支持的Hilbert矩阵阶数更高一点，但是对于阶数过高的Hilbert矩阵求解，SOR迭代法仍然不可取。由于在图中不太容易直观的看出GS迭代法和SOR迭代法谱半径的不同，因此我们绘制出局部的图像进行对比，如图\ref{fig:8}所示，由此图可以看出，SOR迭代法的谱半径要比GS谱半径小，收敛速度较快。 183 | 184 | \begin{figure}[!h] 185 | \centering 186 | \includegraphics[width=0.8\textwidth]{../code/result/sorrho} 187 | \caption{\label{fig:7}SOR迭代矩阵谱半径与阶数n的关系图} 188 | \end{figure} 189 | 190 | \begin{figure}[!h] \centering 191 | \subfigure[GS迭代矩阵谱半径局部图] { \label{fig:8a} 192 | \includegraphics[width=0.48\columnwidth]{../code/result/gsrho2} 193 | } 194 | \subfigure[SOR迭代矩阵谱半径局部图] { \label{fig:8b} 195 | \includegraphics[width=0.48\columnwidth]{../code/result/sorrho2} 196 | } 197 | \caption{GS迭代矩阵和SOR迭代矩阵谱半径局部对比图} 198 | \label{fig:8} 199 | \end{figure} 200 | 201 | \section{实验结果对比} 202 | 在以上几节，我们讨论和分析了求解线性方程组的4种方法以及他们各自的性能，在此节我们将对其求出解得误差进行分析，我们设置阶数$n=2,5,10,20,50,100$，来计算相对误差，其中迭代法采用的初值均为0向量。 203 | 204 | 首先我们求出Jacobi迭代法经过100次迭代之后的相对误差如表\ref{tab:table3}所示，从中我们可以看出Jacobi迭代法对于n>2的Hilbert矩阵便已经不收敛了。 205 | 206 | \begin{table}[htbp] 207 | \centering 208 | \caption{Jacobi迭代法相对误差与阶数关系表格} 209 | \label{tab:table3} 210 | \begin{tabular}{|c|c|c|c|} 211 | \hline 212 | 阶数n & 谱半径$\rho(B_J)$& 迭代次数 &Jacobi迭代法的相对误差 \\ 213 | \hline 214 | 2 &0.866025403784439 &100 & 5.66321656347846e-07\\ 215 | \hline 216 | 5 &3.44414219116595 &100 & 6.18861899881230e+53\\ 217 | \hline 218 | 10 &7.77981513192998 &100 & 1.54832687372472e+89\\ 219 | \hline 220 | 20 &16.4920989837926 &100 & 6.66005666075554e+121\\ 221 | \hline 222 | 50 &42.6768950976645 &100 & 1.30642530458295e+163\\ 223 | \hline 224 | 100 &86.3374902073534 &100 & 5.21053762273348e+193\\ 225 | \hline 226 | \end{tabular} 227 | \end{table} 228 | 229 | 其次我们求出了GS迭代法的相对误差表，我们采用$\frac{||x^{(k)}-x^{(k-1)}s||_2}{||x^{(k-1)}||_2}<1e-6$来停止迭代，定义最大的迭代次数为20000，迭代结果如表\ref{tab:table4}所示。 230 | 231 | \begin{table}[htbp] 232 | \centering 233 | \caption{GS迭代法相对误差与阶数关系表格} 234 | \label{tab:table4} 235 | \begin{tabular}{|c|c|c|c|} 236 | \hline 237 | 阶数n & 谱半径$\rho(B_{GS})$& 迭代次数 &GS迭代法的相对误差 \\ 238 | \hline 239 | 2 &0.750000000000000 &45 & 2.02811359558980e-06\\ 240 | \hline 241 | 5 &0.999957671222958 &7913 & 0.0138028658327348\\ 242 | \hline 243 | 10 &0.999999999997045 &17853 & 0.00873261631894775\\ 244 | \hline 245 | 20 &1.00000000000000 &20000 & 0.00873439429404305\\ 246 | \hline 247 | 50 &1.00000000000000 &20000 & 0.00959066207416708\\ 248 | \hline 249 | 100 &1.00000000000000 &20000 & 0.0101320928739749\\ 250 | \hline 251 | \end{tabular} 252 | \end{table} 253 | 254 | 类似的，我们求出了SOR迭代法的误差表，我们依然采用$\frac{||x^{(k)}-x^{(k-1)}||_\infty}{||x^{(k-1)}||_\infty}<1e-6$来停止迭代，定义最大的迭代次数为20000，迭代结果如表\ref{tab:table5}所示。 