### 数学代写|数值分析代写numerical analysis代考|MATHS7104

statistics-lab™ 为您的留学生涯保驾护航 在代写数值分析numerical analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数值分析numerical analysis代写方面经验极为丰富，各种代写数值分析numerical analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Examples of the effects of roundoff error

To motivate the need to study computer representation of numbers, let us consider first some examples taken from MATLAB-but we note that the same thing happens in C, Java, etc.:

1. The order in which you add numbers on a computer makes a difference!
\begin{aligned} & >1+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16 \ & \text { ans }= \ & \quad 1 \ & >1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \ & \text { ans }= \ & 1.000000000000001 \end{aligned}
Note: AAAeBBB is a common notation for a floating-point number with the value $A A A \times$ $10^{B B B}$. So $1 \mathrm{e}-16=10^{-16}$.

As we will see later in this chapter, the computer stores about 16 base 10 digits for each number; this means we get 15 digits after the first nonzero digit of a number. Hence, if you try to add 1e-16 to 1, there is nowhere for the computer to store the 1e-16 since it is the 17 th digit of a number starting with 1 . It does not matter how many times you add 1e-16; it just gets lost in each intermediate step, since operations are always done from left to right. So even if we add $1 \mathrm{e}-16$ to 1,10 times in a row, we get back exactly 1 . However, if we first add the $1 \mathrm{e}-16$ ‘s together, then add the 1 , these small numbers get a chance to combine to become big enough not to be lost when added to 1 .

$$f(x)=\frac{e^x-e^{-x}}{x}$$
Suppose we wish to calculate
$$\lim _{x \rightarrow 0} f(x) .$$
By L’Hopital’s theorem, we can easily determine the answer to be 2. However, how might one do this on a computer? A limit is an infinite process, and moreover, it requires some analysis to get an answer. Hence on a computer one is seemingly left with the option of choosing small $x$ ‘s and plugging them into $f$. Table $1.1$ shows what we get back from MATLAB by doing so.

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

Computers and software allow us to work in base 10 , but behind the scenes everything is done in base 2. This is because numbers are stored in computer memory (essentially) as “voltage on” (1) or “voltage off” (0). Hence, it is natural to represent numbers in their base 2, or binary, representation. To explain this, let us start with base 10 , or decimal, number system. In base 10 , the number $12.625$ can be expanded into powers of 10 , each multiplied by a coefficient:
$$12.625=1 \times 10^1+2 \times 10^0+6 \times 10^{-1}+2 \times 10^{-2}+5 \times 10^{-3} .$$
It should be intuitive that the coefficients of the powers of 10 must be digits between 0 and 9. Also, the decimal point goes between the coefficients of $10^{\circ}$ and $10^{-1}$.

Base 2 numbers work in an analogous fashion. First, note that it only makes sense to have digits of 0 and 1 , for the same reason that digits in base 10 must be 0 through 9 . Also, the decimal point goes between the coefficients of $2^0$ and $2^{-1}$. Hence in base 2 we have, for example, that
$$(11.001)_{\text {base } 2}=1 \times 2^1+1 \times 2^0+0 \times 2^{-1}+0 \times 2^{-2}+1 \times 2^{-3}=2+1+\frac{1}{8}=3.125 .$$
Converting a base 2 number to a base 10 number is nothing more than expanding it into powers of 2. To get an intuition for this, consider Table $1.3$ that converts the base 10 numbers 1 through 10.

The following algorithm will convert a base 10 number to a base 2 number. Note this is not the most efficient computational algorithm, but perhaps it is the easiest to understand for beginners.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Examples of the effects of roundoff error

1. 您在计算机上添加数字的顺序会有所不同!
$$1+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16+1 \mathrm{e}-16 \quad \text { ans }$$

$$f(x)=\frac{e^x-e^{-x}}{x}$$

$$\lim _{x \rightarrow 0} f(x) .$$

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

$$12.625=1 \times 10^1+2 \times 10^0+6 \times 10^{-1}+2 \times 10^{-2}+5 \times 10^{-3} .$$

$(11.001)_{\text {base } 2}=1 \times 2^1+1 \times 2^0+0 \times 2^{-1}+0 \times 2^{-2}+1 \times 2^{-3}=2+1+\frac{1}{8}=3.125$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。