# 数学代写|数值分析代写numerical analysis代考|MATH3003

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## 数学代写|数值分析代写numerical analysis代考|The QR Decomposition by Triangularization

In the last section we saw that the QR decomposition is a useful direct method for solving overdetermined linear systems; it is also used in algorithms to find eigenvalues. (See Ch. 3.) There are two major approaches to computing the QR decomposition, and we will describe one of them in this section. This is the more commonly employed approach; the second major approach, based on the GramSchmidt process, is discussed in the next section.

We first mention a few brief facts about orthogonal matrices: A $p \times q$ matrix $M$ is said to be orthogonal if $M^T M=I_q$. Then $M M^T=I_p$, and the columns of $M$ are orthonormal (not merely orthogonal). It follows that $|M x|=|x|$ (the transformation by $M$ preserves lengths). If $p=q, M$ has determinant $\pm 1$ and hence is nonsingular, all eigenvalues lie on the unit circle, $|M|=1$ in any natural matrix norm, and $M^{-1}=M^T$.

We will continue to assume (unless stated otherwise) that $A$ is an $m \times n$ matrix with $m \geq n$ and full rank $n$. Consider again the case of the LU decomposition where pivoting is not needed. We started with a matrix $A$ that we wanted to reduce to triangular form using lower triangular matrices, so that we would have $M A=U$ when we were finished, with $M$ lower triangular; we could then solve for $A=L U$ (where $L=M^{-1}$ was also lower triangular). The first step had the form
$$L_1 A=\left[\begin{array}{lll} 1 & 0 & 0 \ X & 1 & 0 \ X & 0 & 1 \end{array}\right]\left[\begin{array}{llll} X & X & X & X \ X & X & X & X \ X & X & X & X \end{array}\right]=\left[\begin{array}{cccc} X & X & X & X \ 0 & X & X & X \ 0 & X & X & X \end{array}\right]$$
(previously we had restricted ourselves to square matrices but the LU decomposition may be applied to nonsquare matrices as well). At the next stage we really need only process the $2 \times 3$ block in the southeast using a $2 \times 2$ lower triangular matrix.

## 数学代写|数值分析代写numerical analysis代考|The QR Decomposition by Orthogonalization

We’ve been placing a greater emphasis on looking at numerical linear algebra algorithms in terms of sub-blocks of the matrices involved. This is important for many reasons: It allows for the efficient design and analysis of algorithms and the maximal use of the BLAS and/or the special architecture of the machine, for example. Often we wish to block problems into chunks of data that fit in the fast cache. In many cases, if we can analyze one step in the iterative process of reducing a matrix to a special form in terms of simple matrix algebra operations, then we can gain a great deal of understanding of the method. Some programming languages allow us to, in effect, lay out the matrix in blocks in memory rather than by rows or by columns.

Putting algorithms in blocked form is not as simple as dividing matrices into blocks. For example, the matrix
$$M=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \end{array}\right]$$
certainly has an LU decomposition (after all, it’s just a permutation of $I_4$ ), but if we block it in the form $$M=\left[\begin{array}{ccccc} 1 & 0 & \vdots & 0 & 0 \ 0 & 0 & \vdots & 0 & 1 \ \cdots & \cdots & \vdots & \cdots & \cdots \ 0 & 0 & \vdots & 1 & 0 \ 0 & 1 & \vdots & 0 & 0 \end{array}\right]$$
then every block is singular. This means that the block version of Gaussian elimination may fail even with complete pivoting.

# 数值分析代考

## 数学代写|数值分析代写数值分析代考|三角化的QR分解

$$L_1 A=\left[\begin{array}{lll} 1 & 0 & 0 \ X & 1 & 0 \ X & 0 & 1 \end{array}\right]\left[\begin{array}{llll} X & X & X & X \ X & X & X & X \ X & X & X & X \end{array}\right]=\left[\begin{array}{cccc} X & X & X & X \ 0 & X & X & X \ 0 & X & X & X \end{array}\right]$$
(以前我们局限于方阵，但LU分解也可以应用于非方阵)。下一阶段我们只需要使用$2 \times 2$下三角矩阵处理东南部的$2 \times 3$块。

## 数学代写|数值分析代写数值分析代考|正交化的QR分解

$$M=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 \ 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \end{array}\right]$$

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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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