255 | 256 | \begin{table}[htbp] 257 | \centering 258 | \caption{SOR迭代法相对误差与阶数关系表格} 259 | \label{tab:table5} 260 | \begin{tabular}{|c|c|c|c|} 261 | \hline 262 | 阶数n & 谱半径$\rho(B_{GS})$& 迭代次数 &GS迭代法的相对误差 \\ 263 | \hline 264 | 2 &0.340000000000000 &16 & 3.67239854881804e-08\\ 265 | \hline 266 | 5 &0.999190149180609 &9751 & 0.000542785068819967\\ 267 | \hline 268 | 10 &0.999999999871931 &12441 & 0.0381269338580335\\ 269 | \hline 270 | 20 &1.00000000000000 &5061 & 0.00712056341512206\\ 271 | \hline 272 | 50 &1.00000000000000 &15924 & 0.00553514159584128\\ 273 | \hline 274 | 100 &1.00000000000000 &9129 & 0.00599821418674224\\ 275 | \hline 276 | \end{tabular} 277 | \end{table} 278 | 279 | 从上述的结果中可以看出，Jacobi迭代法对于n>2的Hilbert矩阵是不收敛的，相对误差越来越大，不能用于此病态线性方程组的求解。GaussSeidel迭代法和SOR迭代法都是收敛的，相对误差并没有特别大。理论上，我们在SOR迭代法采用优化算法求出了最优松弛因子$w$，所以其收敛速度应该要比GS法的收敛速度快，但是从表\ref{tab:table4}和表\ref{tab:table5}中n=5时却出现了相反的情况，究其原因，应该是我们设置的收敛条件$\frac{||x^{(k)}-x^{(k-1)}||_\infty}{||x^{(k-1)}||_\infty}<1e-6$太严格导致会出现迭代次数盲目增加的情况。 280 | 281 | 由于我们知道正确解为$x^*=(1,1,\cdots,1)^T$，因此我们将收敛条件改为$\frac{||x^{(k)}-x^*||_2}{||x^{*}||_2}<1e-2$，将GS迭代法和SOR迭代法的结果列表如表\ref{tab:table6}所示，从这个表中我们便可以看出，SOR法的收敛速度确实要比GS法的收敛速度快。 282 | 283 | \begin{table}[htbp] 284 | \centering 285 | \caption{SOR迭代法与GS迭代法收敛速度对比} 286 | \label{tab:table6} 287 | \begin{tabular}{|c|c|c|c|c|} 288 | \hline 289 | 阶数n & GS法迭代次数& SOR法迭代次数 &GS迭代法的相对误差 & SOR法相对误差\\ 290 | \hline 291 | 2 & 16& 8& 0.00851756855749210& 0.00360255839563727\\ 292 | \hline 293 | 5 & 15527& 6688& 0.00999993017598135& 0.00999579098530880\\ 294 | \hline 295 | 10 & 15428& 9170& 0.00999959518767067& 0.00998574409924582\\ 296 | \hline 297 | 20 & 17301& 1829& 0.00999957638861157& 0.00999716337976600\\ 298 | \hline 299 | 50 & 16516& 3509& 0.00999978378907092& 0.00999838890291187\\ 300 | \hline 301 | 100 & 21622 & 4087& 0.00999998104858494& 0.00999876526546658\\ 302 | \hline 303 | \end{tabular} 304 | \end{table} 305 | 306 | 我们绘制出$n=10,50,100$时，迭代过程中相对误差的变化情况如图\ref{fig:9},\ref{fig:10},\ref{fig:11}所示，从图中我们可以清楚的看出SOR迭代法比GS法先到$10^{-2}$的误差，因此，SOR法收敛比GS法收敛快。 