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

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

Let us develop now a dual idea to triangular orthogonalization, i.e., we will construct a sequence $Q_1, \ldots, Q_n$ of unitary matrices such that
$$\mathrm{Q}_n \cdots \mathrm{Q}_1 \mathrm{~A}=\mathrm{R}$$
with $\mathrm{R}$ upper triangular. If we construct this, the $Q R$ factorization of the matrix $A$ is given by $\mathrm{Q}^{\mathrm{H}}=\mathrm{Q}_n \cdots \mathrm{Q}_1$ and the $\mathrm{R}$ matrix above. This process will be obtained with the help of the so-called Householder reflectors.

Remark $5.44$ (full QR). Notice that in (5.18) we obtain a matrix Q that is unitary, i.e., we are computing a full $Q R$ factorization.

Definition $5.45$ (Householder reflector 3 ). Suppose that $\boldsymbol{w} \in \mathbb{C}^n$ with $|\boldsymbol{w}|_2=1$. The Householder reflector with respect to $\operatorname{span}{\boldsymbol{w}}^{\perp}$ is the matrix
$$\mathrm{H}_w=\mathrm{I}_n-2 w w^{\mathrm{H}} .$$
As illustrated in Figure 5.2, the action of $H_w$ on a point $x \notin \operatorname{span}{w}^{\prime}$ is the mirror reflection of $\boldsymbol{x}$ about span ${\boldsymbol{w}}^{\perp}$.

Proposition $5.46$ (properties of $H_w$ ). Suppose that $\boldsymbol{w} \in \mathbb{C}^n$ with $|\boldsymbol{w}|_2=1$. The reflector $\mathrm{H}_w$ satisfies the following properties.

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

In Chapter 3 we learned that, using direct methods such as Gaussian elimination, one could obtain an exact solution to the linear system of equations $\mathrm{Ax}=\boldsymbol{f}$ with $\mathrm{A} \in \mathbb{C}^{n \times n}$ and $f \in \mathbb{C}^n$. (Of course, we are ignoring the effects of roundoff.) Unfortunately, these algorithms require $\mathcal{O}\left(n^3\right)$ operations, which are frequently too expensive in practice. The high cost begs the following questions: Are there lower cost options? Is an approximation of $\boldsymbol{x}$ good enough? How would such an approximation be generated? As we will see, oftentimes we can find methods that have a much lower cost of computing a good approximate solution to $\boldsymbol{x}$.

As an alternate to the direct methods that we studied in the previous chapters, in the present chapter we will describe the so-called linear iteration methods for constructing sequences, $\left{\boldsymbol{x}k\right}{k=1}^{\infty} \subset \mathbb{C}^n$, with the desire that $\boldsymbol{x}k \rightarrow \boldsymbol{x}=\mathrm{A}^{-1} \boldsymbol{f}$, as $k \rightarrow \infty$. The idea is that, given some $\varepsilon>0$, we look for a $k \in \mathbb{N}$ such that $$\left|x-x_k\right| \leq \varepsilon$$ with respect to some norm. In this context, $\varepsilon$ is called the stopping tolerance. In other words, we want to make certain the error is small in norm. But a word of caution. Usually, we do not have a direct way of approximating the error. The residual is more readily available. Suppose that $\boldsymbol{x}_k$ is an approximation of $\boldsymbol{x}=\mathrm{A}^{-1} \boldsymbol{f}$. The error is $\boldsymbol{e}_k=\boldsymbol{x}-\boldsymbol{x}_k$ and the residual is $\boldsymbol{r}_k=\boldsymbol{f}-\mathrm{A} \boldsymbol{x}_k=\mathrm{A} \boldsymbol{e}_k$. Recall that $$\frac{\left|\boldsymbol{e}_k\right|}{|\boldsymbol{x}|} \leq \kappa(\mathrm{A}) \frac{\left|\boldsymbol{r}_k\right|}{|\boldsymbol{f}|} .$$ Thus, when $\kappa(\mathrm{A})$ is large, $\frac{\left|r_k\right|}{|f|}$, which is easily computable, may not be a good indicator of the size of the relative error $\frac{\left|\boldsymbol{e}{\boldsymbol{k}}\right|}{|\boldsymbol{x}|}$, which is not directly computable. One must be careful when measuring the error.

The material of this section – containing topics such as the Gauss-Seidel method and the (successive) over-relaxation (SOR) method – does not, for the most part, represent the leading edge of research in iterative solvers. We call such methods classical, though not in the sense of a pejorative. Indeed, while workers are not typically applying Gauss-Seidel methods to solve industrial strength problems, understanding such methods is vital to our investigation of more modern methods, like multigrid and conjugate gradient methods and also effective preconditioning strategies. Excellent references for the classical material of this section may be found in the books by Hageman and Young [36] and Young [103]. Another good reference is [81].

# 数值分析代考

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

$$\mathrm{Q}_n \cdots \mathrm{Q}_1 \mathrm{~A}=\mathrm{R}$$

$$\mathbf{H}_w=\mathrm{I}_n-2 w w^{\mathrm{H}} .$$

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

$$\left|x-x_k\right| \leq \varepsilon$$

$$\frac{\left|\boldsymbol{e}_k\right|}{|\boldsymbol{x}|} \leq \kappa(\mathrm{A}) \frac{\left|\boldsymbol{r}_k\right|}{|\boldsymbol{f}|} .$$

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