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

2022年10月10日

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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

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