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数学代写|数值分析代写Numerical analysis代考|MAT12004 Matrix and vector norms and condition number

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数学代写|数值分析代写Numerical analysis代考|Matrix and vector norms and condition number

In order to study how “close” two vectors are, we introduce norms. Some commonly used vector norms are:

Infinity norm: $|\mathbf{v}|_{\infty}=\max _{1 \leq i \leq n}\left|v_i\right|$

Euclidean norm (2-norm): $|\mathbf{v}|_2=\left(\sum_{i=1}^n v_i^2\right)^{1 / 2}$
Although the intuitive measure of distance is the 2-norm, mathematically speaking, any function $|\cdot|$ that satisfies the following four properties can be considered a mathematical measure of distance, which we refer to as a norm:

$|x| \geq 0$

$|x|=0$ if and only if $x=0$

$|\alpha x|=|\alpha||x|$ for all $\alpha \in \mathbb{R}$

$|x+y| \leq|x|+|y|$
The reason why we define a different way to measure the “size” of a vector, as opposed to just using Euclidean norm for everything, is that all measures in finite dimension are equivalent (not equal, however), and so we want to pick one that is easy/cheap to perform calculations with.

To show that $|\mathbf{v}|_{\infty}$ is a norm for vectors in $\mathbb{R}^n$, we must show each of the 4 properties of a norm hold for all vectors $\mathbf{v} \in \mathbb{R}^n$.

数学代写|数值分析代写Numerical analysis代考|Condition number of a matrix

Recall the concept of matrix inverses. If a square matrix $\mathbf{A}$ is nonsingular, then there exists a matrix $\mathbf{A}^{-1}$ such that $\mathbf{A A}^{-1}=\mathbf{I}$ and $\mathbf{A}^{-1} \mathbf{A}=\mathbf{I}$. Recall from linear algebra that $\mathbf{A}^{-1}$ can be found using Gauss-Jordan elimination (like GE, but zero out above the diagonal as well) applied to all columns of the matrix. In general, we do not need to actually calculate $\mathbf{A}^{-1}$; we just need to know it exists.

We now define the condition number of a matrix:
$$\operatorname{cond}(\mathbf{A})=|\mathbf{A}|\left|\mathbf{A}^{-1}\right| \text {. }$$
The condition number satisfies $1 \leq \operatorname{cond}(\mathbf{A}) \leq \infty$, and its value increases as the rows (or columns) of A get closer to being linear dependent (i. e., the matrix is getting closer to being singular). That is, if we think of the rows (or columns) of a matrix as $n$ dimensional vectors, if they are all perpendicular to each other, the 2-condition number is 1 . However, as the smallest angle made by the vectors shrinks, the 2-condition number grows in an inversely proportional manner.

As we will see in the next section, the condition number is the fundamental measure of sensitivity in solving linear systems. If the condition number of $\mathbf{A}$ is small, then we say $\mathbf{A x}=\mathbf{b}$ is a well-conditioned system and we expect an accurate numerical solution by stable algorithms (such as GE with partial pivoting). However, if the condition number of $\mathbf{A}$ is large, then the system is called ill-conditioned and the numerical solution is most likely inaccurate.

数学代写|数值分析代写NUMERICAL ANALYSIS代 考|PIVOTING

GE 在 Step 崩溃 $i$ 如果 $i$ 当前的第 th 个对角线条目 modified系数矩阵，称为主元，为零orcloseto0，因为无法使用零主元诮除非零项。零主元可能 出现在 $G E$ 的任何一步，使算法失败，即使 $\mathbf{A}$ 是非奇异的并且是唯一的解决方案 $\mathbf{A} \mathbf{x}=\mathbf{b}$ 存在。例如，考虑线性方程
$\left(\begin{array}{lll}0 & 11 & 0\end{array}\right)\left(x_1 x_2\right)=(02)$.

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。