commit 3018f44f9d600c40ca77da1d97b36b22bc6080ba parent 4ec62c0901f8c9b5479a8a2e196a7420fff5514b Author: miksa234 <milutin@popovic.xyz> Date: Tue, 5 Apr 2022 17:30:04 +0200 update abgabe Diffstat:
| D | num_ana/build/prb2.aux | | | 39 | --------------------------------------- |
| D | num_ana/build/prb2.fdb_latexmk | | | 257 | ------------------------------------------------------------------------------- |
| D | num_ana/build/prb2.fls | | | 1375 | ------------------------------------------------------------------------------- |
| D | num_ana/build/prb2.log | | | 1143 | ------------------------------------------------------------------------------- |
| D | num_ana/build/prb2.out | | | 19 | ------------------- |
| D | num_ana/build/prb2.toc | | | 19 | ------------------- |
| A | num_ana/build/prb4.pdf | | | 0 | |
| A | num_ana/build/prb5.pdf | | | 0 | |
| A | num_ana/prb4.tex | | | 77 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | num_ana/prb5.tex | | | 238 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| A | num_ana/sheets/sheet_5.pdf | | | 0 |
11 files changed, 315 insertions(+), 2852 deletions(-)
diff --git a/num_ana/build/prb2.aux b/num_ana/build/prb2.aux @@ -1,39 +0,0 @@ -\relax -\providecommand\zref@newlabel[2]{} -\providecommand\hyper@newdestlabel[2]{} -\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} -\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined -\global\let\oldcontentsline\contentsline -\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} -\global\let\oldnewlabel\newlabel -\gdef\newlabel#1#2{\newlabelxx{#1}#2} -\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} -\AtEndDocument{\ifx\hyper@anchor\@undefined -\let\contentsline\oldcontentsline -\let\newlabel\oldnewlabel -\fi} -\fi} -\global\let\hyper@last\relax -\gdef\HyperFirstAtBeginDocument#1{#1} -\providecommand\HyField@AuxAddToFields[1]{} -\providecommand\HyField@AuxAddToCoFields[2]{} -\@writefile{toc}{\contentsline {section}{\numberline {1}Sheet 2}{1}{section.1}\protected@file@percent } -\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Problem 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[4 <./prog/prb2_p.pdf>] -Overfull \hbox (0.80122pt too wide) in paragraph at lines 222--226 -\T1/mlmr/m/n/10 We can show that all the eigen-val-ues of $\OML/mlmm/m/it/10 P[]$ \T1/mlmr/m/n/10 are in $\OT1/mlmr/m/n/10 [0\OML/mlmm/m/it/10 ; \OT1/mlmr/m/n/10 4]$ \T1/mlmr/m/n/10 be-cause $\OML/mlmm/m/it/10 P[]$ \T1/mlmr/m/n/10 is the fi-nite dif-fer-enece/laplacian - [] - -[5] -<./prog/prb2_p.pdf, id=258, page=2, 614.295pt x 794.97pt> -File: ./prog/prb2_p.pdf Graphic file (type pdf) -<use ./prog/prb2_p.pdf, page 2> -Package pdftex.def Info: ./prog/prb2_p.pdf , page2 used on input line 293. -(pdftex.def) Requested size: 418.25368pt x 347.54803pt. -<./prog/prb2_p.pdf, id=260, page=3, 614.295pt x 794.97pt> -File: ./prog/prb2_p.pdf Graphic file (type pdf) -<use ./prog/prb2_p.pdf, page 3> -Package pdftex.def Info: ./prog/prb2_p.pdf , page3 used on input line 295. -(pdftex.def) Requested size: 334.60422pt x 130.80663pt. - [6 <./prog/prb2_p.pdf> <./prog/prb2_p.pdf>] (build/prb2.aux) -Package