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

2022年10月10日

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## 数学代写|数值分析代写numerical analysis代考|Jacobi and Gauss-Seidel Iteration

In the previous chapter we discussed direct methods for the solution of linear systems. These are the methods of choice for moderate size problems. Computing the LU decomposition of an $n \times n$ matrix, for example, requires about $\frac{2}{3} n^3$ floating point operations. If $n=100$ this is about $6.7 \times 10^5$ floating point operations which takes well under a second on a standard desktop computer; if $n=1000$ however this is about $6.7 \times 10^8$ flops which takes considerably longer to finish (due to memory management issues in addition to the thousand-fold extra flops required). Since matrices as large as tens of thousands by tens of thousands are commonly encountered in practice (in problems involving the numerical solution of partial differential equations, such as in fluid dynamics) and matrices as large as hundreds of thousands by hundreds of thousands are not uncommon (for example, applications in data science, or the analysis of genetic data), more efficient methods are clearly needed.

We will have little to say about large, dense matrices. If approximating such a matrix with a simpler matrix is not acceptable, the computation will take a long time. Moving entries of the matrix from memory to the processor(s) will likely be more time-consuming than the actual computations. These are extremely challenging problems-consult an expert.

Fortunately, it is common for large matrices occurring in practice to be sparse. For large, sparse matrices there are a number of iterative methods that-in combination with a smart storage system for the sparse matrix-can lead to considerably shorter computation times. As a rule, direct methods are $O\left(n^3\right)$ and iterative methods are, in the cases in which they are appropriate, $O\left(n^2\right)$. The goal for sparse matrices is always to get a method that is $O(N)$ where $N$ is the number of nonzero entries in a typical row of the matrix.

In this section we will discuss two classical methods for the solution of linear systems by iterative methods. Remember, the implicit assumption of sparsity is standard when we consider these methods for solving $A x=b$-they may not be very efficient otherwise.

Consider a square linear system $A x=b$. We seek an iteration of the form $x^{k+1}=F\left(x^k\right)$ where an initial guess $x^0 \in \mathbb{R}^{\times}$is given and $F$ is simple to compute.

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

The Jacobi, Gauss-Seidel, and SOR methods of Sec. $3.1$ were previously widely employed for the solution of large sparse systems. The modern approach is different: We use such a method to prime another iterative technique. This priming can take one of two forms, and we explore them in this section and the next.

The technique we will discuss here is based on the observation that if $x_0$ is an estimate of the true solution $x$ of $A x=b$ then the residual $r$ satisfies
\begin{aligned} r &=b-A x_0 \ &=A x-A x_0 \ &=A\left(x-x_0\right) \end{aligned}
(since $A x=b$ ). But $e=x-x_0$ is precisely the error in estimating $x$ using $x_0$; we have found a linear system satisfied by the error vector,
and in principle we can solve Eq. (3.1) for $e$ and set $x=x_0+e$ to determine $x$. In practice of course we will have error in $e$ as well but even still $x_1=x_0+e$ may be an improved estimate of $x$. This technique is called iterative refinement (or iterative improvement). It uses the current $x_0$ to predict a correction $e$ to be applied to it.

We could use iterative refinement simply to recover some of the accuracy that is lost in solving $A x=b$ by some other method. For example, it is reasonable to take the approximate solution of $A x=b$ as found by LU decomposition and perform one or two iterations of iterative improvement on it in order to clean it up. As the LU decomposition of $A$ is already known from solving $A x=b$ to get the approximate solution, this can be done efficiently by simply using it in Eq. (3.1). Since iterative improvement requires only $O\left(n^2\right)$ operations if we save the LU factors of $A$ and reuse them, whereas solving $A x=b$ in the first place requires $O\left(n^3\right)$ operations, this is an inexpensive measure to employ to improve a solution. For well-conditioned matrices one iteration will likely suffice. On the other hand, if we have spent $O\left(n^3\right)$ operations and gained an inaccurate solution due to illconditioning, then spending a mere $O\left(n^2\right)$ additional operations to improve it may be wise (we are “saving” the computation, hopefully).

However, as indicated at the start of this section, we can also use it as our solution method. We generate an initial guess $x_0$ that is suitably close to the true solution say, by performing several iterations of Gauss-Seidel iteration to generate this $x_0$-and perform iterative improvement on it until the error ceases to be reduced.

# 数值分析代考

## 数学代写|数值分析代写数值分析代考|迭代细化

Sec. $3.1$的Jacobi, Gauss-Seidel和SOR方法以前被广泛用于求解大型稀疏系统。现代的方法是不同的:我们使用这样的方法来启动另一种迭代技术。这种启动可以采取两种形式之一，我们将在本节和下节探讨它们

\begin{aligned} r &=b-A x_0 \ &=A x-A x_0 \ &=A\left(x-x_0\right) \end{aligned}
(从$A x=b$开始)。但$e=x-x_0$正是用$x_0$估计$x$的误差;我们找到了一个误差向量

## 有限元方法代写

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## 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。