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# 数学代写|高等线性代数Advanced Linear Algebra代考|MATH8722 INTERPOLATION USING SPLINE FUNCTIONS

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## 数学代写|高等线性代数Advanced Linear Algebra代考|INTERPOLATION USING SPLINE FUNCTIONS

To motivate the definition and use of spline functions, we begin with the problem of interpolating the data shown in Table 5.4. The simplest method of interpolation is to connect the node points by straight line segments; the resulting graph is shown in Figure 5.7. This is called piecewise linear interpolation, and the associated interpolating function is denoted by $l(x)$. It agrees with the data, but it has the disadvantage of not having a smooth graph. Most data will represent a smooth curved graph, one without the corners of $y=l(x)$. Consequently, we usually want to construct a smooth curve that interpolates the given data points, but one that follows the shape of $y=l(x)$.

The next choice of interpolation is to use polynomial interpolation. There are seven data points, and thus we consider the interpolating polynomial $P_6(x)$ of degree 6. Its graph is shown in Figure $5.8$ (note the change in vertical scale), and it differs markedly from that of $y=l(x)$. Although it is a smooth graph, it is quite different from that of $y=l(x)$ between some of the interpolation node points, for example, on $0 \leq x \leq 1$.

To pose the problem more generally, suppose $n$ data points $\left(x_i, y_i\right), i=1, \ldots, n$ are given. For simplicity, assume that
$$x_1<x_2<\cdots<x_n$$
and let $a=x_1, b=x_n$. We seek a function $s(x)$ defined on $[a, b]$ that interpolates the data:
$$s\left(x_i\right)=y_i, \quad i=1, \ldots, n$$
For smoothness of $s(x)$, we require that $s^{\prime}(x)$ and $s^{\prime \prime}(x)$ be continuous. In addition, we want the curve to follow the general shape given by the piecewise linear function connecting the data points $\left(x_i, y_i\right)$, as illustrated in Figure 5.7. The standard way in which this has been done has been to ask that the derivative $s^{\prime}(x)$ not change too rapidly between node points. This has been carried out by requiring the second derivative $s^{\prime \prime}(x)$ to be as small as possible and, more precisely, by requiring that
$$\int_a^b\left[s^{\prime \prime}(x)\right]^2 d x$$

be made as small as possible. This may not be a perfect mathematical realization of the idea of a smooth shape-preserving interpolation function for the data $\left{\left(x_i, y_i\right)\right}$, but it usually gives a very good interpolating function from a visual perspective.
There is a unique solution $s(x)$ to this problem, and it satisfies the following:
S1. $s(x)$ is a polynomial of degree $\leq 3$ on each subinterval $\left[x_{j-1}, x_j\right]$, for $j=2,3, \ldots, n$
S2. $s(x), s^{\prime}(x)$, and $s^{\prime \prime}(x)$ are continuous for $a \leq x \leq b$;
S3. $s^{\prime \prime}\left(x_1\right)=s^{\prime \prime}\left(x_n\right)=0$.

.

## 数学代写|线性代数代写线性代数代考|样条插值

$$x_1<x_2<\cdots<x_n$$
，并让$a=x_1, b=x_n$。我们寻找一个在$[a, b]$上定义的函数$s(x)$来插值数据:
$$s\left(x_i\right)=y_i, \quad i=1, \ldots, n$$

$$\int_a^b\left[s^{\prime \prime}(x)\right]^2 d x$$ 来实现的

S1。$s(x)$是在子区间$\left[x_{j-1}, x_j\right]$上的次数为$\leq 3$的多项式，当$j=2,3, \ldots, n$
S2时。$s(x), s^{\prime}(x)$和$s^{\prime \prime}(x)$对于$a \leq x \leq b$是连续的;
S3。$s^{\prime \prime}\left(x_1\right)=s^{\prime \prime}\left(x_n\right)=0$ .

## 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。