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• 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|Gaussian Elimination with Partial Pivoting

As stated previously, three of the most commonly occurring subproblems in scientific computing are the solution of a linear algebraic system, the solution of a nonlinear algebraic system, and the location of minima and maxima. One reason they occur so often is that many more sophisticated computations such as solving a system of differential equations often require that, say, a linear system must be solved for each point. We have discussed nonlinear scalar equations in Ch. 1. We now turn, in this chapter and the next, to the solution of linear systems of equations
$$A x=b$$
where, in general, $A$ is an $m \times n$ matrix (we will assume real entries) and $b \in \mathbb{R}^n$. We will also return to nonlinear equations in this chapter so that we can generalize Newton’s method to systems.

How shall we solve a linear system? In principle we could rewrite Eq. (1.1) in the equivalent root-finding form
$$A x-b=0$$
and apply a method for solving nonlinear systems to $F(x)=A x-b$. Why don’t we? There are a number of reasons. One is that Newton’s method for systems requires the solution of an equation like Eq. (1.1) at every iteration, leading to a circularity problem; we need a method for solving Eq. (1.1) before we can use Newton’s method on a nonlinear system of equations. Another reason is that, especially if $A$ is not square, there may be no solution or there may be infinitely many solutions, and it is important to be able to determine when this is the case. There are also numérical issués, as wê will seeé.

But the over-riding reason that we consider the solution of linear systems separately from the solution of nonlinear systems is size. Linear systems have been studied extensively, and much is known about them; the very use of the term “nonlinear” implies “everything else” (all we know about such problems is that they are not linear). Because of what we know about linear systems from linear algebra, we can solve much larger linear systems, using techniques appropriate for them, than nonlinear systems. Large linear systems, with $n$ in the thousands, occur constantly; handling them conveniently is why MATLAB ${ }^{(\mathbb{R})}$ was developed. They may arise directly, as in a controls problem, but very often arise as a subproblem in a numerical method for a different type of problem, e.g., Newton’s method for systems or the discretization of a problem involving a partial differential equation. Large linear systems are ubiquitous in scientific computing.

## 数学代写|数值分析代写numerical analysis代考|The LU Decomposition

Gaussian elimination with partial pivoting is a good algorithm for small-tomedium sized problems; as usual, it’s difficult to set a value for “small”” or “medium” across a variety of platforms and needs. In addition, most matrices that occur in practice possess some sort of structure. Structure is a generic term that refers either to the pattern of the locations of the zero elements versus nonzero elements within the matrix, referred to as the sparsity structure, or simply the sparsity, of the matrix, or to special properties that the matrix possesses, such as symmetry being triangular. A matrix with mostly zero entries is said to be sparse and one with mostly nonzero entries is said to be dense (or full). There are many special numerical methods for sparse matrices-a typical rule of thumb is $95 \%$ zero entries, which is much more common than you might think-and for matrices with special properties like symmetry.

As a trivial case, if the matrix is already upper triangular then we need only perform the back substitution (and if it is lower triangular, that is, if its transpose is upper triangular, we need only perform forward substitution). Back substitution requires $O\left(n^2\right)$ flops, which is reasonable even for large $n$. (As a rule we can’t expect to do much better than that; after all, a general $n \times n$ matrix has $n^2$ distinct entries, each of which should figure into the solution.) For a general matrix, that is, an unstructured matrix, Gaussian elimination with partial pivoting requires about $2 n^3 / 3$ flops (for large $n$ ): About $n^3 / 3$ additions/subtractions and about $n^3 / 3$ multiplications/divisions. This doesn’t count the cost of the roughly $n^2 / 2$ downthe-column searches for the pivots or any associated bookkeeping operations.

Because it is common to consider matrices so large that $2 n^3 / 3$ flops is unrealistic or at least very inconvenient, and because Gaussian elimination does not take advantage of (nor preserve) any structure the matrix might possess, we’ll need other methods for large matrices. But for small to medium matrices, Gaussian elimination with a pivoting strategy remains the method of choice.

Despite the fact that it does not take advantage of any special structure of the coefficient matrix $A$, Gaussian elimination does have an advantage in the frequently occurring case where we must solve $A x=b$ repeatedly for different choices of $b$. In some cases the $b$ vector plays the role of a forcing vector, analogous to a forcing function for a differential equation, and we load the system represented by $A$ with various choices of $b$; here $A$ represents some system that remains unchanged, and $b$ represents an input that will be changed often.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考| Partial – Pivoting的高斯消去

. .

$$A x=b$$

$$A x-b=0$$
，并将求解非线性系统的方法应用到$F(x)=A x-b$。为什么不呢?原因有很多。一是牛顿系统的方法需要在每次迭代中解一个像Eq.(1.1)这样的方程，这导致了圆度问题;在对非线性方程组使用牛顿法之前，我们需要一种求解式(1.1)的方法。另一个原因是，特别是当$A$不是平方的时候，可能没有解，或者可能有无限多个解，能够确定何时出现这种情况很重要。还有numérical issués，因为wê将seeé.

## 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。

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