### 数学代写|数值分析代写numerical analysis代考|MATH2722

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数值分析代写numerical analysis代考|Two flavors of error

The approximation can be derived by using a Taylor series. Expand $f\left(x_0+h\right)$ around $x_0$ :
$$f\left(x_0+h\right)=f\left(x_0\right)+h f^{\prime}\left(x_0\right)+\frac{h^2}{2} f^{\prime \prime}\left(x_0\right)+O\left(h^3\right)$$
where $O\left(h^3\right)$ denotes the $h^3$ and higher terms in the series. Now solve for $f^{\prime}\left(x_0\right)$ to obtain
$$f^{\prime}\left(x_0\right)=\frac{f\left(x_0+h\right)-f\left(x_0\right)}{h}-\frac{h}{2} f^{\prime \prime}\left(x_0\right)+O\left(h^2\right) .$$
We conclude that
$$D(f, h)=f^{\prime}\left(x_0\right)+\frac{f^{\prime \prime}\left(x_0\right)}{2} h+\cdots .$$
The terms to the right of $f^{\prime}$ are the truncation error – the part of the formula that is dropped to get the approximation. From the leading term of the error (the $h$-term) we see that
$$\text { truncation error } \sim C h \text { as } h \rightarrow 0 .$$
If $f$ were computed exactly, this would also be $E(h)$. But the function $f$, in practice, is not computed exactly (nor is the subtraction in the numerator) so our analysis must take into account such errors. Let’s assume that $f$ is simple and implemented to high accuracy on the computer. Then for any evaluation $\tilde{f}$ of the function,
$$\tilde{f}=f+\delta, \quad|\delta|<\mathrm{u}_m .$$
Then the computed value of $D$ is really
$$\tilde{D}(f, h)=D(f, h)+\frac{\delta_1-\delta_0}{h}$$
which introduces a ’rounding error’ (not part of the truncation error, only present in computation)
$$\mid \text { ’rounding’ error } \mid \leq \frac{\mathrm{u}_m+\mathrm{u}_m}{h}=\frac{2 \mathrm{u}_m}{h} .$$

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

The mathematical theory is motivated by a need to design good numerical methods. There is no perfect algorithm for a given (non-trivial) problem, so each property has a cost – gaining one typically means losing another.

Here are some of the main (overlapping) concerns. The meaning depends on context – different problems demand different properties, and the needs of the user (how accurate or fast does it need to be?) matter also.

• Given a tolerance $\epsilon$, can the algorithm find a solution to within $\epsilon$ ?
• How much time and memory (space) is required to solve the problem at a given accuracy?
• How does the time/memory scale with problem complexity?
• Can the correctness of the solution be verified (reliable error bounds)?
• Robustness/scope
• What is the scope of the algorithm – what problems can it solve? How general is it?
• Can the algorithm adapt to deal with hard cases or does the user need to step in?
• Does the user need to see the inner workings is it a ‘black box’?
• Stability:
• Does the algorithm keep rounding and other errors under control?

Example: Consider Newton’s method
$$x_{n+1}=x_n-\frac{f(x)}{f^{\prime}(x)}, \quad x_0=?$$
which finds a zero $x^$ of a function $f(x)$ (the sequence $x_n$, hopefully, converges to $x^$ ).
On efficiency/accuracy, we want to know whether $x_n$ converges to $x^$ as $n \rightarrow \infty$ and describe how fast the error $\left|x_n-x^\right|$ goes to zero. Given a maximum allowed error $\epsilon$, how many iterations are needed to make the error less than $\epsilon$ ? (answer: the error goes to zero ‘quadratically’; the number of significant digits doubles at each step if the function is nice).

On scope, we want to know for which functions $f(x)$ the algorithm works (answer: $f \in C^2$ or sometimes less, but $f$ must be at least differentiable; only fast when $x^*$ is a simple zero)

On stability, we want to know if an initial error in $x_0$ or errors in evaluating $f$ are amplified as the algorithm progresses. (answer: it works fine).

On robustness, we want to know if the algorithm may fail when given a not-good starting ‘guess’ $x_0$. (answer: not very robust; hard to know how close $x_0$ must be and diverges if too far from the root; requires some human attention).

In short, Newton’s method is extremely accurate/efficient when $x_0$ is chosen close to $x^*$ and the function is smooth and the zero is simple. However, it is not very robust unless coupled with a good scheme for choosing $x_0$ (which is hard to design).

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Two flavors of error

$$f\left(x_0+h\right)=f\left(x_0\right)+h f^{\prime}\left(x_0\right)+\frac{h^2}{2} f^{\prime \prime}\left(x_0\right)+O\left(h^3\right)$$

$$f^{\prime}\left(x_0\right)=\frac{f\left(x_0+h\right)-f\left(x_0\right)}{h}-\frac{h}{2} f^{\prime \prime}\left(x_0\right)+O\left(h^2\right) .$$

$$D(f, h)=f^{\prime}\left(x_0\right)+\frac{f^{\prime \prime}\left(x_0\right)}{2} h+\cdots$$

$$\tilde{f}=f+\delta, \quad|\delta|<\mathbf{u}_m .$$

$$\tilde{D}(f, h)=D(f, h)+\frac{\delta_1-\delta_0}{h}$$

$$\mid \text { ’rounding’ error } \mid \leq \frac{\mathrm{u}_m+\mathrm{u}_m}{h}=\frac{2 \mathrm{u}_m}{h} \text {. }$$

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

• 效率和准确性的权衡:
• 给定公差 $\epsilon$, 算法能否在 $\epsilon$ ?
• 以给定的精度解决问题需要多少时间和内存 (空间) ?
• 时间/内存如何随着问题的复杂性扩展?
• 能否验证解决方案的正确性 (可靠的误差范围) ?
• 稳健性范围
• 该算法的范围是什么一一它可以解决什么问题? 它有多普遍?
• 该算法能否适应处理疑难案例或是否需要用户介入?
• 用户是否需要查看内部工作原理? 它是一个”黑匣子”吗?
• 稳定:
• 算法是否控制舍入和其他错误?
• 稍后我们将对此进行更多讨论!
示例：考虑牛顿法
$$x_{n+1}=x_n-\frac{f(x)}{f^{\prime}(x)}, \quad x_0=?$$
找到一个雿 $\mathrm{x}^{\wedge}$ 个功能 $f(x)$ （序列 $x_n$ ，希望收敛到 $\mathrm{x}^{\wedge}$ ). 定最大允许误差 $\epsilon$ ，需要多少次迭代才能使误差小于 $\epsilon$ ? (答案: 错误“二次”地变为零; 如果函数很好，则 有效数字的数量在每一步都加倍)。
在范围上，我们想知道哪些功能 $f(x)$ 该算法有效（答案: $f \in C^2$ 或有时更少，但是 $f$ 必须至少是可微 的；只有当 $x^$ 是一个简单的零) 关于稳定性，我们想知道初始错误是否在 $x_0$ 或评估错误 $f$ 随着算法的进行而被放大。（答案：它工作正 常)。 在鲁棒性方面，我们想知道当给定一个不好的开始”猜测”时算法是否会失败 $x_0$. (答案: 不是很稳健; 很难 知道有多接近 $x_0$ 必须是并且如果离根太远则发散；需要一些人为的关注)。 简而言之，牛顿法在以下情况下非常准确/高效 $x_0$ 被选择接近 $x^$ 且函数流畅，归零简单。然而，它不是很 健壮，除非加上一个好的选择方案 $x_0$ (这很难设计)。

## 有限元方法代写

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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。