### 数学代写|数值分析代写numerical analysis代考|CIVL5458

statistics-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

## 数学代写|数值分析代写numerical analysis代考|Solving linear systems of equations

The need to solve systems of linear equations $\mathbf{A x}=\mathbf{b}$ arises across nearly all of engineering and science, business, statistics, economics, and many other fields. In a standard undergraduate linear algebra course, we have learned how to solve this problem using Gaussian Elimination (GE). We will show here how such a procedure is equivalent to an LU factorization of the coefficient matrix A, followed by a forward and a back substitution. To achieve stability of the factorization in computer arithmetic, a strategy called pivoting is necessary, which leads to the LU factorization with partial pivoting. This is the standard direct method for solving linear systems where $\mathbf{A}$ is a dense matrix.

Linear systems with large and sparse (most entries are zero) coefficient matrices arise often in numerical solution methods of differential equations, for example, by the finite element and finite difference discretizations. State-of-the-art direct methods can nowadays efficiently solve such linear systems up to an order of a few million, using advanced strategies to keep the LU factors as sparse as possible and the factorization stable. However, problems of ever-increasing dimension need be tackled, and sparse linear systems of order tens of millions to billions have become more routine. To efficiently solve these large systems approximately, iterative methods such as the Conjugate Gradient (CG) method are typically used, and on sufficiently large problems, can be advantageous over direct methods.

This chapter will mainly focus on direct methods but will also discuss the CG method. “Linear solvers” has become a vast field and is a very active research area. We aim here to provide a fundamental understanding of the basic types of solvers, but note that we are just scratching the surface, in particular for iterative methods.

## 数学代写|数值分析代写numerical analysis代考|Solving triangular linear systems

Consider a system of linear equations $\mathbf{A x}=\mathbf{b}$, where the coefficient matrix $\mathbf{A}$ is square and nonsingular. Recall that the GE procedure gradually eliminates all entries in the coefficient matrix below the main diagonal by elementary row operations, until the modified coefficient matrix becomes an upper triangular matrix U. The solution remains unchanged during the entire procedure. In this section, we consider how to solve a linear system where the coefficient matrix is upper or lower triangular. The procedure of elimination will be reviewed and explored in the new perspective of matrix factorization in the next section.

Example 3 (Back substitution for an upper triangular system). Consider the linear system $x_1+2 x_2+3 x_3=2,4 x_2+5 x_3=3$ and $6 x_3=-6$. It can be written in matrix form as $$\left(\begin{array}{lll} 1 & 2 & 3 \ & 4 & 5 \ & & 6 \end{array}\right)\left(\begin{array}{l} x_1 \ x_2 \ x_3 \end{array}\right)=\left(\begin{array}{c} 2 \ 3 \ -6 \end{array}\right)$$
where the coefficient matrix is upper triangular. To solve this linear system, we start from the last equation $6 x_3=-6$, which immediately gives $x_3=\frac{-6}{6}=-1$.

Then, from the second equation $4 x_2+5 x_3=3$, we get $x_2=\frac{3-5 x_3}{4}=2$. Finally, the first equation $x_1+2 x_2+3 x_3=2$ leads to $x_1=2-2 x_2-3 x_3=1$.

This procedure illustrates the general procedure of back substitution. Given an upper triangular linear system with nonzero diagonal entries
$$\mathbf{U x}=\left(\begin{array}{cccc} u_{11} & u_{12} & \ldots & u_{1 n} \ & \ddots & \ddots & \vdots \ & & u_{(n-1)(n-1)} & u_{(n-1) n} \ & & & u_{n n} \end{array}\right)\left(\begin{array}{c} x_1 \ \vdots \ x_{n-1} \ x_n \end{array}\right)=\left(\begin{array}{c} b_1 \ \vdots \ b_{n-1} \ b_n \end{array}\right),$$
we start with the last equation and evaluate $x_n=\frac{b_n}{u_{n n}}$ directly, then substitute it into the previous equation and compute $x_{n-1}=\frac{b_{n-1}-u_{(n-1)} x_{n n}}{u_{(n-1)}(n-1)}$. Assume in general that we have already solved for $x_{i+1}, \ldots, x_n$, then $x_i=\frac{\left.b_i-\sum_{i-i+1}^n u_{i j} x_i-1\right)}{u_{i i}}$ can be evaluated. We continue until the value of $x_1$ is found.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Solving triangular linear systems

$$\left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6 \end{array}\right)\left(\begin{array}{lll} x_1 & x_2 & x_3 \end{array}\right)=\left(\begin{array}{lll} 2 & 3 & -6 \end{array}\right)$$

## 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。