課程名稱︰數值線性代數 課程性質︰數學系選修 課程教師︰薛克民 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2017/01/10 考試時限（分鐘）：110 試題 : 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話 似乎在課堂上聽過的樣子 第02話 那真是太令人絕望了 第03話 已經沒什麼好期望了 第04話 被當、21都是存在的 第05話 怎麼可能會all pass 第06話 這考卷絕對有問題啊 第07話 你能面對真正的分數嗎 第08話 我，真是個笨蛋 第09話 這樣成績，教授絕不會讓我過的 第10話 再也不依靠考古題 第11話 最後留下的補考 第12話 我最愛的學分 --

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