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+\title{Lecture notes 6 -- Introduction to Computational Science} +\author{Micheal Multerer \\ Copied by: Claudio Maggioni} + +\begin{document} +\maketitle + +\section*{2.5: Partial pivoting} +Obviously, Algorithm 2.8 fils if one of the pivots becomes zero. In this case, we need to choose a different pivot element. + +\paragraph{Simple approach:} Column pivoting choose $|a^{(i)}_{k,i}|=\max_{i \leq l \leq n}|a^{(i)}_{l,i}|$ +In order to move the pivot element and the corresponding row we switch row k and row i by a \textit{primitive matrix}: + +\[ +\bar{P}_i := \begin{bmatrix} + 1_1 \\ + & \ddots \\ + && 1_{i-1} \\ + &&& 0 & \hdots & 1_k \\ + &&&\vdots &1_{i+1} \\ + &&& 1_i && 0 \\ + &&&&&& \ddots \\ + &&&&&&& 1_n \\ + \end{bmatrix} +\] + +The following rules apply: +\begin{enumerate} +\item Multiplication by $P_i$ from the left $\Rightarrow$ interchange rows i and k +\item Multiplication by $P_i$ from the right $\Rightarrow$ interchange columns i and k +\item $(\bar{P}_i)^2 = \bar{P}_i \cdot \bar{P}_i = \bar{I}$ +\end{enumerate} + +Then performing an LU decompisition with pivoting can be +written in matrix notation as: +$$A_{i+1} = L_i \cdot \bar{P}_i \cdot A_i$$ + +\paragraph{Note:} For $j < i$, it holds $\bar{P}_i \bar{L}_j = \widetilde{L}_j \bar{P}_i$ where $\widetilde{L}$ is the same matrix as $\bar{L}_j$ except that $[\widetilde{l}_j]_i$ and $[\widetilde{L}_j]_k$ are interchanged: + +\[ +\bar{L}_i := \begin{bmatrix} + 1\\ + &\ddots\\ + &&1\\ + &&&\hat{l}^{(j)}_2\\ + &&&\hat{l}^{(j)}_i&&1\\ + &&&\hat{l}^{(j)}_k&&&\ddots\\ + &&&\hat{l}^{(j)}_n&&&&1\\ + \end{bmatrix} + \text{\hspace{1cm}} +\widetilde{L}_i := \begin{bmatrix} + 1\\ + &\ddots\\ + &&1\\ + &&&\hat{l}^{(j)}_2\\ + &&&\hat{l}^{(j)}_k&&1\\ + &&&\hat{l}^{(j)}_i&&&\ddots\\ + &&&\hat{l}^{(j)}_n&&&&1\\ + \end{bmatrix} +\] + +Resolving (*) then yields: + +$$\bar{A}_{i+1} = \bar{L}_i \bar{P}_i \bar{L}_{i-1} \bar{P}_{i-1} \ldots \bar{L}_1 \bar{P}_1 A$$ +or +$$\bar{L}_{n-1} \bar{P}_{n-1} \bar{L}_{n-2} \bar{P}_{n-2} \ldots \bar{L}_1 \bar{P}_1 \bar{A} = \bar{U}$$ + +Now we can exploit $P_2 \bar{L}_1 \bar{P}_1 = \widetilde{L}_1 \bar{P}_2 \bar{P}_1$ and so on. This yields: + +$$\bar{P}\bar{A} = \bar{L}\bar{U}$$ +with +$$\bar{P} = \bar{P}_{n-1} \bar{P}_{n-2} \ldots \bar{P}_1$$ +and +$$\widetilde{L} = \widetilde{L}^{-1}_1 \ldots \widetilde{L}^{-1}_{n-1}$$ +and $$ \widetilde{L}_{n-1} = \bar{L}_{n-1} $$$$ +\widetilde{L}_{n-2} = \bar{P}_{n-1} \bar{L}_{n-2} \bar{P}_{n-1} $$$$ +\vdots $$$$ +\widetilde{L}_1 = \bar{P}_{n-1} \bar{P}_{n-2} \ldots \bar{P}_{2} \bar{L}_1 \bar{P}_2 \ldots \bar{P}_{n-2} \bar{P}_{n-1} $$ + +\paragraph{Note:} If $\bar{A} \in R^{n \times n}$ is