19th Ave New York, NY 95822, USA

# 数学代写|数值分析代写Numerical analysis代考|MATH2722 Gaussian Elimination

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 数学代写|数值分析代写Numerical analysis代考|Solution of Linear Systems

Gaussian Elimination is a simple, systematic algorithm to solve systems of linear equations. It is the workhorse of linear algebra, and, as such, of absolutely fundamental importance in applied mathematics. In this section, we review the method in the most important case, in which there are the same number of equations as unknowns.
To illustrate, consider an elementary system of three linear equations
$$\begin{array}{r} x+2 y+z=2, \ 2 x+6 y+z=7, \ x+y+4 z=3, \end{array}$$

in three unknowns $x, y, z$. Linearity refers to the fact that the unknowns only appear to the first power, and there are no product terms like $x y$ or $x y z$. The basic solution method is to systematically employ the following fundamental operation:

Linear System Operation #1: Add a multiple of one equation to another equation.
Before continuing, you might try to convince yourself that this operation doesn’t change the solutions to the system. Our goal is to judiciously apply the operation and so be led to a much simpler linear system that is easy to solve, and, moreover has the same solutions as the original. Any linear system that is derived from the original system by successive application of such operations will be called an equivalent system. By the preceding remark, equivalent linear systems have the same solutions.

The systematic feature is that we successively eliminate the variables in our equations in order of appearance. We begin by eliminating the first variable, $x$, from the second equation. To this end, we subtract twice the first equation from the second, leading to
\begin{aligned} x+2 y+z & =2, \ 2 y-z & =3, \ x+y+4 z & =3 . \end{aligned}
Next, we eliminate $x$ from the third equation by subtracting the first equation from it:
$$\begin{array}{r} x+2 y+z=2, \ 2 y-z=3, \ -y+3 z=1 . \end{array}$$

## 数学代写|数值分析代写Numerical analysis代考|Gaussian Elimination — Regular Case

With the basic matrix arithmetic operations in hand, let us now return to our primary task. The goal is to develop a systematic method for solving linear systems of equations. While we could continue to work directly with the equations, matrices provide a convenient alternative that begins by merely shortening the amount of writing, but ultimately leads to profound insight into the structure of linear systems and their solutions.

We begin by replacing the system (3.2) by its matrix constituents. It is convenient to ignore the vector of unknowns, and form the augmented matrix
$$M=(A \mid \mathbf{b})=\left(\begin{array}{cccc|c} a_{11} & a_{12} & \ldots & a_{1 n} & b_1 \ a_{21} & a_{22} & \ldots & a_{2 n} & b_2 \ \vdots & \vdots & \ddots & \vdots & \vdots \ a_{m 1} & a_{m 2} & \ldots & a_{m n} & b_m \end{array}\right)$$
which is an $m \times(n+1)$ matrix obtained by tacking the right hand side vector onto the original coefficient matrix. The extra vertical line is included just to remind us that the last column of this matrix plays a special role. For example, the augmented matrix for the system (4.1), i.e.,
\begin{aligned} x+2 y+z & =2, \ 2 x+6 y+z & =7, \ x+y+4 z & =3, \end{aligned} \quad \text { is } \quad M=\left(\begin{array}{lll|l} 1 & 2 & 1 & 2 \ 2 & 6 & 1 & 7 \ 1 & 1 & 4 & 3 \end{array}\right)

## 数学代写|数值分析代写NUMERICAL ANALYSIS代考|SOLUTION OF LINEAR SYSTEMS

$$x+2 y+z=2,2 x+6 y+z=7, x+y+4 z=3,$$

$$x+2 y+z=2,2 y-z \quad=3, x+y+4 z=3 .$$

## 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。