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+\documentclass{scrartcl}
+\usepackage[utf8]{inputenc}
+\usepackage{graphicx}
+\usepackage{subcaption}
+\usepackage{amsmath}
+\usepackage{pgfplots}
+\pgfplotsset{compat=newest}
+\usetikzlibrary{plotmarks}
+\usetikzlibrary{arrows.meta}
+\usepgfplotslibrary{patchplots}
+\usepackage{grffile}
+\usepackage{amsmath}
+\usepackage{subcaption}
+\usepgfplotslibrary{external}
+\tikzexternalize
+\usepackage[margin=2.5cm]{geometry}
+
+% To compile:
+% sed -i 's#title style={font=\\bfseries#title style={yshift=1ex, font=\\tiny\\bfseries#' *.tex
+% luatex -enable-write18 -shellescape main.tex
+
+\pgfplotsset{every x tick label/.append style={font=\tiny, yshift=0.5ex}}
+\pgfplotsset{every title/.append style={font=\tiny, align=center}}
+\pgfplotsset{every y tick label/.append style={font=\tiny, xshift=0.5ex}}
+\pgfplotsset{every z tick label/.append style={font=\tiny, xshift=0.5ex}}
+
+\setlength{\parindent}{0cm}
+\setlength{\parskip}{0.5\baselineskip}
+
+\title{Optimization methods -- Homework 2}
+\author{Claudio Maggioni}
+
+\begin{document}
+
+\maketitle
+
+\section{Exercise 1}
+
+\subsection{Implement the matrix $A$ and the vector $b$, for the moment, without taking into consideration the
+boundary conditions. As you can see, the matrix $A$ is not symmetric. Does an energy function of
+the problem exist? Consider $N = 4$ and show your answer, explaining why it can or cannot exist.}
+
+Answer is a energy function does not exist. Since A is not symmetric
+(even if it is pd), the minimizer used for the c.g. method
+(i.e. $\frac12 x^T A x - b^T x$ won't work
+since $x^T A x$ might be negative and thus the minimizer does not point to
+the solution of $Ax = b$ necessairly
+
+\subsection{Once the new matrix has been derived, write the energy function related to the new problem
+and the corresponding gradient and Hessian.}
+
+we already enforce x(1) = x(n) = 0, since b(1) = b(n) = 0 and thus
+A(1, :) * x = b(0) = 0 and same for n can be solved only for x(1) = x(n)
+= 0size(A, 1)
+
+The objective is therefore $\phi(x) = (1/2)x^T\overline{A}x - b^x$ with a and b
+defined above, gradient is = $\overline{A}x - b$, hessian is $= \overline{A}$
+
+\subsection{Write the Conjugate Gradient algorithm in the pdf and implement it Matlab code in a function
+called \texttt{CGSolve}.}
+
+See page 112 (133 for pdf) for the algorithm implementation
+
+The solution of this task can be found in Section 1.3 of the script \texttt{main.m}.
+
+\subsection{Solve the Poisson problem.}
+
+The solution of this task can be found in Section 1.4 of the script \texttt{main.m}.
+
+\subsection{Plot the value of energy function and the norm of the gradient (here,
+use semilogy) as functions of the iterations.}
+
+The solution of this task can be found in Section 1.5 of the script \texttt{main.m}.
+
+\subsection{Finally, explain why the Conjugate Gradient method is a Krylov subspace method.}
+
+Because theorem 5.3 holds, which itself holds mainly because of this (5.10, page 106 [127]):
+
+\[r_{k+1} = r_k + a_k * A * p_k\]
+
+\section{Exercise 2}
+
+Consider the linear system $Ax = b$, where the matrix $A$ is constructed in three different ways:
+
+\begin{itemize}
+	\item $A =$ diag([1:10])
+	\item $A =$ diag(ones(1,10))
+	\item $A =$ diag([1, 1, 1, 3, 4, 5, 5, 5, 10, 10])
+	\item $A =$ diag([1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0])
+\end{itemize}
+
+\subsection{How many distinct eigenvalues has each matrix?}
+
+Each matrix has a distinct number of eigenvalues equal to the number of distinct
+elements on its diagonal. So, in order, each A has respectively 10, 1, 5, and 10 distinct eigenvalues.
+
+\subsection{Construct a right-hand side $b=$rand(10,1) and apply the
+Conjugate Gradient method to solve the system for each $A$.}
+
+The solution of this task can be found in section 2.2 of the \texttt{main.m} MATLAB script.
+
+\subsection{Compute the logarithm energy norm of the error for each matrix
+and plot it with respect to the number of iteration.}
+
+The solution of this task can be found in section 2.3 of the \texttt{main.m} MATLAB script.
