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\title{\textbf {2025春计算方法--实验报告 \#2}}
\author{\textbf{\color{red}仅供参考！}}
\date{\today}

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\maketitle

运行环境：[自己给出。。。]
%win11，vscode，py3

\section*{实验内容与要求}

    对函数\textcolor{blue}{$f(x)=\dfrac{3x+1}{x^2-2x+3},x\in[-5,5]$}，构造其$N$次\textcolor{blue}{Lagrange插值函数} $p(x)$，取$\max\limits_{x\in[-5,5]}|f(x)-p(x)|\approx\max\limits_{0\leq i\leq 500}|f(y_i)-p(y_i)|,y_i=\dfrac{i}{50}-5$为\textcolor{blue}{近似误差}。其中\textcolor{red}{$N+1$}个\textcolor{red}{插值节点}取值为：

    \begin{enumerate}
        \item $x_i=-5+\dfrac{10}{N}i,i=0,1,\cdots,N$
        \item $x_i=-5\cos\left(\dfrac{2i+1}{2N+2}\pi\right),i=0,1,\cdots,N$(Chebyshev点)
    \end{enumerate}

    分别取\textcolor{blue}{$N=4,8,16,32$}，比较以上两组节点的插值结果（保留到小数点后12位）


\section{数值结果}


    \begin{table}[H]
        \begin{center}
            \begin{tabular}{|c|c|c|}
                \hline
                $N$ & 方法1 & 方法2 \\
                \hline
                $4$ & $1.108575376458e+00$ & $1.309481298284e+00$ \\
                \hline
                $8$ & $1.868530466062e+00$ & $3.063702120590e-01$ \\
                \hline
                $16$ & $5.529879571592e+00$ & $4.202636883909e-02$ \\
                \hline
                $32$ & $4.059446252049e+02$ & $4.307034883053e-04$ \\
                \hline
            \end{tabular}
        \end{center}
    \end{table}


\section{算法分析}

    略

\section{实验小结}

    略

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