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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
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|量子力学代写quantum mechanics代考|bisection method

There are three basic methods to find the roots of a function: bisection, secant and Newton-Raphson. The process of finding the roots of a given function $f(x)$ means to find the values of $x$ where $f(x)=0$.

In the bisection method, the function is bisected (or divided into small sections) and one uses the recursive process to find the root(s) by increasing step-by-step the value of $\mathrm{x}$, according to a given interval $\mathrm{dx}$, from an initial trial value of $\mathrm{x}$. The recursive process ends when the tolerance (i.e., a given value approaching zero) is reached for a given $x$ in the function.

Let us use the function $f(x)=\mathrm{x}^{3}-9$, where the root is $2.080084$ with single precision. In the code below the trial value for $\mathrm{x}$ is $1.0$, the tolerance is $10^{-6}$ and the initial step size is $0.5$. In this first code, the number of bisections is indefinite.

## 物理代写|量子力学代写quantum mechanics代考|Finding roots of a function: Newton-Raphson method

The Newton-Raphson method is based on the tangent line equation.
Suppose a function $f(\mathrm{x})$ in which we intend to find the tangent line at the point $\mathrm{p}\left(\mathrm{x}{\mathrm{n}}, f\left(\mathrm{x}{\mathrm{n}}\right)\right)$. Let us consider a nearby point $\mathrm{q}\left(\mathrm{x}{\mathrm{n}}+\mathrm{h}, f\left(\mathrm{x}{\mathrm{n}}+\mathrm{h}\right)\right)$ on that curve. The segment uniting these points is the secant line. The slope of this secant line (the ratio $y / x)$ is:
$$\frac{f\left(x_{n}+h\right)-f\left(x_{n}\right)}{h}$$
As the value of $h$ becomes smaller and smaller, the point $q$ approaches $p$ and the secant line tends to become the tangent line when $\mathrm{h}$ is infinitesimally small, the slope assumes a certain value $k$. The coordinates of the tangent line $\left(x_{t}, y_{t}\right)$ can be found as:
$$\frac{y_{t}-f\left(x_{n}\right)}{\left(x_{t}-x_{n}\right)}=k=f^{\prime}\left(x_{n}\right)$$
then:
$$y_{t}=f\left(x_{n}\right)+f^{\prime}\left(x_{n}\right)\left(x_{t}-x_{n}\right)$$
Let us now choose the coordinates of the tangent line $\left(x_{t}, y_{t}\right)$ as the root of the function $f(\mathrm{x})$, that is, at $\mathrm{x}{\mathrm{n}+1}$ and $\mathrm{y}{\mathrm{t}}=0$.
\begin{aligned} &\text { for }: y_{t}=0 \therefore x_{t}=x_{n+1} \ &0=f\left(x_{n}\right)+f^{\prime}\left(x_{n}\right)\left(x_{n+1}-x_{n}\right) \end{aligned}
The value of $x_{n+1}$ is the guess for the root of the function:
$$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}$$
The Newton-Raphson method has a faster convergence than bisection method, but not always converge if the initial point $\mathrm{x}_{0}$ is not within the neighborhood of the root or if the in any iteration point it finds a stationary point or it finds a k-cycle. The Newton-Raphson method is only possible when the derivative of the function is known.

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Finding roots of a function: Newton-Raphson method

Newton-Raphson 方法基于切线方程。

$$\frac{f\left(x_{n}+h\right)-f\left(x_{n}\right)}{h}$$

$$\frac{y_{t}-f\left(x_{n}\right)}{\left(x_{t}-x_{n}\right)}=k=f^{\prime}\left(x_{n}\right)$$

$$y_{t}=f\left(x_{n}\right)+f^{\prime}\left(x_{n}\right)\left(x_{t}-x_{n}\right)$$

$$\text { for : } y_{t}=0 \therefore x_{t}=x_{n+1} \quad 0=f\left(x_{n}\right)+f^{\prime}\left(x_{n}\right)\left(x_{n+1}-x_{n}\right)$$

$$x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}$$
Newton-Raphson 方法比二分法收敛速度更快，但如果初始点不总是收敛 $\mathrm{x}_{0}$ 不在根的邻域内，或者如果在任何迭代点中它找到一个固定点或它找到一个 $k$ 㤧 只有当函数的导数已知时，Newton-Raphson 方法才有可能。

## 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。

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