### 计算机代写|并行计算作业代写Parallel Computing代考|CSC267

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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

## 计算机代写|并行计算作业代写Parallel Computing代考|Numerical Versus Analytical Solutions

Computers are calculators. If we pass them a certain problem like “here are two bodies interacting through gravity”, they yield values as solution: “the bodies end up at position $\mathrm{x}, \mathrm{y}, \mathrm{z}$ “. They yield numerical solutions. This is different to quite a lot of maths we do in school. There, we manipulate formulae and compute expressions like $F(a, b)=\int_a^b 4 x d x=2 b^2-2 a^2$. Indeed, many teachers save us till the very last minute (or until we have pocket calculators) from inserting actual numbers for $b$ and $a$.

In programming languages, we often speak of variables. But these variables still contain data at any point of the program run. They are fundamentally different to variables in a mathematical formula which might or might not hold specific values. We conclude: There are two different ways to handle equations: We can try to solve them for arbitrary input, i.e. find expressions like $F(a, b)=2 b^2-2 a^2$. Once we have such a solution, we can insert different $a$ and $b$ values. The solution is an analytical solution. ${ }^4$ On a computer, we typically work numerically. We hand in numbers, and we get answers for these particular numbers. But we do not get any universal solution.

Definition $3.2$ (Analytical versus numerical solution) If we solve an equation via formula rewrites such as integration rules, we obtain an analytical solution over the variables. Analytical solutions describe a generic system behaviour. If we solve it for one particular set of initial values right from the start, we strive for a numerical solution.

Computers yield numerical solutions. This statement is not $100 \%$ correct. There are computer programs which yield symbolic solutions. They are called computer algebra systems. While they are very powerful, they cannot find an analytical solution always and obviously do not yield a result if there is no analytical solution. We walk down the numerics route in this course.

Analogous to this distinction of numerical and analytical solutions, we can also distinguish how we manipulate formulae. If you want to determine the derivative $\partial_t y(t)$ of $y(t)=t^2$, you know $\partial_t y(t)=2 t$ from school. You manipulate the formulae symbolically. In a computer, you can also evaluate $\partial_t y(t)$ only for a given input. This often comprises some algorithmic approximations. In this case, you again tackle the expression numerically.

## 计算机代写|并行计算作业代写Parallel Computing代考|Fixed Point Formats

The simplest scheme one might come up with for a machine is a fixed point storage format. In fixed point notation, we write down all numbers with a certain number of digits before the decimal point and a certain number after the decimal point. With four digits (decimal) and two leading digits, we can, for example, write the number three as 0300 . This means $03.00$. Fixed point storage is the first variant I have sketched in our introductory thought experiment.

We immediately see that such a representation is not a fit for scientific computing. Let $x=3$ in the representation be divided by three. The result $x / 3=1.00$ suits our data structure. However, once we divide by three once more, we start to run into serious trouble. Indeed $1 / 3 \approx 0.33$ – which is the closest value to $1 / 3$ we can numerically encode – is already off the real result by $0.00 \overline{3}$.

You might be tempted to accept that you have a large error. But as a computational scientist, you neither can accept that most of your bits soon start to hold zeroes, i.e. no information at all, nor that your storage format is only suited to hold numbers from a very limited range (basically from $0.01$ to 30 and even here with quite some error).

Definition $4.1$ (Relative and absolute error) Computers always yield wrong results. To quantify this effect, we distinguish the absolute from the relative error. Let $x_M$ be the machine’s representation of $x$. The absolute round-off error then is given by
$$e=\left|x_M-x\right| \text {. }$$
The relative round-off error is given by
$$\epsilon=\frac{e}{|x|}=\left|\frac{x_M-x}{x}\right|$$
The reason behind inaccurate representations of numbers is that we work numerically. Each number in the computer is mapped onto a sequence of bits. This sequence is finite. So at one point, we have to cut the bits of the real number off. We introduce an error.

## 计算机代写|并行计算作业代写Parallel Computing代考|Fixed Point Formats

$$e=\left|x_M-x\right| .$$

$$\epsilon=\frac{e}{|x|}=\left|\frac{x_M-x}{x}\right|$$

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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。