# 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|AEM6061

#### Doug I. Jones

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## 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|Modulo 2 Linear Generators

Good random generators must have very large state spaces. For a linear congruential generator, this means that the modulus $m$ must be a large integer. However, for multiple recursive generators, it is not necessary to take a large modulus, as the period length can be as large as $m^k-1$. Because binary operations are in general faster than floating point operations (which are in turn faster than integer operations), it makes sense to consider MRGs and other random number generators that are based on linear recurrences modulo 2. A general framework for such random number generators is given in [10], where the state is a $k$-bit vector $\mathbf{X}t=\left(X{t, 1}, \ldots, X_{t, k}\right)^{\top}$ that is mapped via a linear transformation to a $w$-bit output vector $\mathbf{Y}t=\left(Y{t, 1}, \ldots, Y_{t, w}\right)^{\top}$, from which the random number $U_t \in(0,1)$ is obtained by bitwise decimation as follows: Here, $A$ and $B$ are $k \times k$ and $w \times k$ binary matrices, respectively, and all operations are performed modulo 2. In particular, addition corresponds to the bitwise XOR operation (in particular, $1+1=0$ ). The integer $w$ can be thought of as the word length of the computer (i.e., $w=32$ or 64 ). Usually (but there are exceptions, see [10]) $k$ is taken much larger than $w$.

A popular modulo 2 generator was introduced by Matsumoto and Nishimura [16]. The dimension $k$ of the state vector $\mathbf{X}t$ in Algorithm 2.2.1 is in this case $k=w n$, where $w$ is the word length (default 32) and $n$ a large integer (default 624). The period length for the default choice of parameters can be shown to be $2^{w(n-1)+1}-1=2^{19937}-1$. Rather than take the state $\mathbf{X}_t$ as a $w n \times 1$ vector, it is convenient to consider it as an $n \times w$ matrix with rows $\mathbf{x}_t, \ldots, \mathbf{x}{t+n-1}$. Starting from the seed rows $\mathbf{x}0, \ldots, \mathbf{x}{n-1}$, at each step $t=0,1,2, \ldots$ the $(t+n)$-th row is calculated according to the following rules:

1. Take the first $r$ bits of $\mathbf{x}t$ and the last $w-r$ bits of $\mathbf{x}{t+1}$ and concenate them together in a binary vector $\mathbf{x}$.
2. Apply the following binary operation to $\mathbf{x}=\left(x_1, \ldots, x_w\right)$ to give a new binary vector $\tilde{\mathbf{x}}$ :
$$\widetilde{\mathbf{x}}= \begin{cases}\mathbf{x} \gg 1 & \text { if } x_w=0 \ (\mathbf{x} \gg 1) \oplus \mathbf{a} & \text { if } x_w=1\end{cases}$$
3. Let $\mathbf{x}{t+n}=\mathbf{x}{t+m} \oplus \widetilde{\mathbf{x}}$
Here $\oplus$ stands for the XOR operation and $\gg 1$ for the rightshift operation (shift the bits one position to the right, adding a 1 from the left). The binary vector a and the numbers $m$ and $r$ are specified by the user (see below).

## 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|Inverse-Transform Method

Let $X$ be a random variable with cdf $F$. Since $F$ is a nondecreasing function, the inverse function $F^{-1}$ may be defined as
$$F^{-1}(y)=\inf {x: F(x) \geqslant y}, \quad 0 \leqslant y \leqslant 1 .$$
(Readers not acquainted with the notion inf should read min.) It is easy to show that if $U \sim \mathrm{U}(0,1)$, then
$$X=F^{-1}(U)$$
has cdf $F$. That is to say, since $F$ is invertible and $\mathbb{P}(U \leqslant u)=u$, we have
$$\mathbb{P}(X \leqslant x)=\mathbb{P}\left(F^{-1}(U) \leqslant x\right)=\mathbb{P}(U \leqslant F(x))=F(x) .$$
Thus, to generate a random variable $X$ with cdf $F$, draw $U \sim \mathrm{U}(0,1)$ and set $X=F^{-1}(U)$. Figure $2.1$ illustrates the inverse-transform method given by the following algorithm:

