### 统计代写|数值分析和优化代写numerical analysis and optimazation代考|The Power Method

statistics-lab™ 为您的留学生涯保驾护航 在代写数值分析和优化numerical analysis and optimazation方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数值分析和优化numerical analysis and optimazation方面经验极为丰富，各种代写数值分析和优化numerical analysis and optimazation相关的作业也就用不着说。

• 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

## 统计代写|数值分析和优化代写numerical analysis and optimazation代考|The Power Method

The power method forms the basic of many iterative algorithms for the calculation of eigenvalues and eigenvectors. It generates a single eigenvector and eigenvalue of $A$.

1. Pick a starting vector $\mathbf{x}^{(0)} \in \mathbb{R}^{n}$ satisfying $\left|\mathbf{x}^{(0)}\right|=1$. Set $k=0$ and choose a tolerance $\epsilon>0$.
2. Calculate $\mathbf{x}^{(k+1)}=A \mathbf{x}^{(k)}$ and find the real number $\lambda$ that minimizes

$\left|\mathbf{x}^{(k+1)}-\lambda \mathbf{x}^{(k)}\right|$. This is given by the Rayleigh quotient
$$\lambda=\frac{\mathbf{x}^{(k)^{T}} A \mathbf{x}^{(k)}}{\mathbf{x}^{(k)^{T}} \mathbf{x}^{(k)}} .$$

1. Accept $\lambda$ as an eigenvalue and $\mathbf{x}^{(k)}$ as an eigenvector , if $| \mathbf{x}^{(k+1)}-$ $\lambda \mathbf{x}^{(k)} | \leq \epsilon$.
2. Otherwise, replace $\mathbf{x}^{(k+1)}$ by $\mathbf{x}^{(k+1)} /\left|\mathbf{x}^{(k+1)}\right|$, increase $k$ by one, and go back to step $2 .$

## 统计代写|数值分析和优化代写numerical analysis and optimazation代考|Inverse Iteration

The method described in this section is called inverse iteration and is very effective in practice. It is similar to the power method with shifts, except that, instead of $\mathbf{x}^{(k+1)}$ being a multiple of $(A-s I) \mathbf{x}^{(k)}$, it is calculated as a scalar multiple of the solution to
$$(A-s I) \mathbf{x}^{(k+1)}=\mathbf{x}^{(k)}, \quad k=0,1, \ldots$$
where $s$ is a scalar that may depend on $k$. Thus the inverse power method is the power method applied to the matrix $(A-s I)^{-1}$. If $s$ is close to an eigenvalue, then the matrix $A-s I$ has an eigenvalue close to zero, but this implies that $(A-s I)^{-1}$ has a very large eigenvalue and we have seen that in this case the power method converges fast.

In every iteration $\mathbf{x}^{(k+1)}$ is scaled such that $\left|\mathbf{x}^{(k+1)}\right|=1$. We see that the calculation of $\mathbf{x}^{(k+1)}$ requires the solution of an $n \times n$ system of equations.
If $s$ is constant in every iteration such that $A-s I$ is nonsingular, then $\mathbf{x}^{(k+1)}$ is a multiple of $(A-s I)^{-k-1} \mathbf{x}^{(0)}$. As before we let $\mathbf{x}^{(0)}=\sum_{j=1}^{n} c_{j} \mathbf{v}{j}$, where $\mathbf{v}{j}, j=1, \ldots, n$, are the linearly independent eigenvectors. The eigenvalue equation then implies $(A-s I) \mathbf{v}{j}=\left(\lambda{j}-s\right) \mathbf{v}{j}$. For the inverse we have then $(A-s I)^{-1} \mathbf{v}{j}=\left(\lambda_{j}-s\right)^{-1} \mathbf{v}{j}$. It follows that $\mathbf{x}^{(k+1)}$ is a multiple of $$(A-s I)^{-k-1} \mathbf{x}^{(0)}=\sum{j=0}^{n} c_{j}(A-s I)^{-k-1} \mathbf{v}{j}=\sum{j=0}^{n} c_{j}\left(\lambda_{j}-s\right)^{-k-1} \mathbf{v}{j}$$ Let $m$ be the index of the smallest number of $\left|\lambda{j}-s\right| \cdot j=1, \ldots, n$. If $c_{m}$ is nonzero then $\mathbf{x}^{(k+1)}$ tends to be multiple of $\mathbf{v}_{m}$ as $k$ tends to infinity. The speed of convergence can be excellent if $s$ is close to $\lambda_{m}$. It can be improved even more if $s$ is adjusted during the iterations as the following implementation shows.

