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

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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|Error Analysis

In this chapter, we intend to investigate and analyze the important complicated problems and points that occur in numerical calculations or calculations based on the numerical algorithms. As we know, in the numerical analysis, most numerical methods are iterative. This means that their formulation is in the form of difference equations. Therefore, given one or more initial values, the next values must be calculated. Usually, the initial values are not accurately available and are approximate; or due to the structure of the mathematical model, the calculations performed using iterative methods produce approximate results, that is, whether the initial values are approximate or not, the difference model may also have error factors. Obviously, two points were always considered in the computer or the numerical calculations. One is the speed of calculations and the other is the memory occupied by numerical results. Due to the advances in science and technology, the second factor has been ignored in the presentation of structured algorithms, but the first factor is considered as an advantage for the presentation of numerical algorithms. Given that each computational device has its own computational accuracy, it can be said that the zero of each computing device or computer is different from that of another computer, that is, the smallest positive number of one machine is different from that of another machine.

Therefore, an algorithm performed in two machines will have different results. But in both cases,there is a computational error which is less in one than the other. Currently, due to the advances in technology and the construction of advanced satellites and long-range air-to-air missiles and missiles with nuclear warheads, an approximate estimate of the target with the lowest error rate and the calculations of missile or satellite launch with the least amount of error is important. This is because the missile is trying to hit a specified target over a distance of, for example, thousands of kilometers, which may also be approximate. However, how to launch the missile, initial speed, initial acceleration, traveled distance, obstacles in the path of the missile such as air resistance, winds blowing from lateral directions, etc., and how to hit the target, all are factors that required to be considered, and obviously none of these factors can be accessed accurately and without an error. Therefore, taking into account these factors and problems, hitting the target with a missile should be done with an error of, for example, a maximum of $0.01$. Obviously, all models related to this process are in the form of mathematical models, for example, the differential equations with the initial conditions, the integral equations, the differential equations with partial derivatives, the calculations of integral series, and so on. So we need to examine the errors of such models and estimate the upper and lower bounds of such errors. In this regard, some problems about error analysis are presented.

## 数学代写|数值分析代写numerical analysis代考|Errors in an Algorithm

Suppose that $Y=\phi(X)$, where $\phi$ is a combination of all the steps of the algorithm. For this purpose, we define:
$$\phi: D \rightarrow \mathbb{R}^{m}$$
where $D$ is an open subset of $\mathbb{R}^{n}$. We also assume that $X^{t}=\left(x_{1}, \ldots, x_{n}\right)$ and $Y^{t}=\left(y_{1}, \ldots, y_{m}\right)$ are the input and output vectors of the algorithm, respectively. It is clear that:
$$y_{i}=\varphi_{i}\left(x_{1}, \ldots, x_{n}\right), \quad i=1, \ldots, m$$
that is:
$$Y=\left[\begin{array}{c} y_{1} \ \vdots \ y_{m} \end{array}\right]=\left[\begin{array}{c} \varphi_{1}\left(x_{1}, \ldots, x_{n}\right) \ \vdots \ \varphi_{\mathrm{m}}\left(x_{1}, \ldots, x_{11}\right) \end{array}\right]$$
If we want to specify $\phi$ for an algorithm that has $r+1$ operators (steps), we have:
$$\varphi^{(i)}: D_{i} \rightarrow D_{i+1,} \quad i=0, \ldots, r, D_{i} \subseteq \mathbb{R}^{n_{i}}, n_{i} \in \mathbb{Z}$$

$$\phi=\varphi^{(r)} \ldots \varphi^{(0)}, D_{0}=D, D_{r+1} \subseteq \mathbb{R}^{n_{r}+1}=\mathbb{R}^{m}$$
To calculate $\phi$ in each algorithm, we have the ordered series of $\varphi^{(i)}$ operators with the sum equal to $\phi$, so that the output of one operator will be the input of the next operator, and finally the output of the last operator will be $Y$.
If in $i$ th step of the algorithm, the vector $X^{(i)}$ has $n_{i}$ inputs for the operator $\varphi^{(i)}$, then we have:
$$\varphi^{(i)}: D_{i} \rightarrow \mathbb{R}^{n_{i}+1}, D_{i} \subseteq \mathbb{R}^{n_{i}}$$
So that
$$\varphi^{(i)}\left(X^{(i)}\right)=X^{(i+1)}$$
$$\begin{gathered} \phi: D \rightarrow \mathbb{R}^{m}, D \subseteq \mathbb{R}^{n} \ \phi(X)=\left[\begin{array}{c} \varphi_{1}\left(x_{1}, \ldots, x_{n}\right) \ \vdots \ \varphi_{m}\left(x_{1}, \ldots, x_{n}\right) \end{array}\right] \end{gathered}$$

## 数学代写|数值分析代写numerical analysis代考|Errors in an Algorithm

$$\phi: D \rightarrow \mathbb{R}^{m}$$

$$y_{i}=\varphi_{i}\left(x_{1}, \ldots, x_{n}\right), \quad i=1, \ldots, m$$

$$Y=\left[y_{1} \vdots y_{m}\right]=\left[\varphi_{1}\left(x_{1}, \ldots, x_{n}\right) \vdots \varphi_{\mathrm{m}}\left(x_{1}, \ldots, x_{11}\right)\right]$$

$$\begin{gathered} \varphi^{(i)}: D_{i} \rightarrow D_{i+1,} \quad i=0, \ldots, r, D_{i} \subseteq \mathbb{R}^{n_{i}}, n_{i} \in \mathbb{Z} \ \phi=\varphi^{(r)} \ldots \varphi^{(0)}, D_{0}=D, D_{r+1} \subseteq \mathbb{R}^{n_{r}+1}=\mathbb{R}^{m} \end{gathered}$$

$$\varphi^{(i)}: D_{i} \rightarrow \mathbb{R}^{n_{i}+1}, D_{i} \subseteq \mathbb{R}^{n_{i}}$$

$$\varphi^{(i)}\left(X^{(i)}\right)=X^{(i+1)}$$

$$\phi: D \rightarrow \mathbb{R}^{m}, D \subseteq \mathbb{R}^{n} \phi(X)=\left[\varphi_{1}\left(x_{1}, \ldots, x_{n}\right) \vdots \varphi_{m}\left(x_{1}, \ldots, x_{n}\right)\right]$$

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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。