## 数学代写|数值分析代写numerical analysis代考|MATHS 7104

2022年7月25日

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## 数学代写|数值分析代写numerical analysis代考|Spline Interpolation

Splines were first introduced in 1946 by a person named I. J. Schoenbery. If we want to describe this type of interpolator, we must say that it is one of the best interpolators considering the properties and characteristics of interpolation that will be discussed, because it is both a best approximation and a most accurate approximation. The best in the sense that the behavior of the interpolation points is approximated very smoothly, that is, it can be said that it is a functional approximation. Also, it is a most accurate approximation in the sense that in different points, it has approximations with much less error. So, it can be claimed that splines approximate the behavior of the function more accurately. In fact, there are two reasons why this interpolator is superior to other interpolators.

First, in addition to interpolating points $\left(x_{i}, f_{i}\right),\left(x_{i}, f_{i}\right)$, and also $\left(x_{i}, f_{i}\right)$, this function can be said to approximate higher order derivatives, too. Second, this interpolator has a uniform convergence. As we have already seen, other interpolators did not have such properties. Although it can be claimed that the Hermite-type interpolation also has these two properties, but with a difference that in Hermite interpolation, points must exist in the form of $\left(x_{i}, f_{i}^{(k)}\right)$. This means that consecutive derivatives of the function muse exist as the width of the points, and this is a disadvantage, because this is not always possible. On the other hand, spline works with the points $\left(x_{i}, f_{i}\right)$ and does not need Hermite interpolation points, but interpolating convergence to the function and interpolating derivatives to the function derivatives are made in the spline conditions. Therefore, it can be said that if Hermite interpolation conditions are satisfied, Hermite interpolator has both So, it can be said that spline is preferable to Hermite. Due to the good features mentioned about spline, many scientific applications can be enumerated for it, for example civil works, surveying, construction of international airport it, for example civil works, surveying, construction of international airport neering software, etc.

## 数学代写|数值分析代写numerical analysis代考|Lemma

Suppose that $v \in V$ and the cluster point $u^{}$ form a minimum sequence. If $u^{} \in T$, then it is the best approximation of $v$ out of $T$.
Proof: Suppose that $\left(u_{i}\right)$ is a minimum sequence that
$$\lim {i \rightarrow \infty}\left|v-u{i}\right|=E_{T}(v)$$
Also assume that subsequence $\left(u_{i(i)}\right)$ converges to $u^{} \in T$. In this case, given that $$\lim {i \rightarrow \infty}\left|v-u{i}\right|=E_{T}(v), \quad \lim {j \rightarrow \infty}\left|u{i}-u^{i}\right|=0$$
we can say that for every $j$, we have:
$$\left|u-u^{}\right| \leq\left|v-u_{i}\right|+\left|u_{j}-u^{}\right|, \quad\left|v-u^{}\right| \leq E_{T}(v)$$
For every $u \in T$, we have
$$E_{T}(v) \leq|v-u|$$
So, it can be concluded that
$$\left|v-u^{}\right|=E_{T}(v)$$ and $u^{}$ is the best approximation.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Lemma

$$\lim i \rightarrow \infty|v-u i|=E_{T}(v)$$

$$\lim i \rightarrow \infty|v-u i|=E_{T}(v), \quad \lim j \rightarrow \infty\left|u i-u^{i}\right|=0$$

$$|u-u| \leq\left|v-u_{i}\right|+\left|u_{j}-u\right|, \quad|v-u| \leq E_{T}(v)$$

$$E_{T}(v) \leq|v-u|$$

$$|v-u|=E_{T}(v)$$

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## 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。