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数学代写|数值分析代写Numerical analysis代考|MATH408 The Practical Implementation of the QR Method

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数学代写数值分析代写Numerical analysis代考|The Practical Implementation of the QR Method

In its original form (6.6.4.3), the $Q R$ method has some serious drawbacks that make it hardly competitive with the methods considered so far:
(a) The method is expensive: a complete step $A_i \rightarrow A_{i+1}$ for a dense $n \times n$ matrix $A$ requires $O\left(n^3\right)$ operations.
(b) Convergence is very slow if some quotients $\left|\lambda_j / \lambda_k\right|, j \neq k$, of the eigenvalues of $A$ are close to 1 .
However, there are remedies for these disadvantages.
(a) Because of the amount of work, one applies the $Q R$ method only to reduced matrices $A$, namely matrices of Hessenberg form, or in the case of Hermitian matrices, to Hermitian tridiagonal matrices (i.e., Hermitian Hessenberg matrices). A general matrix $A$, therefore must first be reduced to one of these forms by means of the methods described in Section 6.5. For this procedure to make sense, one must show that these special matrices are invariant under the $Q R$ transformation: if $A_i$ is a (possibly Hermitian) Hessenberg matrix then so is $A_{i+1}$. This invariance is easily established. From Theorem $(6.6 .4 .4)(\mathrm{a})$, the matrix $A_{i+1}=Q_i^H A_i Q_i$ is unitarily similar to $A_i$ so that $A_{i+1}$ is Hermitian if $A_i$ is. If $A_i$ is an $n \times n$ Hessenberg matrix, then $A_{i+1}$ can be computed as follows: First, reduce the subdiagonal elements of $A_i$ to 0 by means of suitable Givens matrices of type $\Omega_{12}, \ldots, \Omega_{n-1, n}$ [see Section 4.9]
$$\begin{gathered} \Omega_{n-1, n} \cdots \Omega_{23} \Omega_{12} A_i=R_i=\left[\begin{array}{ccc} • & \cdots & * \ & \ddots & \vdots \ 0 & & * \end{array}\right] \ A_i=Q_i R_i, \quad Q_i:=\Omega_{12}^H \Omega_{23}^H \cdots \Omega_{n-1, n}^H \end{gathered}$$
and then compute $A_{i+1}$ by means of
$$A_{i+1}=R_i Q_i=R_i \Omega_{12}^H \Omega_{23}^H \cdots \Omega_{n-1, n}^H$$

数学代写|数值分析代写Numerical analysis代考|Computation of the Singular Values of a Matrix

The singular values (6.4.6) of an $m \times n$ matrix $A$ and its singular-value decomposition (6.4.11) can be computed rapidly, and in a numerically stable manner, by a method due to Golub and Reinsch (1971), which is closely related to the $Q R$ method. We assume, without loss of generality, that $m \geq n$ (otherwise, replace $A$ by $A^H$ ). The decomposition (6.4.11) can then be written in the form $(6.7 .1)$
$$A=U\left[\begin{array}{c} D \ 0 \end{array}\right] V^H, \quad D:=\operatorname{diag}\left(\sigma_1, \ldots, \sigma_n\right), \quad \sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_n \geq 0$$
where $U$ is a unitary $m \times m$ matrix, $V$ is a unitary $n \times n$ matrix, and $\sigma_1$, $\ldots, \sigma_n$ are the singular values of $A$, i.e., $\sigma_i^2$ are the eigenvalues of $A^H A$. In principle, therefore, one could determine the singular values by solving the eigenvalue problem for the Hermitian matrix $A^H A$, but this approach can be subject to loss of accuracy: for the matrix
$$A:=\left[\begin{array}{ll} 1 & 1 \ \varepsilon & 0 \ 0 & \varepsilon \end{array}\right], \quad|\varepsilon|<\sqrt{\mathrm{eps}}, \quad \text { eps }=\text { machine precision }$$
for example, the matrix $A^H A$ is given by
$$A^H A=\left[\begin{array}{cc} 1+\varepsilon^2 & 1 \ 1 & 1+\varepsilon^2 \end{array}\right],$$
and $A$ has the singular values $\sigma_1(A)=\sqrt{2+\varepsilon^2}, \sigma_2(A)=|\varepsilon|$. In floatingpoint arithmetic, with precision eps, instead of $A^H A$ one obtains the matrix
$$\mathrm{fl}\left(A^H A\right)=: B=\left[\begin{array}{ll} 1 & 1 \ 1 & 1 \end{array}\right],$$
with eigenvalues $\tilde{\lambda}_1=2$ and $\tilde{\lambda}_2=0 ; \sigma_2(A)=|\varepsilon|$ does not agree to machine precision with $\sqrt{\tilde{\lambda}_2}=0$.

数学代写数值分析代写Numerical analysis代考|The Practical Implementation of the QR Method

(a) 该方法很昂贵: 一个完整的步乑 $A_i \rightarrow A_{i+1}$ 对于密集的 $n \times n$ 矩阵 $A$ 需要 $O\left(n^3\right)$ 操作。
(b) 如果某些商数收敛很慢 $\left|\lambda_j / \lambda_k\right|, j \neq k$ ，的特征值 $A$ 接近 1。

(a) 由于工作量大，一个人应用 $Q R$ 仅适用于简化矩轴的方法 $A$ ，即 Hessenberg 形式的矩阵，或者在 Hermitian 矩伡的情况

Hessenberg 矩阵那么也是 $A_{i+1}$. 这种不变性很容易建立。从定理 $(6.6 .4 .4)(\mathrm{a})$ ，矩阵 $A_{i+1}=Q_i^H A_i Q_i$ 单一地类似于于 $A_i$ 以便
$A_{i+1}$ 是 Hermitian 如果 $A_i$ 是。如果 $A_i$ 是一个 $n \times n$ Hessenberg 矩阵，则 $A_{i+1}$ 可以计算如下: 首先，减少对角线元溸 $A_i$ 通过

\$\$
\begin } { \text { gathered } }
\Omega_{n-1, $n} \backslash$ cdots \Omega_ ${23} \backslash$ \Omega_{12 $} A_{-} \mathrm{i}=\mathrm{R}_{-} \mathrm{i}=\backslash$ left $[\backslash$ begin ${$ array $}{c c c}$

\& $|\operatorname{cdots} \& *|$
$\& \mid$ ddots \& $\mid$ vdots $\mid$
$0 \& \& *$
lend{array} \right] $\backslash$
$A_{-} i=Q_{-} i R_{-} i$, \quad Q_i:=\Omega_{12 $} \wedge H$ \Omega_{23 $} \wedge H$ \cdots \Omega_ ${n-1, n} \wedge H$ lend{聚集 $}$
andthencompute $\$ A_{i+1} \$$bymeansof A_{-}{i+1}=R_{-} i Q_{-} i=R_{-} i \backslash Omega_ {12} \wedge H \backslash Omega_ {23} \wedge H Ccdots \Omega_ {n-1, n} \wedge H \\ 数学代写|数值分析代写Numerical analysis代考|Computation of the Singular Values of a Matrix 的奇异值 (6.4.6) m \times n 矩阵 A 及其奇异值分解 (6.4.11) 可以通过 Golub 和 Reinsch (1971) 的方法以数值稳定的方式快速计算， 该方法与 Q R 方法。我们假设，不失一般性， m \geq n (否则，更换 A 经过 A^H ). 分解 (6.4.11) 可以写成以下形式(6.7.1)$$
A=U[D 0] V^H, \quad D:=\operatorname{diag}\left(\sigma_1, \ldots, \sigma_n\right), \quad \sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_n \geq 0


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