307 | 308 | \begin{figure}[!h] 309 | \centering 310 | \includegraphics[width=0.8\textwidth]{../code/result/er10} 311 | \caption{\label{fig:9}n=10时迭代过程中相对误差的变化情况} 312 | \end{figure} 313 | 314 | \begin{figure}[!h] 315 | \centering 316 | \includegraphics[width=0.8\textwidth]{../code/result/er50} 317 | \caption{\label{fig:10}n=50时迭代过程中相对误差的变化情况} 318 | \end{figure} 319 | 320 | \begin{figure}[!h] 321 | \centering 322 | \includegraphics[width=0.8\textwidth]{../code/result/er100} 323 | \caption{\label{fig:11}n=100时迭代过程中相对误差的变化情况} 324 | \end{figure} 325 | 326 | 此外，我们还绘制出了最优松弛因子关于阶数$n$的曲线如图\ref{fig:12}所示，从图中可以看出，阶数在13和14之间的突变是非常不合理的，考虑到我们在计算最优松弛因子时进行了$(D-wL)^{-1}$操作，由此可能引起极大的误差，因此在图\ref{fig:12}中只有当阶数n较小时的曲线才是正确的。 327 | 328 | 综上所述，对于Hilbert矩阵构成的病态线性方程组，Gauss消去法在阶数小于10时误差较小，可以求解，阶数较大时不能正确求解；Jacobi迭代法由于当阶数n>2时不收敛，因此不能用于求解；Gauss-Seidel迭代法和SOR迭代法可以用于求解此病态方程组，但是由于当阶数n较大时，收敛速度很慢，迭代次数很多，选取最优松弛因子的SOR迭代法比Gauss-Seidel迭代法收敛速度较快。 329 | 330 | \begin{figure}[!h] 331 | \centering 332 | \includegraphics[width=0.8\textwidth]{../code/result/bestw} 333 | \caption{\label{fig:12}最优松弛因子与阶数n的关系图} 334 | \end{figure} 335 | 336 | \section{实验结论} 337 | Hilbert 矩阵的 2-条件数随着阶数 n 的增大而呈指数增长，因此高阶 338 | Hilbert 矩阵的病态性是毋庸置疑的，且阶数越高，解的误差应该越大，迭代法 339 | 收敛速度应该越慢。但是实验结果中常常在阶数 n 从 12 变为 13 时出现巨变，似 340 | 乎与理论不符合。事实上，无论是 Gauss 消去、Gauss-Seidel 迭代还是 SOR 迭 341 | 代，在阶数高时的求解步骤中就可能存在类似对病态矩阵求逆的操作，从而早成 342 | 高阶下的很多实验结果本身就有很大误差。而阶数 12 恰好又是这个临界点，因 343 | 此这种突变便可以理解了。寻找 Hilbert 矩阵 2-条件数和阶数 n 的关系时，一 344 | 开始直接用 cond 函数也发现在 n=12 出现了突变，但用专门的 invhilb 函数时便 345 | 能得到正确地结果，可见这种猜想是正确的。 346 | 根据上述实验结果，发现对于 Hilbert 矩阵构成的病态线性方程组，Gauss 347 | 消去法无法准确求解。当矩阵的阶数不断增加时，解的相对误差也迅速增长。 348 | 349 | 用 Jacobi 迭代法求解 Hilbert 矩阵构成的病态线性方程，在阶数高于 2 时 350 | 便不再收敛。Jacobi 迭代矩阵的谱半径随阶数 n 的增大呈现近似线性增长，因 351 | 此 Jacobi 迭代法无法用于该病态方程组的求解。 352 | 353 | Gauss-Seidel 法和 SOR 法的迭代矩阵的谱半径在阶数 n 不断增长得过程中 354 | 趋近与 1，基于此不便于判断迭代法是否收敛。事实上，Hilbert矩阵是对称正定阵。令$[-1,1]$上的n次多项式$P(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$，则式\ref{eq:1}。其中$H_n$为n阶Hilbert矩阵，因此可以看出$H_n$是对称正定的。由于我们有定理\ref{theorem1}，所以Gauss-Seidel法和SOR法都是收敛的。但是由于在高阶（高于 5 阶）情况下，两者迭代矩阵的谱半 355 | 径都非常接近于 1，导致收敛速度很慢。前面的实验结果已经提到，要达到1e-2的相对误差，G-S 法要迭代上万次，SOR 法也要迭代几千次。至于阶数从 5、 356 | 10、20、50 、100变化过程中，收敛速度似乎波动稳定，原因也是矩阵本身的病态性导 357 | 致迭代法本身就是有较大误差的。事实上，收敛速度应该随着阶数增大而变慢。 358 | 359 | 360 | \begin{equation}\label{eq:1} 361 | \int_{-1}^{1}[P(x)]^2=[a_0,a_1,\cdots,a_2]H_n\left[ 362 | \begin{array}{c} 363 | a_0 \\ 364 | a_1 \\ 365 | \vdots \\ 366 | a_n 367 | \end{array} 368 | \right] 369 | \end{equation} 370 | 371 | \begin{theorem} 372 | \label{theorem1} 373 | 设$A\in \mathbb{R}^{n\times n}$是对称正定矩阵，且$012）的情况，只有采用迭代法才能保证精度， 380 | 但是收敛速度非常慢，迭代次数非常高，这也是病态线性方程组令人头疼的地方 381 | 所在吧。 382 | \end{document} 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