Generate a random variable from the pdf
$$f(x)= \begin{cases}2 x, & 0 \leqslant x \leqslant 1 \ 0 & \text { otherwise }\end{cases}$$
The cdf is
$$F(x)= \begin{cases}0, & x<0 \\ \int_0^x 2 y \mathrm{~d} y=x^2, & 0 \leqslant x \leqslant 1 \\ 1, & x>1\end{cases}$$
Applying (2.5), we have
$$X=F^{-1}(U)=\sqrt{U}$$
Therefore, to generate a random variable $X$ from the pdf (2.7), first generate a random variable $U$ from $\mathrm{U}(0,1)$ and then take its square root.

# 模拟和蒙特卡洛方法代写

## 数学代写|模拟和蒙特卡洛方法作业代写模拟和蒙特卡罗方法代考|Modulo 2线性发生器

Matsumoto和Nishimura介绍了一种流行的模2发生器。算法2.2.1中状态向量$\mathbf{X}t$的维度$k$在本例中是$k=w n$，其中$w$是单词长度(默认32)，$n$是大整数(默认624)。默认参数选择的周期长度可以显示为$2^{w(n-1)+1}-1=2^{19937}-1$。与其将状态$\mathbf{X}_t$作为$w n \times 1$向量，不如将其视为包含行$\mathbf{x}_t, \ldots, \mathbf{x}{t+n-1}$的$n \times w$矩阵。从种子行$\mathbf{x}0, \ldots, \mathbf{x}{n-1}$开始，在每一步$t=0,1,2, \ldots$，根据以下规则计算$(t+n)$ -th行:

1. 拿第一个 $r$ 一些 $\mathbf{x}t$ 最后一个 $w-r$ 一些 $\mathbf{x}{t+1}$ 把它们集中在一个二元向量中 $\mathbf{x}$.
2. Apply the following binary operation to $\mathbf{x}=\left(x_1, \ldots, x_w\right)$ 给出一个新的二元向量 $\tilde{\mathbf{x}}$ :
$$\widetilde{\mathbf{x}}= \begin{cases}\mathbf{x} \gg 1 & \text { if } x_w=0 \ (\mathbf{x} \gg 1) \oplus \mathbf{a} & \text { if } x_w=1\end{cases}$$
3. Let $\mathbf{x}{t+n}=\mathbf{x}{t+m} \oplus \widetilde{\mathbf{x}}$
这里 $\oplus$ 表示异或操作和 $\gg 1$ 对于右移操作(将位向右移动一个位置，从左边增加一个1)。二元向量a和这些数字 $m$ 和 $r$ 由用户指定(见下面)。

## 数学代写|模拟和蒙特卡洛方法作业代写模拟和蒙特卡罗方法代考|反变换方法

.

$$F^{-1}(y)=\inf {x: F(x) \geqslant y}, \quad 0 \leqslant y \leqslant 1 .$$
(不熟悉inf概念的读者应该读min。)很容易表明，如果$U \sim \mathrm{U}(0,1)$，则
$$X=F^{-1}(U)$$

$$\mathbb{P}(X \leqslant x)=\mathbb{P}\left(F^{-1}(U) \leqslant x\right)=\mathbb{P}(U \leqslant F(x))=F(x) .$$

$$f(x)= \begin{cases}2 x, & 0 \leqslant x \leqslant 1 \ 0 & \text { otherwise }\end{cases}$$
cdf是
$$F(x)= \begin{cases}0, & x<0 \\ \int_0^x 2 y \mathrm{~d} y=x^2, & 0 \leqslant x \leqslant 1 \\ 1, & x>1\end{cases}$$

$$X=F^{-1}(U)=\sqrt{U}$$

## 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。

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