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## 计算机代写|基础编程代写Fundamental of Programming代考|Analysis of an Algorithm

Let us now apply some of the techniques of the preceding sections to the study of a typical algorithm.

Algorithm M (Find the maximum). Given $n$ elements $X[1], X[2], \ldots, X[n]$, we will find $m$ and $j$ such that $m=X[j]=\max _{1 \leq i \leq n} X[i]$, where $j$ is the largest index that satisfies this relation.

M1. [Initialize.] Set $j \leftarrow n, k \leftarrow n-1, m \leftarrow X[n]$. (During this algorithm we will have $m=X[j]=\max _{k<i \leq n} X[i]$.)
M2. [All tested?] If $k=0$, the algorithm terminates.
M3. [Compare.] If $X[k] \leq m$, go to M5.
M4. [Change $m$.] Set $j \leftarrow k, m \leftarrow X[k]$. (This value of $m$ is a new current maximum.)
M5. [Decrease $k$. .] Decrease $k$ by one and return to M2. I
This rather obvious algorithm may seem so trivial that we shouldn’t bother to analyze it in detail; but it actually makes a good demonstration of the way in which more complicated algorithms may be studied. Analysis of algorithms is quite important in computer programming, because there are usually several algorithms available for a particular application and we would like to know which is best.

## 计算机代写|基础编程代写Fundamental of Programming代考|Asymptotic Representations

We often want to know a quantity approximately, instead of exactly, in order to compare it to another. For example, Stirling’s approximation to $n$ ! is a useful representation of this type, when $n$ is large, and we have also made use of the fact that $H_n \approx \ln n+\gamma$. The derivations of such asymptotic formulas generally involve higher mathematics, although in the following subsections we will use nothing more than elementary calculus to get the results we need.
*1.2.11.1. The $O$-notation. Paul Bachmann introduced a very convenient notation for approximations in his book Analytische Zahlentheorie (1894). It is the $O$-notation, which allows us to replace the ” $”$ sign by “=” and to quantify the degree of accuracy; for example,
$$H_n=\ln n+\gamma+O\left(\frac{1}{n}\right) .$$
(Read, “H sub $n$ equals the natural $\log$ of $n$ plus Euler’s constant [pronounced ‘Oiler’s constant’] plus big-oh of one over $n . “)$

In general, the notation $O(f(n))$ may be used whenever $f(n)$ is a function of the positive integer $n$; it stands for a quantity that is not explicitly known, except that its magnitude isn’t too large. Every appearance of $O(f(n))$ means precisely this: There are positive constants $M$ and $n_0$ such that the number $x_n$ represented by $O(f(n))$ satisfies the condition $\left|x_n\right| \leq M|f(n)|$, for all integers $n \geq n_0$. We do not say what the constants $M$ and $n_0$ are, and indeed those constants are usually different for each appearance of $O$.

## 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。