## 统计代写|数值分析和优化代写numerical analysis and optimazation代考|Deflation

Suppose we have found one solution of the eigenvector equation $A \mathbf{v}=\lambda \mathbf{v}$ (or possibly a pair of complex conjugate eigenvalues with their corresponding eigenvectors), where $A$ is an $n \times n$ matrix. Deflation constructs an $(n-1) \times$ $(n-1)$ (or $(n-2) \times(n-2)$ ) matrix, say $B$, whose eigenvalues are the other $n-1$ (or $n-2$ ) eigenvalues of $A$. The concept is based on the following theorem.
Theorem 2.12. Let $A$ and $S$ be $n \times n$ matrices, $S$ being nonsingular. Then $\mathbf{v}$ is an eigenvector of $A$ with eigenvalue $\lambda$ if and only if $S \mathbf{v}$ is an eigenvector of $S A S^{-1}$ with the same eigenvalue. $S$ is called a similarity transformation.
Proof.
$$A \mathbf{v}=\lambda \mathbf{v} \Leftrightarrow A S^{-1}(S \mathbf{v})=\lambda \mathbf{v} \Leftrightarrow\left(S A S^{-1}\right)(S \mathbf{v})=\lambda(S \mathbf{v})$$
Let’s assume one eigenvalue $\lambda$ and its corresponding eigenvector have been found. In deflation we apply a similarity transformation $S$ to $A$ such that the first column of $S A S^{-1}$ is $\lambda$ times the first standard unit vector $\mathbf{e}{1}$, $$\left(S A S^{-1}\right) \mathbf{e}{1}=\lambda \mathbf{e}{1} .$$ Then we can let $B$ be the bottom right $(n-1) \times(n-1)$ submatrix of $S A S^{-1}$. We see from the above theorem that it is sufficient to let $S$ have the property $S \mathbf{v}=a \mathbf{e}{1}$, where $a$ is any nonzero scalar.

If we know a complex conjugate pair of eigenvalues, then there is a twodimensional eigenspace associated with them. Eigenspace means that if $A$ is applied to any vector in the eigenspace, then the result will again lie in the eigenspace. Let $\mathbf{v}{1}$ and $\mathbf{v}{2}$ be vectors spanning the eigenspace. For example these could have been found by the two-stage power method. We need to find a similarity transformation $S$ which maps the eigenspace to the space spanned by the first two standard basis vectors $\mathbf{e}{1}$ and $\mathbf{e}{2}$. Let $S_{1}$ such that $S_{1} \mathbf{v}{1}=a \mathbf{e}{1}$. In addition let $\hat{\mathbf{v}}$ be the vector composed of the last $n-1$ components of $S_{1} \mathbf{v}{2}$. We then let $S{2}$ be of the form
$$S_{2}=\left(\begin{array}{cccc} 1 & 0 & \cdots & 0 \ 0 & & & \ \vdots & & \hat{S} & \ 0 & & & \end{array}\right),$$

## 统计代写|数值分析和优化代写numerical analysis and optimazation代考|The Power Method

1. 选择一个起始向量X(0)∈Rn令人满意的|X(0)|=1. 放ķ=0并选择一个公差ε>0.
2. 计算X(ķ+1)=一种X(ķ)并找到实数λ最小化

|X(ķ+1)−λX(ķ)|. 这是由瑞利商给出的
λ=X(ķ)吨一种X(ķ)X(ķ)吨X(ķ).

1. 接受λ作为特征值和X(ķ)作为一个特征向量，如果|X(ķ+1)− λX(ķ)|≤ε.
2. 否则，更换X(ķ+1)经过X(ķ+1)/|X(ķ+1)|， 增加ķ减一，然后返回步骤2.

## 统计代写|数值分析和优化代写numerical analysis and optimazation代考|Inverse Iteration

(一种−s一世)X(ķ+1)=X(ķ),ķ=0,1,…

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。