### 数学代写|数值分析代写numerical analysis代考|MATH3003

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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|bit floating-point numbers

By far the most common computer number representation system is the 64-bit “double” floating-point number system. This is the default used by all major mathematical and computational software. In some cases, it makes sense to use 32 or 128 bit number systems, but that is a discussion for later (later, as in “not in this book”), as first we must learn the basics. Each “bit” on a computer is a 0 or a 1, and each number on a computer is represented by 640 ‘s and 1’s. If we assume each number is in standard binary form, then the important information for each number is (i) sign of the number, (ii) exponent, and (iii) the digits after the decimal point. Note that the number 0 is an exception and is treated as a special case for the number system.
The IEEE standard divides up the 64 bits as follows:

• 1 bit sign: 0 for positive, 1 for negative;
• 11 bit exponent: the base 2 representation of (standard binary form exponent + 1023);
• 52 bit mantissa: the first 52 digits after decimal point from standard binary form.
The reason for the “shift” (sometimes also called bias) of 1023 in the exponent is so that the computer does not have to store a sign for the exponent (more numbers can be stored this way). The computer knows internally that the number is shifted, and knows how to handle it.

With the bits from above denoted as sign $s$, exponent $E$, and mantissa $b_1, \ldots, b_{52}$ the corresponding number is standard binary form is $(-1)^s \cdot 1 . b_1 \ldots b_{52} \times 2^{E-1023}$.
Example 2. Convert the base 10 number $d=11.5625$ to 64 bit double floating-point representation.

From a previous example, we know that $11.5625=(1011.1001)_{\text {base2 }}$, and so has standard binary representation of $1.0111001 \times 2^3$. Hence, we immediately know that
\begin{aligned} & \text { sign bit }=0 \ & \text { mantissa }=0111001000000000000000000000000000000000000000000000 \end{aligned}

As we saw in Example 1 in this chapter, if we add 1 to $10^{-16}$, it does not change the 1 at all. Additionally, the next computer representable number after 1 is $1+2^{-52}=$ $1+2.22 \times 10^{-16}$. Since $1+10^{-16}$ is closer to 1 than it is to $1+2.22 \times 10^{-16}$, it gets rounded to 1 , leaving the $10^{-16}$ to be lost forever.

We have seen this effect in the example at the beginning of this chapter when repeatedly adding $10^{-16}$ to 1 . Theoretically speaking, addition in floating-point computation is not associative, meaning $(A+B)+C=A+(B+C)$ may not hold, due to rounding.

One way to minimize this type of error when adding several numbers is to add from smallest to largest (if they all have the same sign) and to use factorizations that lessen the problem. There are other more complicated ways to deal with this kind of error that is out of the scope of this book, for example, the “Kahan Summation Formula.”

The issue here is that insignificant digits can become significant digits, and the problem is illustrated in Example 2, earlier in this chapter. Consider the following MATLAB command and output:
\begin{aligned} & \gg 1+1 e-15-1 \ & \text { ans }= \ & \text { 1. } 110223024625157 \mathrm{e}-15 \ & \end{aligned}
Clearly, the answer should be $10^{-15}$, but we do not get that, as we observe error in the second significant digit. It is true that the digits of accuracy in the subtraction operation is 16 , but there is a potential problem with the “garbage” digits 110223024625157 (these digits arise from rounding error). If we are calculating a limit, for example, they could play a role.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|bit floating-point numbers

• 1位符号: 0为正, 1 为负；
• 11位指数： (标准二进制形式指数 $+1023$ ) 的2进制表示;
• 52 位尾数：标准二进制形式的小数点后的前 52 位。
指数中 1023 的”移位” (有时也称为偏差) 的原因是计算机不必为指数存储符号 (可以通过这种方式 存储更多数字) 。计算机内部知道数字被移动了，并且知道如何处理它。
上面的位表示为符号 $s$ ，指数 $E$ ，和尾数 $b_1, \ldots, b_{52}$ 相应的数字是标准的二进制形式是 $(-1)^s \cdot 1 . b_1 \ldots b_{52} \times 2^{E-1023}$.
示例 2. 转换以 10 为底的数字 $d=11.5625$ 到 64 位双浮点表示。
从前面的例子我们知道 $11.5625=(1011.1001)_{\text {base22 }}$ ，所以有标准的二进制表示 $1.0111001 \times 2^3$. 因此，我们立即知道
$$\text { sign bit }=0 \quad \text { mantissa }=011100100000000000000000000000000000000000000000$$

$$\gg 1+1 e-15-1 \quad \text { ans }=1.110223024625157 \mathrm{e}-15$$

## 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。