## 数学代写|数值分析代写numerical analysis代考|COSC2500

2023年1月4日

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|Accuracy in solving linear systems

An important question which we now consider is whether numerical solutions to linear systems are accurate. Some systems are very sensitive to small changes in data or roundoff error, and thus their answers are potentially inaccurate. Other systems are not sensitive, and their solutions are likely good. We will quantify the sensitivity and accuracy of systems with the notion of matrix conditioning.

There are two major sources of error that arise when solving linear systems of equations. The first comes from poor representation of the equations in the computer. This arises in the 16th digit from rounding error, but also if the equations are created from experiments. then likely there is measurement error in the fourth (or so) digit in each entry of $\mathbf{A}$ and $\mathbf{b}$. Hence although one wants to solve $\mathbf{A x}=\mathbf{b}$, one is really solving $\hat{\mathbf{A}} \hat{\mathbf{x}}=\hat{\mathbf{b}}$. The question then arises, how close is $\hat{\mathbf{x}}$ to $\mathbf{x}$ ? Note that this type of error is not from numerical calculations, but from error in the representation of the linear system.
The second source of error comes from the calculations that produce a numerical solution to $\mathbf{A x}-\mathbf{b}$. When GE (or some variant of it) is used as the linear solver, the numerical error produced may be in the last few digits of the solution components (i. e., relative error is small). With other types of solvers such as CG, we may accept an approximate solution when the relative residual drops to $10^{-6}$. We will aim to quantify this phenomenon in this chapter, too.

## 数学代写|数值分析代写numerical analysis代考|Error and residual in linear system solving

Assume now that we can represent $\mathbf{A}$ and $\mathbf{b}$ exactly, and let us consider a different type of error. All numerical methods for solving $\mathbf{A x}-\mathbf{b}$ introduce error; that is, they almost surely find $\hat{\mathbf{x}} \neq \mathbf{x}$. Unfortunately, we usually never know $\mathbf{x}$, but we still want to have an idea of the size of the error $\mathbf{e}=\hat{\mathbf{x}}-\mathbf{x}$. What we do know, if given an approximation $\hat{\mathbf{x}}$, is the residual $\mathbf{r}=\mathbf{b}-\mathbf{A} \hat{\mathbf{x}}$. Residual and error are different, but related:
$$\mathbf{A e}=\mathbf{A}(\hat{\mathbf{x}}-\mathbf{x})=\mathbf{A} \hat{\mathbf{x}}-\mathbf{A x}=\mathbf{A} \hat{\mathbf{x}}-\mathbf{b}=\mathbf{r} .$$
Multiplying both sides of $\mathbf{A e}=\mathbf{r}$ by $\mathbf{A}^{-1}$ gives $\mathbf{e}=\mathbf{A}^{-1} \mathbf{r}$, and then taking norms of both sides yield
$$|\mathbf{e}|=\left|\mathbf{A}^{-1}\right||\mathbf{r}| \leq\left|\mathbf{A}^{-1}\right||\mathbf{r}|,$$
where the last inequality came from a property of matrix norms. Dividing both sides by $|\hat{\boldsymbol{x}}|$, and multiplying the right-hand side by $\frac{|\mathbf{A}|}{|\mathbf{A}|}$ yield
$$\frac{|\mathbf{e}|}{|\hat{\mathbf{x}}|} \leq \operatorname{cond}(A) \frac{|\mathbf{r}|}{|\mathbf{A}| \hat{\mathbf{x}} |} .$$
The left-hand side is the relative error of the solution, and the right-hand side is the condition number of $\mathbf{A}$ times the relative residual $\frac{|\vec{r}|}{|\mathbf{A}| \hat{\mathbf{x} \mid} \mid}$. With $\hat{\mathbf{x}}$ computed by numerically stable algorithms such as GE with partial pivoting, the relative residual $\frac{|\mathbf{r}|}{|\mathbf{A}| \mathbf{x} |}$ is on the order of machine epsilon. ${ }^6$ Overall, direct solvers for sparse matrices typically produce approximations that have very small relative residuals, for example, smaller than $10^{-12}$. Iterative solvers, such as CG or GMRES, often use relative residual size as a stopping criteria, and usually on the order of $10^{-6}$ or $10^{-8}$. Hence, if there is a large condition number compared to the relative residual, then the error $\frac{|\hat{\mathbf{x}}-\mathbf{x}|}{|\hat{\mathbf{x}}|}$ may be large.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Error and residual in linear system solving

$$\mathbf{A e}=\mathbf{A}(\hat{\mathbf{x}}-\mathbf{x})=\mathbf{A} \hat{\mathbf{x}}-\mathbf{A} \mathbf{x}=\mathbf{A} \hat{\mathbf{x}}-\mathbf{b}=\mathbf{r}$$

$$|\mathbf{e}|=\left|\mathbf{A}^{-1}\right||\mathbf{r}| \leq\left|\mathbf{A}^{-1}\right||\mathbf{r}|,$$

$$\frac{|\mathbf{e}|}{|\hat{\mathbf{x}}|} \leq \operatorname{cond}(A) \frac{|\mathbf{r}|}{|\mathbf{A}| \hat{\mathbf{x}} \mid}$$

## 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。