%% Assignment 2
% Name: Claudio Maggioni
%
% Date: 19/3/2019
%
% This is a template file for the first assignment to get started with running
% and publishing code in Matlab.  Each problem has its own section (delineated
% by |%%|) and can be run in isolation by clicking into the particular section
% and pressing |Ctrl| + |Enter| (evaluate current section).
%
% To generate a pdf for submission in your current directory, use the following
% three lines of code at the command window:
%
%  >> options.format = 'pdf'; options.outputDir = pwd; publish('assignment2.m', options)
%

%% Problem 3
format rational
A = [4 3 2 1; 8 8 5 2; 16 12 10 5; 32 24 20 11];
[L,U,P] = pivotedOuterProductLU(A)

function [L,U,P] = pivotedOuterProductLU(A)
    dimensions = size(A);
    n = dimensions(1);
    
    p = 1:n;
    L = zeros(n);
    U = zeros(n);

    for i = 1:n
        values = A(:,i);
        values(values == 0) = -Inf;
        [~, p_k] = max(values);
        k = find(p == p_k);
        
        p(k) = p(i);
        p(i) = p_k;
        
        L(:,i) = A(:,i) / A(p(i),i);
        U(i,:) = A(p(i),:);
        A = A - L(:,i) * U(i,:);
    end
    
    I = eye(n);
    P = zeros(n);
    for i = 1:n
        P(:,i) = I(:,p(i));
    end
    P = transpose(P);
    L
    L = P * L;   
end