rerunfilecheck Info: File `prb2.out' has not changed. -(rerunfilecheck) Checksum: DBF7E22537B42492E7219F09DF8D3F93;1383. - 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409 PDF objects out of 1000 (max. 8388607) - 315 compressed objects within 4 object streams - 85 named destinations out of 1000 (max. 500000) - 198 words of extra memory for PDF output out of 10000 (max. 10000000) - diff --git a/num_ana/build/prb2.out b/num_ana/build/prb2.out @@ -1,19 +0,0 @@ -\BOOKMARK [1][-]{section.1}{\376\377\000S\000h\000e\000e\000t\000\040\0002}{}% 1 -\BOOKMARK [2][-]{subsection.1.1}{\376\377\000P\000r\000o\000b\000l\000e\000m\000\040\0001}{section.1}% 2 -\BOOKMARK [3][-]{subsubsection.1.1.1}{}{subsection.1.1}% 3 -\BOOKMARK [3][-]{subsubsection.1.1.2}{}{subsection.1.1}% 4 -\BOOKMARK [3][-]{subsubsection.1.1.3}{}{subsection.1.1}% 5 -\BOOKMARK [2][-]{subsection.1.2}{\376\377\000P\000r\000o\000b\000l\000e\000m\000\040\0002}{section.1}% 6 -\BOOKMARK [3][-]{subsubsection.1.2.1}{}{subsection.1.2}% 7 -\BOOKMARK [3][-]{subsubsection.1.2.2}{}{subsection.1.2}% 8 -\BOOKMARK [2][-]{subsection.1.3}{\376\377\000P\000r\000o\000b\000l\000e\000m\000\040\0003}{section.1}% 9 -\BOOKMARK [3][-]{subsubsection.1.3.1}{}{subsection.1.3}% 10 -\BOOKMARK [3][-]{subsubsection.1.3.2}{}{subsection.1.3}% 11 -\BOOKMARK [3][-]{subsubsection.1.3.3}{}{subsection.1.3}% 12 -\BOOKMARK [2][-]{subsection.1.4}{\376\377\000P\000r\000o\000b\000l\000e\000m\000\040\0004}{section.1}% 13 -\BOOKMARK [3][-]{subsubsection.1.4.1}{}{subsection.1.4}% 14 -\BOOKMARK [3][-]{subsubsection.1.4.2}{}{subsection.1.4}% 15 -\BOOKMARK [3][-]{subsubsection.1.4.3}{}{subsection.1.4}% 16 -\BOOKMARK [2][-]{subsection.1.5}{\376\377\000P\000r\000o\000b\000l\000e\000m\000\040\0005}{section.1}% 17 -\BOOKMARK [3][-]{subsubsection.1.5.1}{}{subsection.1.5}% 18 -\BOOKMARK [3][-]{subsubsection.1.5.2}{}{subsection.1.5}% 19 diff --git a/num_ana/build/prb2.toc b/num_ana/build/prb2.toc @@ -1,19 +0,0 @@ -\contentsline {section}{\numberline {1}Sheet 2}{1}{section.1}% -\contentsline {subsection}{\numberline {1.1}Problem 1}{1}{subsection.1.1}% -\contentsline {subsubsection}{\numberline {1.1.1}}{1}{subsubsection.1.1.1}% -\contentsline {subsubsection}{\numberline {1.1.2}}{2}{subsubsection.1.1.2}% -\contentsline {subsubsection}{\numberline {1.1.3}}{2}{subsubsection.1.1.3}% -\contentsline {subsection}{\numberline {1.2}Problem 2}{3}{subsection.1.2}% -\contentsline {subsubsection}{\numberline {1.2.1}}{3}{subsubsection.1.2.1}% -\contentsline {subsubsection}{\numberline {1.2.2}}{3}{subsubsection.1.2.2}% -\contentsline {subsection}{\numberline {1.3}Problem 3}{4}{subsection.1.3}% -\contentsline {subsubsection}{\numberline {1.3.1}}{4}{subsubsection.1.3.1}% -\contentsline {subsubsection}{\numberline {1.3.2}}{4}{subsubsection.1.3.2}% -\contentsline {subsubsection}{\numberline {1.3.3}}{4}{subsubsection.1.3.3}% -\contentsline {subsection}{\numberline {1.4}Problem 4}{5}{subsection.1.4}% -\contentsline {subsubsection}{\numberline {1.4.1}}{5}{subsubsection.1.4.1}% -\contentsline {subsubsection}{\numberline {1.4.2}}{5}{subsubsection.1.4.2}% -\contentsline {subsubsection}{\numberline {1.4.3}}{5}{subsubsection.1.4.3}% -\contentsline {subsection}{\numberline {1.5}Problem 5}{6}{subsection.1.5}% -\contentsline {subsubsection}{\numberline {1.5.1}}{6}{subsubsection.1.5.1}% -\contentsline {subsubsection}{\numberline {1.5.2}}{6}{subsubsection.1.5.2}% diff --git a/num_ana/build/prb4.pdf b/num_ana/build/prb4.pdf Binary files differ. diff --git a/num_ana/build/prb5.pdf b/num_ana/build/prb5.pdf Binary files differ. diff --git a/num_ana/prb4.tex b/num_ana/prb4.tex @@ -0,0 +1,77 @@ +\include{./preamble.tex} + +\begin{document} +\maketitle +\section{Sheet 4} +\subsection{Problem 1} +Consider a linear system of equations $Ax = b$, where +\begin{align} + A = + \begin{pmatrix} + 2 & -1 & 0 \\ + -1 & 2 & -1\\ + 0 & -1 & 2 + \end{pmatrix}, \qquad + b = + \begin{pmatrix} + 4 \\ 0 \\ 0 + \end{pmatrix}, +\end{align} +we carry out iterations of the CG method by hand until we reach the +solution with an initial guess $x_0 = \begin{pmatrix} 0 & 0 & 0 +\end{pmatrix}^T$. For the sake of completeness the CG method has the following +iteration at the $k$-th step +\begin{align} + \alpha_k &= \frac{r_k^Tr_k}{p_k^TAp_k}\\ + x_{k+1} &= x_k + \alpha_k p_k\\ + r_{k+1} &= r_k - \alpha_k A p_k \\ + \beta_{k} &= \frac{r_{k+1}^Tr_{k+1}}{r_{k}^T r_k}\\ + p_{k+1} &= r_{k+1} + \beta_{k}p_k \\ +\end{align} +For $k=0$ we have +\begin{align} + r_0 &= b - Ax_0 = b,\\ + p_0 &= r_0 = b = \begin{pmatrix} 4 & 0 & 0 \end{pmatrix}^T. +\end{align} +For k=1 we have +\begin{align} + \alpha_0 &= \frac{1}{2}, \quad x_1=\begin{pmatrix} 2 \\ 0 \\0 + \end{pmatrix}, \quad r_1 = \begin{pmatrix} 0 \\ 2 \\0 + \end{pmatrix},\\ + \beta_0 &= \frac{1}{4},\quad p_1 = \begin{pmatrix} 1\\2\\0 + \end{pmatrix}. +\end{align} +For k=2 we have +\begin{align} + \alpha_1 &= \frac{2}{3}, \quad x_2=\frac{1}{3}\begin{pmatrix} 8 \\ 4 \\0 + \end{pmatrix}, \quad r_2 = \frac{1}{3}\begin{pmatrix} 0 \\ 0 \\4 + \end{pmatrix},\\ + \beta_1 &= \frac{4}{9},\quad p_2 = \frac{1}{9}\begin{pmatrix} 4\\8\\12 + \end{pmatrix}. +\end{align} +For k=3 we have +\begin{align} + \alpha_2 &= \frac{3}{4}, \quad x_3=\begin{pmatrix} 1 \\ 2 \\3 + \end{pmatrix}, \quad r_3 = \begin{pmatrix} 0 \\ 0 \\0 + \end{pmatrix},\\ + \beta_2 &= 0,\quad p_3 = \begin{pmatrix} 0\\0\\0 + \end{pmatrix}. +\end{align} +Since $r_3 = \textbf{0}$ we can stop here, and $x_3 = x$ is the unique +solution. The Krylov space of $\mathcal{K}_k(A, b)$ is defined for $k=3$ as +\begin{align} + \mathcal{K}_3(A,b) = \text{span}\left\{b, Ab, A^2b \right\} = \text{span} \left\{ \begin{pmatrix} 0\\0\\4 \end{pmatrix}, + \begin{pmatrix} 8\\-4\\0 \end{pmatrix}, + \begin{pmatrix}26\\-16\\4\end{pmatrix}\right\} +\end{align} +the rank of the span of $\mathcal{K}_k(A,b)$ is full thereby the +$\dim(\mathcal{K}_k(A,b)) = 3$. Furthermore the residuals $r_0,\ldots, +r_{k-1}$ form an orthogonal basis for $\mathcal{K}_k(A,b)$. This can be +verified by checking that `key' elements in $\mathcal{K}_k(A, b)$ can be +expressed as a linear combination of $r_0, r_1, r_2$. +\begin{align} + b = 3\cdot r_2,\quad Ab = 2r_0-2r_1,\quad A^2b=6r_0-8r_1+3r_2. +\end{align} +\subsection{Exercise 