### 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
• 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|FLOATING-POINT ARITHMETIC

Scientific calculations are usually carried out in floating-point arithmetic. An $n$-digit floating-point number in base $\beta$ has the form
$$x=\pm\left(. d_1 d_2 \cdots d_n\right)\beta \beta^e$$ where $\left(. d_1 d_2 \cdots d_n\right)\beta$ is a $\beta$-fraction called the mantissa, and $e$ is an integer called the exponent. Such a floating-point number is said to be normalized in case $d_1 \neq 0$, or else $d_1=d_2=\cdots=d_n=0$.

For most computers, $\beta=2$, although on some, $\beta=16$, and in hand calculations and on most desk and pocket calculators, $\beta=10$.

The precision or length $n$ of floating-point numbers on any particular computer is usually determined by the word length of the computer and may therefore vary widely (see Fig. 1.1). Computing systems which accept FORTRAN programs are expected to provide floating-point numbers of two different lengths, one roughly double the other. The shorter one, called single precision, is ordinarily used unless the other, called double precision, is specifically asked for. Calculation in double precision usually doubles the storage requirements and more than doubles running time as compared with single precision.

The exponent $e$ is limited to a range
$$m<e<M$$
for certain integers $m$ and $M$. Usually, $m=-M$, but the limits may vary widely; see Fig. $1.1$.

There are two commonly used ways of translating a given real number $x$ into an $n \beta$-digit floating-point number $f l(x)$, rounding and chopping. In

## 数学代写|数值分析代写numerical analysis代考|LOSS OF SIGNIFICANCE AND ERROR PROPAGATION

If the number $x^$ is an approximation to the exact answer $\mathrm{x}$, then we call the difference $x-x^$ the error in $x^$; thus Exact $=$ approximation $+$ error The relative error in $x^$, as an approximation to $\mathrm{x}$, is defined to be the number $\left(x-x^\right) / x$. Note that this number is close to the number $(x-$ $\left.x^\right) / x^$ if it is at all small. [Precisely, if $\alpha=\left(x-x^\right) / x$, then $(x-$ $x ) / x^=\alpha /(1-\alpha)$.]

Every floating-point operation in a computational process may give rise to an error which, once generated, may then be amplified or reduced in subsequent operations.

One of the most common (and often avoidable) ways of increasing the importance of an error is commonly called loss of significant digits. If $x^$ is an approximation to $\mathrm{x}$, then we say that $x^$ approximates $x$ to $r$ significant $\beta$-digits provided the absolute error $\left|x-x^\right|$ is at most $\frac{1}{2}$ in the $r$ th significant $\beta$-digitof $x$. This can be expressed in a formula as $$\left|x-x^\right| \leq \frac{1}{2} \beta^{s-r+1}$$
with $s$ the largest integer such that $\beta^s \leq|x|$. For instance, $x^=3$ agrees with $x=\pi$ to one significant (decimal) digit, while $x^=\frac{22}{7}=3.1428 \cdots$ is correct to three significant digits (as an approximation to $\pi$ ). Suppose now that we are to calculate the number
$$z=x-y$$
and that we have approximations $x^$ and $y^$ for $x$ and $y$, respectively, available, each of which is good to $r$ digits. Then
$$z^=x^-y^*$$
is an approximation for $z$, which is also good to $r$ digits unless $x^$ and $y^$ agree to one or more digits. In this latter case, there will be cancellation of digits during the subtraction, and consequently $z^*$ will be accurate to fewer than $r$ digits.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|FLOATING-POINT ARITHMETIC

$$x=\pm\left(. d_1 d_2 \cdots d_n\right) \beta \beta^e$$

$$m<e<M$$

## 数学代写|数值分析代写numerical analysis代考|LOSS OF SIGNIFICANCE AND ERROR PROPAGATION

$$z=x-y$$

$$z^{=} x^{-} y^*$$

## 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。