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# 计算机代写|基础编程代写Fundamental of Programming代考|Harmonic Numbers

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## 计算机代写|基础编程代写Fundamental of Programming代考|Harmonic Numbers

The following sum will be of great importance in our later work:
$$H_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\sum_{k=1}^n \frac{1}{k}, \quad n \geq 0 .$$
This sum does not occur very frequently in classical mathematics, and there is no standard notation for it; but in the analysis of algorithms it pops up nearly every time we turn around, and we will consistently call it $H_n$. Besides $H_n$, the notations $h_n$ and $S_n$ and $\psi(n+1)+\gamma$ are found in mathematical literature. The letter $H$ stands for “harmonic,” and we speak of $H_n$ as a harmonic number because (1) is customarily called the harmonic series. Chinese bamboo strips written before 186 B.C. already explained how to compute $H_{10}=7381 / 2520$, as an exercise in arithmetic. [See C. Cullen, Historia Math. 34 (2007), 10-44.]
It may seem at first that $H_n$ does not get too large when $n$ has a large value, since we are always adding smaller and smaller numbers. But actually it is not hard to see that $H_n$ will get as large as we please if we take $n$ to be big enough, because
$$H_{2^m} \geq 1+\frac{m}{2}$$
This lower bound follows from the observation that, for $m \geq 0$, we have
\begin{aligned} H_{2^{m+1}} & =H_{2^m}+\frac{1}{2^m+1}+\frac{1}{2^m+2}+\cdots+\frac{1}{2^{m+1}} \ & \geq H_{2^m}+\frac{1}{2^{m+1}}+\frac{1}{2^{m+1}}+\cdots+\frac{1}{2^{m+1}}=H_{2^m}+\frac{1}{2} \end{aligned}

## 计算机代写|基础编程代写Fundamental of Programming代考|Fibonacci Numbers

The sequence
$$0,1,1,2,3,5,8,13,21,34, \ldots,$$
in which each number is the sum of the preceding two, plays an important role in at least a dozen seemingly unrelated algorithms that we will study later. The numbers in the sequence are denoted by $F_n$, and we formally define them as
$$F_0=0 ; \quad F_1=1 ; \quad F_{n+2}=F_{n+1}+F_n, \quad n \geq 0 .$$
This famous sequence was published in 1202 by Leonardo Pisano (Leonardo of Pisa), who is sometimes called Leonardo Fibonacci (Filius Bonaccii, son of Bonaccio). His Liber Abaci (Book of the Abacus) contains the following exercise: “How many pairs of rabbits can be produced from a single pair in a year’s time?” To solve this problem, we are told to assume that each pair produces a new pair of offspring every month, and that each new pair becomes fertile at the age of one month. Furthermore, the rabbits never die. After one month there will be 2 pairs of rabbits; after two months, there will be 3 ; the following month the original pair and the pair born during the first month will both usher in a new pair and there will be 5 in all; and so on.

Fibonacci was by far the greatest European mathematician of the Middle Ages. He studied the work of al-Khwārizm̄̄ (after whom “algorithm” is named, see Section 1.1) and he added numerous original contributions to arithmetic and geometry. The writings of Fibonacci were reprinted in 1857 [B. Boncompagni, Scritti di Leonardo Pisano (Rome, 1857-1862), 2 vols.; $F_n$ appears in Vol. 1, 283285]. His rabbit problem was, of course, not posed as a practical application to biology and the population explosion; it was an exercise in addition. In fact, it still makes a rather good computer exercise about addition (see exercise 3);

Fibonacci wrote: “It is possible to do [the addition] in this order for an infinite number of months.”

Before Fibonacci wrote his work, the sequence $\left\langle F_n\right\rangle$ had already been discussed by Indian scholars, who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes or syllables. The number of such rhythms having $n$ beats altogether is $F_{n+1}$; therefore both Gopāla (before 1135) and Hemacandra (c. 1150) mentioned the numbers 1, 2, 3, 5, 8, 13, 21 , 34, … explicitly. [See P. Singh, Historia Math. 12 (1985), 229-244; see also exercise $4.5 .3-32$.

## 计算机代写|基础编程代写Fundamental of Programming代考|Harmonic Numbers

$$H_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\sum_{k=1}^n \frac{1}{k}, \quad n \geq 0 .$$

$$H_{2^m} \geq 1+\frac{m}{2}$$

\begin{aligned} H_{2^{m+1}} & =H_{2^m}+\frac{1}{2^m+1}+\frac{1}{2^m+2}+\cdots+\frac{1}{2^{m+1}} \ & \geq H_{2^m}+\frac{1}{2^{m+1}}+\frac{1}{2^{m+1}}+\cdots+\frac{1}{2^{m+1}}=H_{2^m}+\frac{1}{2} \end{aligned}

## 计算机代写|基础编程代写Fundamental of Programming代考|Fibonacci Numbers

$$0,1,1,2,3,5,8,13,21,34, \ldots,$$

$$F_0=0 ; \quad F_1=1 ; \quad F_{n+2}=F_{n+1}+F_n, \quad n \geq 0 .$$

## 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。