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Quadrature rules based on equispaced nodes seem natural, as one would wonder what we can achieve if nonequispaced nodes are used. In fact, using carefully chosen nonequispaced nodes may develop highly efficient and accurate quadrature rules, which converge to the integrals of $f \in C^{\infty}([a, b])$ exponentially as the number of nodes increases, asymptotically outperforming all previously discussed quadrature rules with errors on the order of $\mathcal{O}\left(h^p\right)$ for any integer $p>0$. We shall discuss two rules: Clenshaw-Curtis and Gauss. 1
Clenshaw-Curtis Quadrature. The Clenshaw-Curtis quadrature for $\int_a^b f(x) d x$ is
$$Q_{2 C}^n(f)=\int_a^b p_n(x) d x=\sum_{k=0}^n w_k f\left(x_k\right)$$
where $p_n(x)$ is the Chebyshev interpolant for $f(x)$, and the quadrature nodes $x_k=$ $-\cos \left(\frac{k \pi}{n}\right) \frac{b-a}{2}+\frac{a+b}{2}$ are the Chebyshev points. For an odd $n$, the weights $w_k=\int_a^b L_k(x) d x$ satisfy
$$w_k= \begin{cases}\frac{b-a}{2 n^2} & k=0, n, \ \frac{b-a}{n}\left{1-\sum_{j=1}^{\frac{n-1}{2}} \frac{2}{4 j^2-1} \cos \left(\frac{2 k j \pi}{n}\right)\right} & 1 \leq k \leq n-1,\end{cases}$$
and for an even $n$,
$$w_k= \begin{cases}\frac{b-a}{2\left(n^2-1\right)} & k=0, n \ \frac{b-a}{n}\left{1-\frac{(-1)^k}{n^2-1}-\sum_{j=1}^{\frac{n}{2}-1} \frac{2}{4 j^2-1} \cos \left(\frac{2 k j \pi}{n}\right)\right} & 1 \leq k \leq n-1\end{cases}$$

Another type of quadrature that converges exponentially for analytic integrand is the Gauss quadrature. Though Gauss quadrature is much more well-known than Clenshaw-Curtis, they are comparable in many aspects.

Consider the quadrature $I_f=\int_a^b f(x) \rho(x) d x$, where $\rho(x)>0$ is a weight function. A quadrature $Q(f)=\sum_{k=0}^n w_k f\left(x_k\right)$ (note that it does not evaluate $\rho(x)$ anywhere) is an $(n+1)$-node Gauss quadrature if its degree of accuracy is $2 n+1$. The nodes and weights of Gauss quadrature can be constructed directly: we set up the nonlinear system of equations $\int_a^b x^{\ell} \rho(x) d x=\sum_{k=0}^n w_k x_k^{\ell}$ for $0 \leq \ell \leq 2 n+1$, and solve for all nodes and weights, for example, by Newton’s method. This approach is fine for small $n$, but not a good choice for large $n$, because each iteration of Newton’s method needs to solve a linear system of equations involving $2 n+2$ unknowns, taking $\mathcal{O}\left(n^3\right)$ flops per iteration.

Example 67. Determine the 2-point $(n=1)$ Gauss quadrature rule for $\rho(x)=1$ on $[-1,1]$ by hand. Let the quadrature rule be $Q=w_0 f\left(x_0\right)+w_1 f\left(x_1\right)$, which is exact for polynomial integrand of degree $\leq 2 n+1=3$. Therefore, we have the following equations for the unknowns $w_0, w_1, x_1$, and $x_2$ :
\begin{aligned} & w_0+w_1=\int_{-1}^1 1 d x=2 \quad w_0 x_0+w_1 x_1=\int_{-1}^1 x d x=0 \ & w_0 x_0^2+w_1 x_1^2=\int_{-1}^1 x^2 d x=\frac{2}{3} \quad w_0 x_0^3+w_1 x_1^3=\int_{-1}^1 x^3 d x=0 \end{aligned}
This system of nonlinear equations seem hard to solve by hand. However, we note that the quadrature rule should be symmetric with respect to the origin; that is, $x_0=-x_1$ and $w_0=w_1$. This observation easily leads to $w_0=w_1=1, x_0=-\frac{\sqrt{3}}{3}$, and $x_1=\frac{\sqrt{3}}{3}$. We can determine the 3-point Gauss quadrature similarly.

## 数值分析代写

$$Q_{2 C}^n(f)=\int_a^b p_n(x) d x=\sum_{k=0}^n w_k f\left(x_k\right)$$

\left 缺少或无法识别的分隔符

$\backslash$ left 缺少或无法识别的分隔符

$Q=w_0 f\left(x_0\right)+w_1 f\left(x_1\right)$ ，这对于次数的多项式被积函数是精确的 $\leq 2 n+1=3$. 因此，对于末知数，我 们有以下方程 $w_0, w_1, x_1$ ，和 $x_2$ :
$$w_0+w_1=\int_{-1}^1 1 d x=2 \quad w_0 x_0+w_1 x_1=\int_{-1}^1 x d x=0 \quad w_0 x_0^2+w_1 x_1^2=\int_{-1}^1 x^2 d x=\frac{2}{3} \quad w_0 x_0^3+w_1 x_1^3=\int_{-1}^1 x^3 d x=0$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。