hw3: done 2 and 3, 1 still to transcribe
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%% Assignment 2
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% Name: Claudio Maggioni
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%
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% Date: 19/3/2019
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%
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% This is a template file for the first assignment to get started with running
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% and publishing code in Matlab. Each problem has its own section (delineated
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% by |%%|) and can be run in isolation by clicking into the particular section
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% and pressing |Ctrl| + |Enter| (evaluate current section).
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%
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% To generate a pdf for submission in your current directory, use the following
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% three lines of code at the command window:
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%
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% >> options.format = 'pdf'; options.outputDir = pwd; publish('assignment2.m', options)
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%
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%% Problem 3
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format rational
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A = [4 3 2 1; 8 8 5 2; 16 12 10 5; 32 24 20 11];
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[L,U,P] = pivotedOuterProductLU(A)
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function [L,U,P] = pivotedOuterProductLU(A)
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dimensions = size(A);
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n = dimensions(1);
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p = 1:n;
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L = zeros(n);
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U = zeros(n);
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for i = 1:n
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values = A(:,i);
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values(values == 0) = -Inf;
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[~, p_k] = max(values);
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k = find(p == p_k);
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p(k) = p(i);
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p(i) = p_k;
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L(:,i) = A(:,i) / A(p(i),i);
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U(i,:) = A(p(i),:);
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A = A - L(:,i) * U(i,:);
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end
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I = eye(n);
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P = zeros(n);
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for i = 1:n
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P(:,i) = I(:,p(i));
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end
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P = transpose(P);
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L
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L = P * L;
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end
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% vim: set ts=2 sw=2 et tw=80:
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\documentclass[12pt,a4paper]{article}
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\usepackage[utf8]{inputenc} \usepackage[margin=2cm]{geometry}
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\usepackage{amstext} \usepackage{amsmath} \usepackage{array}
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\newcommand{\lra}{\Leftrightarrow}
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\title{Howework 3 -- Introduction to Computational Science}
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\author{Claudio Maggioni}
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\begin{document} \maketitle
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\section*{Question 1}
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to be transcribed
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\section*{Question 2}
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\[i = 1 \hspace{1cm} k = 4 \hspace{1cm} \begin{bmatrix}4&2&3&1\\\end{bmatrix}\]
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\[
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l_1 =
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\begin{bmatrix}
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1/8 \\
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1/4 \\
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1/2 \\
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1 \\
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\end{bmatrix}
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\hspace{1cm}
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u_1 = \begin{bmatrix} 32 & 24 & 10 & 11\end{bmatrix}
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\hspace{1cm}
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\]\[
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A_2 = \begin{bmatrix}4 &3 &2& 1\\ 8& 8& 5& 2\\ 16& 12& 10& 5\\ 32& 24& 20 &11 \\\end{bmatrix} - \begin{bmatrix}
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4 & 3 & 5/2 & 11/8 \\
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8 & 6 & 5 & 11/4 \\
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16 & 12 & 10 & 11/2 \\
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32 & 24 & 20 & 11 \\
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\end{bmatrix} =
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\begin{bmatrix}
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0 & 0 & -1/2 & -3/8 \\
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0 & 2 & 0 & -3/4 \\
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0 & 0 & 0 & -1/2 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix}
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\]
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\[i = 2 \hspace{1cm} k = 2 \hspace{1cm} p = \begin{bmatrix}4&2&3&1\\\end{bmatrix}\]
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\[
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l_2 =\begin{bmatrix}
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0\\1\\0\\0\\
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\end{bmatrix}
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\hspace{1cm}
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u_2 =\begin{bmatrix} 0 & 2 & 0 & -3/4 \end{bmatrix}\]
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\[
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A_3 = \begin{bmatrix}
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0 & 0 & -1/2 & -3/8 \\
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0 & 2 & 0 & -3/4 \\
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0 & 0 & 0 & -1/2 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix} -
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\begin{bmatrix}
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0 & 0 & 0 & 0 \\
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0 & 2 & 0 & -3/4 \\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix}
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=
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\begin{bmatrix}
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0 & 0 & -1/2 & -3/8 \\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & -1/2 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix}
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\]
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\[ i = 3 \hspace{1cm} k = 4 \hspace{1cm} p = \begin{bmatrix}
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4&2&1&3\\
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\end{bmatrix}
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\]
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\[
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l_3 = \begin{bmatrix}
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1 \\0\\0\\0\\
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\end{bmatrix}
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\hspace{1cm}
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u_3 = \begin{bmatrix} 0 & 0& -1/2 & -3/8\\\end{bmatrix}
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\]\[
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A_4 = \begin{bmatrix}
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0 & 0 & -1/2 & -3/8 \\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & -1/2 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix} -
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\begin{bmatrix}
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0 & 0 & -1/2& -3/8\\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix} =
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\begin{bmatrix}
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 \\
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0 & 0 & 0 & -1/2 \\
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0 & 0 & 0 & 0 \\
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\end{bmatrix}
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\]
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\[ i =4 \hspace{1cm} k = 4 \hspace{1cm} p = \begin{bmatrix}
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4&2&1&3\\
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\end{bmatrix}\]
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\[l_4 = \begin{bmatrix} 0 \\0\\1\\0\\\end{bmatrix}
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u_4 = \begin{bmatrix}0&0&0&-1/2\end{bmatrix}
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\]
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\[
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P =
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\begin{bmatrix}
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0 & 0 & 0 & 1 \\
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0 & 1 & 0 & 0 \\
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1 & 0 & 0 & 0 \\
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0 & 0 & 1 & 0 \\
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\end{bmatrix}
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\]
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\[
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L = P *
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\begin{bmatrix}
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1/8 & 0 & 1 & 0 \\
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1/4 & 1 & 0 & 0 \\
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1/2 & 0 & 0 & 1 \\
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1 & 0 & 0 & 0 \\
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\end{bmatrix}
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=
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\begin{bmatrix}
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1 & 0 & 0 & 0 \\
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1/4 & 1 & 0 & 0 \\
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1/8 & 0 & 1 & 0 \\
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1/2 & 0 & 0 & 1 \\
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\end{bmatrix}
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\]
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\[
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U =
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\begin{bmatrix}
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32 & 24 & 20 & 11 \\
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0 & 2 & 0 & -3/4 \\
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0 & 0 & -1/2 & -3/8 \\
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0 & 0 & 0 & -1/2 \\
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\end{bmatrix}
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\]
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\end{document}
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