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# 数学代写|高等线性代数Advanced Linear Algebra代考|MATH2200 POLYNOMIAL INTERPOLATION

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Many people are introduced to interpolation in elementary algebra, where it is used to extend tables of logarithms or trigonometric functions to argument values not given in the table. Others are introduced to interpolation when they study the trapezoidal method for numerical integration. In both cases, the form of interpolation used is called linear interpolation, and it will be used here as an introduction to more general polynomial interpolation. Our approach will not be the same as that given in most beginning algebra texts, but later we will show the connection between the two approaches.

Given two points $\left(x_0, y_0\right)$ and $\left(x_1, y_1\right)$ with $x_0 \neq x_1$, draw a straight line through them, as in Figure 5.1. The straight line is the graph of the linear polynomial
$$P_1(x)=\frac{\left(x_1-x\right) y_0+\left(x-x_0\right) y_1}{x_1-x_0}$$
The reader should check that the graph of this function is the straight line determined by $\left(x_0, y_0\right)$ and $\left(x_1, y_1\right)$. We say that this function interpolates the value $y_i$ at the point $x_i, i=0,1$; or
$$P_1\left(x_i\right)=y_i, \quad i=0,1$$

5.1, for $f(x)=e^x$. To use linear interpolation to estimate the value of $f(x)$ when $x$ is not in the table, locate two values $x_0$ and $x_1$ that enclose $x$ in the table. Construct $P_1(x)$ to interpolate $f\left(x_i\right)$ at $x_i, i=0,1$. Then use $P_1(x)$ to estimate $f(x)$.

Most data arise from graphs that are curved rather than straight. To better approximate such behavior, we look at polynomials of degree greater than one. Assume that three data points $\left(x_0, y_0\right),\left(x_1, y_1\right)$, and $\left(x_2, y_2\right)$ are given, with $x_0, x_1, x_2$ distinct points. We construct the quadratic polynomial that passes through these points as follows:
$$P_2(x)=y_0 L_0(x)+y_1 L_1(x)+y_2 L_2(x)$$
with
\begin{aligned} &L_0(x)=\frac{\left(x-x_1\right)\left(x-x_2\right)}{\left(x_0-x_1\right)\left(x_0-x_2\right)}, \quad L_1(x)=\frac{\left(x-x_0\right)\left(x-x_2\right)}{\left(x_1-x_0\right)\left(x_1-x_2\right)} \ &L_2(x)=\frac{\left(x-x_0\right)\left(x-x_1\right)}{\left(x_2-x_0\right)\left(x_2-x_1\right)} \end{aligned}
Formula (5.8) is called Lagrange’s formula for the quadratic interpolating polynomial; and the polynomials $L_0, L_1$, and $L_2$ are called the Lagrange interpolation basis functions.

## 数学代写|线性代数代写线性代数代考|多项式插值

$$P_1(x)=\frac{\left(x_1-x\right) y_0+\left(x-x_0\right) y_1}{x_1-x_0}$$

$$P_1\left(x_i\right)=y_i, \quad i=0,1$$

5.1，为$f(x)=e^x$。当$x$不在表中时，要使用线性插值来估计$f(x)$的值，请在表中找到包含$x$的两个值$x_0$和$x_1$。构造$P_1(x)$在$x_i, i=0,1$上插值$f\left(x_i\right)$。然后使用$P_1(x)$估计$f(x)$。

## 数学代写|线性代数代写线性代数代考|二次插值

$$P_2(x)=y_0 L_0(x)+y_1 L_1(x)+y_2 L_2(x)$$
with
\begin{aligned} &L_0(x)=\frac{\left(x-x_1\right)\left(x-x_2\right)}{\left(x_0-x_1\right)\left(x_0-x_2\right)}, \quad L_1(x)=\frac{\left(x-x_0\right)\left(x-x_2\right)}{\left(x_1-x_0\right)\left(x_1-x_2\right)} \ &L_2(x)=\frac{\left(x-x_0\right)\left(x-x_1\right)}{\left(x_2-x_0\right)\left(x_2-x_1\right)} \end{aligned}

## 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。