+
+\subsection{Comment on the convergence of the method for the different matrices. What can you say observing
+the number of iterations obtained and the number of clusters of the eigenvalues of the related
+matrix?}
+
+The method converges quickly for each matrix. The fastest convergence surely happens for $A2$, which is
+the identity matrix and therefore makes the $Ax = b$ problem trivial.
+
+For all the other matrices, we observe the energy norm of the error decreasing exponentially as the iterations
+increase, eventually reaching $0$ for the cases where the method converges exactly (namely on matrices $A1$ and $A3$).
+
+Other than for the fourth matrix, the number of iterations is exactly equal
+to the number of distinct eigenvalues for the matrix. That exception on the fourth matrix is simply due to the
+tolerance termination condition holding true for an earlier iteration, i.e. we terminate early since we find an
+approximation of $x$ with residual norm below $10^{-8}$.
+
+\end{document}
diff --git a/Claudio_Maggioni_2/ex1.m b/Claudio_Maggioni_2/ex1.m
deleted file mode 100644
index 6703782..0000000
--- a/Claudio_Maggioni_2/ex1.m
+++ /dev/null
@@ -1,100 +0,0 @@
-%% Homework 2 - Optimization Methods
-% Author: Claudio Maggioni
-%
-% Note: exercises are not in the right order due to matlab constraints of
-% functions inside of scripts.
-%
-% Sources:
-
-clear
-clc
-close all
-
-%% 1.4 - Solution for 1D Poisson for N=1000 using CG
-
-n = 1000;
-[A, b] = build_poisson(n);
-
-A(2, 1) = 0;
-A(n-1, n) = 0;
-
-[x, ys, gnorms] = CGSolve(A, b, zeros(n,1), 1000, 1e-8);
-display(x);
-
-%% 1.5 - Plots for the 1D Poisson solution
-
-plot(ys);
-sgtitle("Objective function values per iteration");
-
-figure;
-semilogy(gnorms);
-sgtitle("Log of gradient norm per iteration");
-
-
-
-
-%% 1.1 - Build the Poisson matrix A and vector b (check this)
-
-% Answer is a energy function does not exist. Since A is not symmetric
-% (even if it is pd), the minimizer used for the c.g. method
-% (i.e. (1/2)x^TAx - b^x) won't work
-% since x^TAx might be negative and thus the minimizer does not point to
-% the solution of Ax=B necessairly
-
-function [A,b] = build_poisson(n)
-    A = diag(2 * ones(1,n));
-    
-    A(1,1) = 1;
-    A(n,n) = 1;
-    
-    for i = 2:n-1
-        A(i, i+1) = -1;
-        A(i, i-1) = -1;
-    end
-    
-    h = 1 / (n - 1);
-    
-    b = h^2 * ones(n, 1);
-    b(1) = 0;
-    b(n) = 0;
-end
-
-%% 1.2
-
-% we already enforce x(1) = x(n) = 0, since b(1) = b(n) = 0 and thus
-% A(1, :) * x = b(0) = 0 and same for n can be solved only for x(1) = x(n)
-% = 0
-%
-% The objective is therefore \phi(x) = (1/2)x^T\overline{A}x - b^x with a and b
-% defined above, gradient is = \overline{A}x - b, hessian is = \overline{A} 
-
-%% 1.3 - Implementation of Conjugate Gradient
-
-function [x, ys, gnorms] = CGSolve(A, b, x, max_itr, tol)
-    ys = zeros(max_itr, 1);
-    gnorms = zeros(max_itr, 1);
-    
-    r = b - A * x;
-    d = r;
-    delta_old = dot(r, r);
-    
-    for i = 1:max_itr
-        ys(i) = 1/2 * dot(A*x, x) - dot(b, x);
-        gnorms(i) = sqrt(delta_old);
-        
-        s = A * d;
-        alpha = delta_old / dot(d, s);
-        x = x + alpha * d;
-        r = r - alpha * s;
-        delta_new = dot(r, r);
-        beta = delta_new / delta_old;
-        d = r + beta * d;
-        delta_old = delta_new;
-        
-        if delta_new / norm(b) < tol
-            ys = [ys(1:i); 1/2 * dot(A*x, x) - dot(b, x)];
-            gnorms = [gnorms(1:i); sqrt(delta_old)];
-            break
-        end
-    end
-end
\ No newline at end of file
diff --git a/Claudio_Maggioni_2/main.m b/Claudio_Maggioni_2/main.m
new file mode 100644
index 0000000..2bff263
--- /dev/null
+++ b/Claudio_Maggioni_2/main.m
@@ -0,0 +1,132 @@
+%% Homework 2 - Optimization Methods
+% Author: Claudio Maggioni
+%
+% Note: exercises are not in the right order due to matlab constraints of
+% functions inside of scripts.
