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# 数学代写|数值分析代写Numerical analysis代考|MAT12004 Lagrange interpolation

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## 数学代写|数值分析代写Numerical analysis代考|Lagrange interpolation

Given that $n$ is a nonnegative integer, let $\mathcal{P}_n$ denote the set of all (realvalued) polynomials of degree $\leq n$ defined over the set $\mathbb{R}$ of real numbers. The simplest interpolation problem can be stated as follows: given $x_0$ and $y_0$ in $\mathbb{R}$, find a polynomial $p_0 \in \mathcal{P}_0$ such that $p_0\left(x_0\right)=y_0$. The solution to this is, trivially, $p_0(x) \equiv y_0$. The purpose of this section is to explore the following more general problem.

Let $n \geq 1$, and suppose that $x_i, i=0,1, \ldots, n$, are distinct real numbers (i.e., $x_i \neq x_j$ for $i \neq j$ ) and $y_i, i=0,1, \ldots, n$, are real numbers; we wish to find $p_n \in \mathcal{P}_n$ such that $p_n\left(x_i\right)=y_i, i=0,1, \ldots, n$.

To prove that this problem has a unique solution, we begin with a useful lemma.

Lemma 6.1 Suppose that $n \geq 1$. There exist polynomials $L_k \in \mathcal{P}n$, $k=0,1, \ldots, n$, such that $$L_k\left(x_i\right)= \begin{cases}1, & i=k, \ 0, & i \neq k,\end{cases}$$ for all $i, k=0,1, \ldots, n$. Moreover, $$p_n(x)=\sum{k=0}^n L_k(x) y_k$$
satisfies the above interpolation conditions; in other words, $p_n \in \mathcal{P}_n$ and $p_n\left(x_i\right)=y_i, i=0,1, \ldots, n$.

## 数学代写|数值分析代写Numerical analysis代考|Convergence

An important theoretical question is whether or not a sequence $\left(p_n\right)$ of interpolation polynomials for a continuous function $f$ converges to $f$ as $n \rightarrow \infty$. This question needs to be made more specific, as $p_n$ depends on the distribution of the interpolation points $x_j, j=0,1, \ldots, n$, not just on the value of $n$. Suppose, for example, that we agree to choose equally spaced points, with
$$x_j=a+j(b-a) / n, \quad j=0,1, \ldots, n, \quad n \geq 1 .$$
The question of convergence then clearly depends on the behaviour of $M_{n+1}$ as $n$ increases. In particular, if
$$\lim {n \rightarrow \infty} \frac{M{n+1}}{(n+1) !} \max {x \in[a, b]}\left|\pi{n+1}(x)\right|=0,$$
then, by $(6.10)$,
$$\lim {n \rightarrow \infty} \max {x \in[a, b]}\left|f(x)-p_n(x)\right|=0,$$
and we say that the sequence of interpolation polynomials $\left(p_n\right)$, with equally spaced points on $[a, b]$, converges to $f$ as $n \rightarrow \infty$, uniformly on the interval $[a, b]$.

## 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。