课程名称︰数值线性代数 课程性质︰数学系选修 课程教师︰薛克民 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2017/01/10 考试时限（分钟）：110 试题 : Instructions: ● Total points 100 ● Open books, notes, and laptop ● Answer the questions thoroughly and justify all your answers 1. (35 points) Given an arbitrary 2×2 real symmetric matrix written in the form \[A = \begin{bmatrix} w+z & \varepsilon\\ \varepsilon & z \end{bmatrix}.\] (a) (25 points) Perform the following shifted QR step: \[A-zI = QR, \bar A = RQ+zI.\] Show that \[\bar A = \begin{bmatrix} \bar x+\bar z & \bar\varepsilon\\ \bar\varepsilon & \bar z \end{bmatrix},\] where \[ \bar z = z - \frac{\varepsilon^2w}{w^2+\varepsilon^2}, \bar w = w + 2\frac{\varepsilon^2w}{w^2+\varepsilon^2}, \bar\varepsilon = \frac{\varepsilon^3}{w^2+\varepsilon^3}. \] (b) (10 points) What does the result shown in (a) tell you about the conver- gence of the QR-iteration for this type of matrix? 2. (20 points) Let $A \in \mathbf C^{m\times m}$, $x \in \mathbf C^m$, and $X = \begin{bmatrix}x, Ax, \ldots, A^{m-1}x\end{bmatrix}$. If X is nonsingular, show that $X^{-1}AX$ is an upper Hessenberg matrix. 3. (45 points) Let $A \in \mathbf R^{m\times m}$ be symmetric, $T_n = Q_n^TAQ_n \in \mathbf R^{n\times n}$ be the projection of A onto the Krylov subspace $\mathcal K_n$ (computed via Lanczos algorithm), and $r_n = b - Ax_n \in \mathbf R^m$ be the residual where $x_n \in \mathcal K_n$ gives an approxi- mate solution of Ax = b at the iteration step n. Assume that A is positive definite also, i.e., $\langle v, Av\rangle = v^TAv > 0$, for any vector v and $T_n$ is nonsingular. (a) (15 points) Show that $x_n = Q_nT_n^{-1}e_1\|b\|_2$ minimizes $\|r_n\|^2_{A^{-1}} = r_n^TA^{-1}r_n$, where $e_1 = \begin{bmatrix}1, 0, \ldots, 0\end{bmatrix}$ is an n×1 vector. (b) (15 points) Show that the minimization of $\|r_n\|_{A^{-1}}$ in (a) is equivalent to minimizing the error in A-norm, i.e., $\|x-x_n\|_A$. (c) (15 points) Show that $Q_n^Tr_n = 0$. -- 第01话 似乎在课堂上听过的样子 第02话 那真是太令人绝望了 第03话 已经没什麽好期望了 第04话 被当、21都是存在的 第05话 怎麽可能会all pass 第06话 这考卷绝对有问题啊 第07话 你能面对真正的分数吗 第08话 我，真是个笨蛋 第09话 这样成绩，教授绝不会让我过的 第10话 再也不依靠考古题 第11话 最後留下的补考 第12话 我最爱的学分 --

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