non-singular, the pivoted LU decomposition $\bar{P}\bar{A} = \bar{L}\bar{U}$ always exists. +\\\\ +We can easily add column priority in Algorithm 2.8: + +\paragraph{Algorithm 2.9}(Outer product LU decomposition with column pivoting) \\ +input: matrix $\bar{A} = [a_{i,j}]^n_{i,j = 1} \in R^{n\times n}$ \\ +output: pivoted LU decomposition $ \bar{L}\bar{U} = \bar{P}\bar{A}$ + +\begin{enumerate} +\item Set $\bar{A}_1 = \bar{A}, \bar{p} = [1,2,\ldots,n]$ +\item For $i = 1,2,\ldots,n$ \begin{itemize} +\item compute: k = $\argmax_{1 \leq j \leq n} | a^{(i)}_{p_j,i} |$ \% find pivot +\item swap: $ p_i \gets\to p_k$ +\item $\bar{l}_i := \bar{a}^{(i)}_{:,i} / a^{(i)}_{p_i,p_i}$ +\item $\bar{u}_i := a_{p_i,:}$ +\item compute: $\bar{A}_{i+1} = \bar{A}_i - \bar{l}_i \cdot \bar{u}_i$ +\end{itemize} +\item set $\bar{P} := [\bar{l}_{p_1},\bar{l}_{p_2},\ldots, \bar{l}_{p_n}]^T$ \% $\bar{e}_i$ is i-th unit vector +\item set $bar{L} = bar{P}[\bar{l}_1,\bar{l}_2,\cdots,\bar{l}_n]$ +\end{enumerate} + +\paragraph{Example 2.10} \textit{(omitted)} + +\section*{2.6: Cholesky decomposition} +If $\bar{A}$ is symmetric and positive definite, i.e. all eigenvalues of $\bar{A}$ are bigger than zero or equivalently $bar{x}^T\bar{A}\bar{x} > 0$ for all $\bar{x} \neq 0$, we can compute a symmetric decomposition of $\bar{A}$. + +\paragraph{Note:} if $\bar{A}$ is symmetric and positive definite, then the \textit{Schur complement} $\bar{S}:=\bar{A}_{2:n,2:n} - (\bar{a}_{2:n},1/a_{1,1})\bar{a}^T_{2:n,1}$, is symmetric and positive definite as well. In particular, it holds $s_{i,i} > 0$ and $a_{i,i} > 0$ ! + +\paragraph{Definition 2.11} A decomposition $\bar{A} = \bar{L}\bar{L}^T$ with a lower triangular matrix $\bar{L}$ with positive diagonal elements is called \textit{Cholesky decomposition of $\bar{A}$}. + +\paragraph{Note:} A Cholesky decomposition exists, if $\bar{A}$ is symmetric and positive definite. + +\paragraph{Algorithm 2.12} (outer product of Cholesky decomposition) \\ +input: matrix $\bar{A}$ symmetric and positive definite \\ +output: Cholesky decomposition $\bar{A} = \bar{L}\bar{L}^T = +[\bar{l}_1,\bar{l}_2,\ldots,\bar{l}_n] [\bar{l}_1,\bar{l}_2,\ldots,\bar{l}_n]^T$ + +\begin{enumerate} +\item set: $\bar{A}_1 := \bar{A}$ +\item for $i = 1,2,\ldots,n$ \begin{itemize} +\item set: $\bar{l}_i := a^{(i)}_{:,i} / \sqrt{a^{(i)}_{i,i}}$ +\item set: $\bar{A}_{i+1} := \bar{A}_{i} - \bar{l}_i\bar{l}^T_i$ +\end{itemize} +\item set: $\bar{L} = [\bar{l}_1,\bar{l}_2,\ldots,\bar{l}_n]$ +\end{enumerate} + +The computational cost is $\frac{1}{6}n^{3}+O(n^{2})$ and thus only half the cost of LU decomposition. + +\paragraph{Example 2.13} \textit{(omitted)} + +\end{document} \ No newline at end of file