3, 4} +Not important see notes is not easy +\end{document} diff --git a/num_ana/prb5.tex b/num_ana/prb5.tex @@ -0,0 +1,238 @@ +\include{./preamble.tex} + +\begin{document} +\maketitle +\section{Sheet 5} +\subsection{Problem 1} +Let $A \in \mathbb{R}^{n \times n}$ be an SPD matrix with an additive +decomposition $A = D + L + L^T$, where $D$ consists of the diagonal entries +of $A$, $L$ and $L^T$ of the lower diagonal and upper-diagonal entries of $A$ +respectively. For $\omega \in(0,2)$ the SSOR preconditioner is defined as +follows +\begin{align} + C_\omega = \frac{1}{2-\omega}\left( \frac{1}{\omega} D+L\right) + \left( \frac{1}{\omega} D \right)^{-1}\left( \frac{1}{\omega}D + L^T + \right). +\end{align} +We can rewrite $C_\omega = KK^T$, where $K$ is an invertible lower-triangular +matrix by a simple splitting of the diagonal entries $D = +D^{\frac{1}{2}}D^{\frac{1}{2}}$. We get +\begin{align} + C_\omega &= \frac{1}{2-\omega}\left( \frac{1}{\omega}D +L \right)\omega + D^{-1}\left( \frac{1}{\omega}D + L^T \right) \\ + &= \frac{1}{2-\omega}\left( D^{\frac{1}{2}} + \omega + LD^{-\frac{1}{2}}\left( D^{\frac{1}{2}}+\omega LD^{-\frac{1}{2}} \right) +\right)^T = KK^T, +\end{align} +where +\begin{align} + K := \frac{1}{\sqrt{1-\omega} }\left( D^{\frac{1}{2}} + \omega LD^{- + \frac{1}{2}} \right) +\end{align} +The Matrix $C_\omega$ is a good approximation for the inverse of $A$ because +\begin{align} + H^{\text{SSOR}}_\omega = I - C^{-1}_\omega A\\ + \rho(H^{\text{SSOR}}_\omega) < 1 +\end{align} +for a right choice of $\omega$. Furthermore the general idea is that we +multiply the system $Ax=b$ with a preconditioning matrix $P \in +\text{GL}_n\left(\mathbb{R}\right)$ with inevitability, then we get the +system +\begin{align} + \underbrace{PA}_{\approx I}x&= Pb\\ +\end{align} +The thing is that $\text{cond}_2(A) = v(n)$ bound by the curse of +dimensionality and $\text{cond}_2 = s \ll v(n)$ not dependent and thereby $P$ +would be an optimal preconditioner. +\subsection{Exercise 2} +Let $m, n \in \mathbb{N}$, $I \in \mathbb{R}^{m\times m}$ the identity in +$\mathbb{R}^{m\times m}$ and $Q$ be the banded matrix +\begin{align} + Q = + \begin{pmatrix} + 4 & - 1 && & \\ + -1 & 4& -1& & \\ + &\ddots& \ddots & \ddots & \\ + & & -1 & 4 & -1\\ + & & & & 4 \\ + \end{pmatrix} +\end{align} +The eigenvalues of the matrix $Q$ lie in $\sigma(A) \subset [4-2, 4+2]$ by +the Gershorin disk theorem. Since no eigenvalue is $0$, then $Q$ is +invertible. +Now consider the matrix $A \in \mathbb{R}^{nm \times nm}$ +\begin{align} + Q = + \begin{pmatrix} + Q & - I & & & \\ + -I & Q& -I& & \\ + &\ddots& \ddots & \ddots & \\ + & & -I & Q & -I\\ + & & & & Q \\ + \end{pmatrix} +\end{align} +We can consider the separation wrt. addition $A = D + L + L^T$ (Like in +Exercise 1). The Jacobi-Method iteration matrix is $J = - D^{-1}(L + L^T)$, +where $L$ is the lower triangular with \textbf{-1 or 0 entries}. Further more +the Gershorin theorem states that $\sigma(A) < 1$. All in all the matrix $I - +J$ is by the geometric(Neumann) series +\begin{align} + (I-J)^{-1} = \sum_{n=0}^{\infty}J^k +\end{align} +and