# 数学代写|数值分析代写numerical analysis代考|MA1020

#### Doug I. Jones

Lorem ipsum dolor sit amet, cons the all tetur adiscing elit

couryes-lab™ 为您的留学生涯保驾护航 在代写数值分析numerical analysis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numerical analysis代写方面经验极为丰富，各种代写数值分析numerical analysis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
couryes™为您提供可以保分的包课服务

## 数学代写|数值分析代写numerical analysis代考|Preconditioning

Iterative improvement is one way in which we can prime an iterative method using the ideas of the Jacobi iteration
$$x^{k+1}=-D^{-1}(U+L) x^k+D^{-1} b$$
or the Gauss-Seidel iteration
$$x^{k+1}=-(L+D)^{-1} U x^k+(L+D)^{-1} b$$
(where $A=U+L+D) \mathrm{R}$. However, there is another way that is standard nowadays (and is an area of active research): Preconditioning, where we take a linear system $A x=b$ and modify it to an equivalent system of the form
$$M_1 A M_2 y=\tilde{b}$$
(where $y=M_2^{-1} x$ and $\tilde{b}=M_1 b$ ) for which
$$\kappa\left(M_1 A M_2\right)<<\kappa(A)$$
so that $M_1 A M_2$ is significantly better conditioned than $A$ itself ${ }^5$. We refer to $M_1$ as a left preconditioner and $M_2$ as a right preconditioner. If the costs involved in finding and using $M_1$ and $M_2$ are not too severe compared to what is gained from the reduction in the condition number, this can be very beneficial.

Preconditioning followed by an iterative method is the standard approach for large sparse linear systems. In fact, an iteration or two of Jacobi or Gauss-Seidel iteration followed by iterative improvement (see Sec. 3.3) is often considered an example of this approach, with the Jacobi or Gauss-Seidel iteration being roughly like a form of preconditioning.

In some cases it is important to use both a right preconditioner and a left preconditioner; for example, if $A$ is positive definite then $M_1 A$ may not be but $M_1 A M_2$ can be made to be positive definite, and this is likely to be desirable.

## 数学代写|数值分析代写numerical analysis代考|Krylov Space Methods

Many iterative methods for solving linear systems $A x=b$ and for finding eigenvalues and eigenvectors of a matrix $A$ are based on the Krylov space
$$K_k=\operatorname{span}\left{w, A w, A^2 w, \ldots, A^{k-1} w\right}$$
associated with $A$ and a given vector $w$, which is a subspace of $\mathbb{R}^n$. We call it the Krylov subspace generated by $A$ and $w$. We’ll focus on using it for the solution of linear systems rather than eigenvalue problems. The general idea is this: Given the linear system $A x=b$ we first try to find an approximate solution $w_1$ that is a multiple of $w$. We then look for a better approximate solution $w_2$ that was a linear

combination of $w$ and $A w$, that is, one that is an element of $K_2$. We continue in this way, so that the $k$ th approximation $w_k$ is an element of $K_k$. Since
$$K_n=\mathbb{R}^n$$
(for a typical $A$ and $w$ ) it appears that if a solution exists, then it is in $K_n$. We hope to find a good approximate solution in $K_k$ for $k \ll n$ because in the cases of interest $n$ will be very large (and $A$ will be sparse).

There are many ways to build such a method. The typical choice for $w$ is either $b$ or the residual $b-A w_0$ for some initial guess $w_0$. We’ll look at a method that uses $w=b$, so that Eq. (5.1) becomes
$$K_k=\operatorname{span}\left{b, A b, A^2 b, \ldots, A^{k-1} b\right} .$$
Choose an initial guess $w_0$ as to the solution of $A x=b$. Then define
$$V_k=w_0+K_k$$
by which we mean that $V_h$ is the set of all vectors of the form $w_0+w$, where $w_0$ is the initial guess and $w$ is any element of $K_k$. (Frequently we take $w_0=0$, in which case $V_k=K_k$.) We say that $V_k$ is an affine space, meaning a vector space shifted by a particular vector (here, $w_0$ ). Note that if $w_0 \notin K_k$ then $V_k$ is not a vector space.

# 数值分析代考

## 数学代写|数值分析代写数值分析代考|预处理

$$x^{k+1}=-D^{-1}(U+L) x^k+D^{-1} b$$

$$x^{k+1}=-(L+D)^{-1} U x^k+(L+D)^{-1} b$$
(其中$A=U+L+D) \mathrm{R}$。然而，现在还有另一种标准的方法(也是一个活跃的研究领域):预处理，我们取一个线性系统$A x=b$，并将其修改为一个等价的系统，形式为
$$M_1 A M_2 y=\tilde{b}$$
(其中$y=M_2^{-1} x$和$\tilde{b}=M_1 b$)，其中
$$\kappa\left(M_1 A M_2\right)<<\kappa(A)$$
，因此$M_1 A M_2$明显优于$A$本身${ }^5$。我们将$M_1$作为左侧预处理条件，将$M_2$作为右侧预处理条件。如果发现和使用$M_1$和$M_2$所涉及的成本与减少条件数量所获得的成本相比不是太严重，那么这将是非常有益的

## 数学代写|数值分析代写数值分析代考|Krylov空间方法

$$K_k=\operatorname{span}\left{w, A w, A^2 w, \ldots, A^{k-1} w\right}$$

$w$和$A w$的组合，即一个是$K_2$的元素。我们继续这样做，使$k$ th近似$w_k$是$K_k$的一个元素。由于
$$K_n=\mathbb{R}^n$$
(对于典型的$A$和$w$)，如果存在一个解决方案，那么它似乎在$K_n$中。我们希望在$K_k$中为$k \ll n$找到一个好的近似解，因为在感兴趣的情况下$n$将非常大(而$A$将是稀疏的)。

$$K_k=\operatorname{span}\left{b, A b, A^2 b, \ldots, A^{k-1} b\right} .$$

$$V_k=w_0+K_k$$
，这意味着$V_h$是所有形式为$w_0+w$的向量的集合，其中$w_0$是初始猜测，$w$是$K_k$的任意元素。(通常我们使用$w_0=0$，在这种情况下是$V_k=K_k$。)我们说$V_k$是一个仿射空间，这意味着一个由特定向量(这里是$w_0$)移位的向量空间。请注意，如果$w_0 \notin K_k$，那么$V_k$不是一个向量空间

## 有限元方法代写

tatistics-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。

Days
Hours
Minutes
Seconds

# 15% OFF

## On All Tickets

Don’t hesitate and buy tickets today – All tickets are at a special price until 15.08.2021. Hope to see you there :)