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# 数学代写|数值分析代写Numerical analysis代考|STAT434 LOSS OF SIGNIFICANCE

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## 数学代写|数值分析代写Numerical analysis代考|LOSS OF SIGNIFICANCE

An advantage of knowing the details of computer arithmetic is that we are therefore in a better position to understand potential pitfalls in computer calculations. One major problem that arises in many forms is the loss of significant digits that results from subtracting nearly equal numbers. In its simplest form, this is an obvious statement. Assume that through considerable effort, as part of a long calculation, we have determined two numbers correct to seven significant digits, and now need to subtract them:
$$\begin{array}{r} 123.4567 \ -123.4566 \ \hline 000.0001 \end{array}$$
The subtraction problem began with two input numbers that we knew to seven-digit accuracy, and ended with a result that has only one-digit accuracy. Although this example is quite straightforward, there are other examples of loss of significance that are more subtle, and in many cases this can be avoided by restructuring the calculation.

## 数学代写|数值分析代写Numerical analysis代考|REVIEW OF CALCULUS

Some important basic facts from calculus will be necessary later. The Intermediate Value Theorem and the Mean Value Theorem are important for solving equations in Chapter 1. Taylor’s Theorem is important for understanding interpolation in Chapter 3 and becomes of paramount importance for solving differential equations in Chapters 6,7 , and 8 .

The graph of a continuous function has no gaps. For example, if the function is positive for one $x$-value and negative for another, it must pass through zero somewhere. This fact is basic for getting equation solvers to work in the next chapter. The first theorem, illustrated in Figure $0.1(\mathrm{a})$, generalizes this notion.

(Intermediate Value Theorem) Let $f$ be a continuous function on the interval $[a, b]$. Then $f$ realizes every value between $f(a)$ and $f(b)$. More precisely, if $y$ is a number between $f(a)$ and $f(b)$, then there exists a number $c$ with $a \leq c \leq b$ such that $f(c)=y$.

Show that $f(x)=x^2-3$ on the interval $[1,3]$ must take on the values 0 and 1 .
Because $f(1)=-2$ and $f(3)=6$, all values between -2 and 6 , including 0 and 1 , must be taken on by $f$. For example, setting $c=\sqrt{3}$, note that $f(c)=$ $f(\sqrt{3})=0$, and secondly, $f(2)=1$
(Continuous Limits) Let $f$ be a continuous function in a neighborhood of $x_0$, and assume $\lim {n \rightarrow \infty} x_n=x_0$. Then $$\lim {n \rightarrow \infty} f\left(x_n\right)=f\left(\lim _{n \rightarrow \infty} x_n\right)=f\left(x_0\right)$$
In other words, limits may be brought inside continuous functions.

## 数学代写|数值分析代写NUMERICAL ANALYSIS代 考|LOSS OF SIGNIFICANCE

$\backslash$ begin ${$ array}{r} $123.4567 \backslash-123.4566 \backslash \backslash h l i n e 000.0001 \backslash$ end{array $}$

## 数学代写|数值分析代写NUMERICAL ANALYSIS代 考|REVIEW OF CALCULUS

IntermediateValueTheorem让 $f$ 是区间上的连续函数 $[a, b]$. 然后 $f$ 实现之间的每一个价值 $f(a)$ 和 $f(b)$. 更准确地说，如果 $y$ 是介于 $f(a)$ 和 $f(b)$, 那 么存在一个数

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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。