we have the identity +\begin{align} + J = I - D^{-1} A \quad \Rightarrow \quad (I-J)^{-1} = DA^{-1}. +\end{align} +Thereby the sum transforms to +\begin{align} + A^{-1} = D^{-1}\sum_{n=0}^{\infty} J^k. +\end{align} +The entries of $D$ are all $4$ and thereby non-negative, the matrix is also +invertible. The matrix $L$ has only -1 or 0 entries which get compensated +with the minus sing in $J = -D^{-1}(L +L^T) = D^{-1}(-L - L^T)$, thereby all +entries of $J^k$ are positive for all $k$. Finally we arrive at the +conclusion, that all entries of $A^{-1}$ are non-negative and $A$ is a +$M$-matrix or `(inverse) monotone' matrix. +\subsection{Exercise 3} +Let $A \in \mathbb{R}^{n \times n}$ be an SPD matrix and $b \in +\mathbb{R}^{n}$ be a right hand side of a linear system. Suppose we apply the +CG method for solving $Ax = b$. The $k$-th iterate $x_k$ of the CG method +then satisfies the $A$-norm optimality condition +\begin{align} + \|x_k - x\|_A = \min_{y\in x_0 + B_k} \|y - x\|_A, +\end{align} +where +\begin{align} + B_k = \text{span}\left\{ p_0,\ldots,p_{k-1} \right\} = \text{span}\left\{ + r_0, Ar_0, \ldots,A^{k-1}r_0\right\} +\end{align} +is the Krylov space. The search directions $p_k$ form an $A$-orthogonal +system. +\newline +Now if the spectrum of $A$, $\sigma(A) = [a, b] \subset (0, \infty)$ then +for any polynomial $p \in \mathbb{P}_k^{0, 1}:= \left\{p \in \mathbb{P}: p(0) += 1\right\} $ we have that +\begin{align} + \|x_k - x\|_A \le \left( \sup_{t\in[a,b]} \mid p(t) \mid \right) + \|x_0 - x\|_A. +\end{align} +To show this we have that for all $y \in x_0 + B_k$ the representation +\begin{align} + y = \sum_{j=0}^{k-1} c_j A^j r_0 = x_0 + q_y(A)r_0 +\end{align} +for suitable $c_j$'s and a polynomial $q_y \in \mathbb{P}_{k-1}$. Now +\begin{align} + y - x &= x_0 + x - q_y(A) r_0 = x_0 - x + q_y(A) \left( b - Ax_0 + \right)\\ + &= x_0 - x + q_y(A) \left(Ax - Ax_0 \right)\\ + &=\underbrace{\left(I - q_y(A)A \right)}_{=: p_y(A) \in + \mathbb{P}_k^{0, 1}} (x-x_0). +\end{align} +With this information we may consider the norm +\begin{align} + \|x_k - x\|_A \leq \|p_y(A)(x- x_0)\|_A \qquad \forall y \in x_0+B_k. +\end{align} +Now we use the fact that $A$ is SPD thereby there is an orthogonal matrix $Q$ +diagonalizing $A = Q^T \Lambda Q$, with $\Lambda = \text{diag}(\lambda_1, \ldots, +\lambda_n)$ consisting of eigenvalues of $A$, then +\begin{align} + A^k = Q^T\Lambda Q \cdots Q^T \Lambda Q = Q^T \Lambda^k Q +\end{align} +With this we can transform the polynomial $p_y(A)$ and the geometric +(Neumann) series +\begin{align} + p_y(A) = \sum_{j=0}^{\infty} c_j A^j = Q^T \left(\sum_{j=0}^{\infty} c_j + \Lambda^j\right) +\end{align} +The norm becomes then +\begin{align} + \|p(A) (x-x_0)\|^2_a + &= \langle Ap(A) (x-x_0), p(A)(x-x_0)\rangle =\\ + &= \langle Q^T\Lambda Q Q^Tp(\Lambda)Q (x-x_0), + Q^Tp(\Lambda)Q(x-x_0)\rangle =\\ + &= \langle Q^T\Lambda p(\Lambda)Q (x-x_0), + Q^Tp(\Lambda)Q(x-x_0)\rangle =\\ + &= \langle \Lambda p(\Lambda)Q (x-x_0), + p(\Lambda)Q(x-x_0)\rangle =\\ + &= \langle \Lambda^{\frac{1}{2}} p(\Lambda)Q (x-x_0), + \Lambda^{\frac{1}{2}} p(\Lambda)Q(x-x_0)\rangle =\\ + &= \|\Lambda^{\frac{1}{2}}p(\Lambda)Q(x-x_0)\|_2\\ + &= \|p(\Lambda)\Lambda^{\frac{1}{2}}Q(x-x_0)\|_2\\ + &\leq \|p(\Lambda)\|_2 \|\Lambda^{\frac{1}{2}}Q(x-x_0)\|_2. +\end{align} +The Norm of the polynomial is the maximal eigenvalue thereby +\begin{align} + \|p(\Lambda\|_2 = \max_{\lambda \in \sigma(A)} \mid p(\lambda) \mid \leq + \sup_{t\in[a,b]} \mid p(t)\mid, +\end{align} +we can do the supremum boundary because $\lambda \in [a, b]$. As for the +second part +\begin{align} + \|\Lambda^{\frac{1}{2}}Q(x-x_0)\|_2^2 &= (x-x_0)^T Q^T + \Lambda^{\frac{1}{2}}\Lambda^{\frac{1}{2}}Q(x-x_0)\\ + &= (x-x_0)^T A (x-x_0) \\ + &= \|x-x_0\|_A^2. +\end{align} +And finally we get the result +\begin{align} + \|x_k - x\| \le \sup_{t\in[a,b]} \mid p(t) \mid \|x-x_0\|_A^2. +\end{align} +The last approximation can be done because $\sup_{t\in[a,b]} \mid p(t) \mid$ +holds for \textbf{all} $p \in \mathbb{P}_k^{0,1}$ thereby we can bound by an +infimum over all the polynomials in $\mathbb{P}_k^{0,1}$ and we get +\begin{align} + \sup_{t\in[a,b]} \mid p(t) \mid \le \inf_{p \in \mathbb{P}_k^{0,1}} + \|p\|_{C([0,1])}. +\end{align} +\subsubsection{Exercise 4} +We can do subsequently the as in the last exercise wit the GMRES method. So +we let $A \in \mathbb{R}^{n \times n}$ be an SPD and $b \in \mathbb{R}^{n}$ +be the right hand side of the linear system. The iterates of $x_k$ of the CG +method satisfy the $A^{-1}$-norm optimality +\begin{align} + \|Ax_k - b\|_{A^{-1}} = \min_{y\in x_0 + C_k} \|Ay - b\|_{A^{-1}}, +\end{align} +with $C_k = \text{span}\left\{ p_0, Ap_0, \ldots , A^{k-1}p_0 \right\}$. The +`generalized minimal residual', short GMRES method, instead, formally +constructs a sequence of iterates $x_k^G$ by +\begin{align} + \|Ax^G_2 - b \|_2 = \min_{y \in x_0 + C_k}\|Ay - b\|_2. +\end{align} +The GMRES method allows for an error inequality similar to the one observed +in the CG method +\begin{align} + \|Ax_k^G - b\|_2 \le \inf_{p \in \mathbb{P}_k^{0,1}} \|p(A)\|_2 \|Ax_0 + -b\|_2. +\end{align} +To show this we start off by minimizing over a $z \in C_k$ +\begin{align} + \|Ax_k^G - b\|_2 = \min_{y\in x_k + C_k} \|Ay - b\|_2 = \min_{z \in + C_k}\|Az + Ax_0 -b\|_2. +\end{align} +Then for all $z \in C_k$, there exists a $\pi_k \in \mathbb{P}_{k-1}$ such that +\begin{align} + z = \pi_k(A) p_0 = \pi_k(A) r_0, +\end{align} +Then the minimization can be bounded +\begin{align} + \min_{z \in C_k} \|Az + Ax_0 - b\|_2 &= + \min_{\pi_k \in \mathbb{P}_k}\|A\pi_k(A)r_0 + Ax_0 -b\|_2\\ + &\le \|A\pi_k(A) (b-Ax_0) + Ax_0 +b\|_2\\ + &= \|(Ax_0 -b)\underbrace{(I-A\pi_k(A)}_{=:p \in + \mathbb{P}_k^{0,1}}\|_2\\ + &= \|(Ax_0-b)p(A)\|\\ + &\le \|p(A)\|_2 \|Ax_0 -b\|_2. +\end{align} +Following the same argumentation as in Exercise 3 we get for the norm of the +polynomial +\begin{align} + \|p(A)\|_2 &= \max_{\lambda \in \sigma(A)} \mid p(\lambda) \mid\\ + &\le \sup_{t\in [a,b]} \mid p(t) \mid\\ + &\le \inf_{p \in \mathbb{P}_k^{0,1}} \|p\|_{C\left([a,b] \right) }. +\end{align} + + + +\end{document} + + diff --git a/num_ana/sheets/sheet_5.pdf b/num_ana/sheets/sheet_5.pdf Binary files differ.