+
+clear
+clc
+close all
+plots = 1;
+
+%% 1.4 - Solution for 1D Poisson for N=1000 using CG
+
+n = 1000;
+[A, b] = build_poisson(n);
+
+A(2, 1) = 0;
+A(n-1, n) = 0;
+
+[x, ys, gnorms] = CGSolve(A, b, zeros(n,1), n, 1e-8);
+display(x);
+
+%% 1.5 - Plots for the 1D Poisson solution
+
+if plots
+    plot(0:(size(ys,1)-1), ys);
+    sgtitle("Objective function values per iteration");
+    axis([-1 500 -inf +inf]);
+
+    figure;
+    semilogy(0:(size(gnorms,1)-1), gnorms);
+    sgtitle("Log of gradient norm per iteration");
+    axis([-1 500 -inf +inf]);
+end
+%% 2.1 - Matrix definitions
+
+A1 = diag([1:10]);
+A2 = diag(ones(1,10));
+A3 = diag([1, 1, 1, 3, 4, 5, 5, 5, 10, 10]);
+A4 = diag([1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0]);
+
+%% 2.2 - Application of CG for each AX/b couple
+
+rng(0); % random seed fixed due to reproducibility purposes
+b = rand(10,1);
+display(b);
+
+n = 10;
+[x1, ys1, gnorms1, xs1] = CGSolve(A1, b, zeros(n,1), n, 1e-8);
+[x2, ys2, gnorms2, xs2] = CGSolve(A2, b, zeros(n,1), n, 1e-8);
+[x3, ys3, gnorms3, xs3] = CGSolve(A3, b, zeros(n,1), n, 1e-8);
+[x4, ys4, gnorms4, xs4] = CGSolve(A4, b, zeros(n,1), n, 1e-8);
+
+%% 2.3 - Logarithm energy norm of the error computation
+
+if plots 
+    enl_plot(x1, xs1, A1);
+    sgtitle("Log energy norm of the error per iter. (matrix A1)");
+    enl_plot(x2, xs2, A2);
+    sgtitle("Log energy norm of the error per iter. (matrix A2)");
+    enl_plot(x3, xs3, A3)
+    sgtitle("Log energy norm of the error per iter. (matrix A3)");
+    enl_plot(x4, xs4, A4);
+    sgtitle("Log energy norm of the error per iter. (matrix A4)");
+end
+
+function enl_plot(xsol, xs, A)
+    enls = zeros(size(xs, 2), 1);
+    for i = 1:size(xs, 2)
+        x = xs(:, i);
+        enls(i) = log((xsol - x)' * A * (xsol - x));
+    end
+    
+    figure;
+    plot(0:(size(xs, 2)-1), enls, '-k.');
+    axis([-1 11 -35 +2]);
+end
+
+%% 1.1 - Build the Poisson matrix A and vector b
+
+function [A,b] = build_poisson(n)
+    A = diag(2 * ones(1,n));
+    
+    A(1,1) = 1;
+    A(n,n) = 1;
+    
+    for i = 2:n-1
+        A(i, i+1) = -1;
+        A(i, i-1) = -1;
+    end
+    
+    h = 1 / (n - 1);
+    
+    b = h^2 * ones(n, 1);
+    b(1) = 0;
+    b(n) = 0;
+end
+
+%% 1.3 - Implementation of Conjugate Gradient
+
+function [x, ys, gnorms, xs] = CGSolve(A, b, x, max_itr, tol)
+    ys = zeros(max_itr + 1, 1);
+    gnorms = zeros(max_itr + 1, 1);
+    xs = zeros(size(x, 1), max_itr + 1);
+    
+    r = A * x - b;
+    p = -r;
+    
+    gnorms(1) = norm(p, 2);
+    xs(:, 1) = x;
+    ys(1) = 1/2 * dot(A*x, x) - dot(b, x);
+    
+    for i = 1:(max_itr+1)
+        
+        alpha = -r' * p / (p' * A * p);
+        x = x + alpha * p;
+        r = A * x - b;
+        beta = (r' * A * p) / (p' * A * p);  
+        p = -r + beta * p;
+        
+        gnorms(i+1) = norm(p, 2);
+        xs(:, i+1) = x;
+        ys(i+1) = 1/2 * dot(A*x, x) - dot(b, x);
+        
+        if gnorms(i+1) < tol
+            ys = ys(1:(i+1));
+            gnorms = gnorms(1:(i+1));
+            xs = xs(:, 1:(i+1));
+            break
+        end
+    end
+end