mp5: 3.3 correction

mp5/Project_5_Maggioni_Claudio.tex

@@ -20,7 +20,7 @@
 \setassignment
 \setduedate{Wednesday, December 02, 2020, 11:59 PM}
 
-\serieheader{Numerical Computing}{2020}{Student: Claudio Maggioni}{Discussed with: --}{Solution for Project 5}{}
+\serieheader{Numerical Computing}{2020}{Student: Claudio Maggioni}{Discussed with: Gianmarco De Vita (3.3)}{Solution for Project 5}{}
 \newline
 
 \assignmentpolicy
@@ -98,11 +98,7 @@ The plot of the squared residual 2-norms over all iterations can be found in Fig
 condition number and convergence rate.}
 
 The eigenvalues of A can be found in figure
-\ref{fig:plot2}. The condition number for matrix $A$ according to \texttt{rcond(...)} is $\approx 3.2720 \cdot 10^7$,
+\ref{fig:plot2}.
-which is very low without sitting in the denormalized range (i.e. $< \text{eps}$) and thus very good for the Conjugate Gradient algorithm.
-This well conditioning is also
-reflected in the eigenvalue plot, which shows a not so
-drastic increase of the first eigenvalues ordered in increasing order.
 
 \begin{figure}[h]
 	\centering
@@ -110,6 +106,24 @@ drastic increase of the first eigenvalues ordered in increasing order.
 	\caption{Semilog plot of the eigenvalues of A}\label{fig:plot2}
 \end{figure}
 
+The condition number for matrix $A$ according to \texttt{cond(...)} is $\approx 1.3700 \cdot 10^6$,
+which is rather ill conditioned. The eigenvalue plot agrees with the condition number, by showing that there are significant differences in the magnitude of the eigenvalues of \texttt{A\_test}.
+
+The two methods agree due to the fact that the condition number is computed by using singular values,
+which in turn are derived by eigenvalues. This fact was demonstrated practically by Dr. Edoardo Vecchi, \textit{future PhD}
+using this MATLAB snippet.
+
+\begin{verbatim}
+X = A_test'*A_test;
+singular_values = sqrt(eig(X));
+norm(sort(singular_values,'descend') - svd(A_test))/numel(A_test)
+\end{verbatim}
+
+The final norm is less than $10^{12}$, showning that the definition of
+\texttt{singular\_values}, which is a generalization of the computation of
+eigenvalues for rectangular matrices, is equivalent to MATLAB's \texttt{svd(...)} other than approximation errors.
+
+
 \section{Debluring problem [40 points]}
 \subsection{ Solve the debluring problem for the blurred image matrix
 \texttt{B.mat} and transformation matrix \texttt{A.mat} using

mp5/Project_5_Maggioni_Claudio/code_template.m

@@ -0,0 +1,78 @@
+close all;
+clear; clc;
+addpath /Users/maggicl/Git/matlab2tikz/src/;
+
+%% Load Default Img Data
+load('blur_data/B.mat');
+B=double(B);
+load('blur_data/A.mat');
+A=double(A);
+
+% Show Image
+figure
+im_l=min(min(B));
+im_u=max(max(B));
+imshow(B,[im_l,im_u])
+title('Blured Image')
+
+% Vectorize the image (row by row)
+b=B';
+b=b(:);
+
+AT = A' * A;
+bt = A' * b;
+
+IL = ichol(AT, struct('type', 'nofill', 'diagcomp', 0.01));
+
+[x, rvec] = myCG(AT, bt, zeros(size(b)), 200, 1e-6);
+[x2, ~, ~, ~, rvec2] = pcg(AT, bt, 1e-6, 200, IL, IL');
+
+figure;
+semilogy(rvec / norm(bt));
+hold on;
+semilogy(rvec2 / norm(bt));
+hold off;
+title('Residual norms over iteration (y is log)')
+matlab2tikz('showInfo', false, '../res_log.tex');
+
+
+X = zeros(250, 250);
+for i = 0:249
+    for j = 1:250
+        X(i + 1, j) = x(i * 250 + j);
+    end
+end
+
+figure;
+im_l=min(min(X));
+im_u=max(max(X));
+imshow(X,[im_l,im_u])
+title('Sharp Image (myCG)')
+matlab2tikz('showInfo', false, '../img_my.tex');
+
+
+X2 = zeros(250, 250);
+for i = 0:249
+    for j = 1:250
+        X2(i + 1, j) = x2(i * 250 + j);
+    end
+end
+
+figure;
+im_l=min(min(X2));
+im_u=max(max(X2));
+imshow(X2,[im_l,im_u])
+title('Sharp Image (rcg)')
+matlab2tikz('showInfo', false, '../img_rcg.tex');
+
+
+
+%% Validate Test values
+load('test_data/A_test.mat');
+load('test_data/x_test_exact.mat');
+load('test_data/b_test.mat');
+
+%res=||x^*-A^{-1}b||
+res=x_test_exact-inv(A_test)*b_test;
+norm(res);
+%(Now do it with your CG and Matlab's PCG routine!!!)

mp5/img_my.tex

@@ -7,8 +7,8 @@ width=1.736in,
 height=1.736in,
 at={(0.75in,0.653in)},
 scale only axis,
-point meta min=-0.0235938830592792,
+point meta min=-0.0268505831942635,
-point meta max=0.99393991750956,
+point meta max=1.01130200411968,
 axis on top,
 xmin=0.5,
 xmax=250.5,

mp5/img_rcg.tex

@@ -7,8 +7,8 @@ width=1.736in,
 height=1.736in,
 at={(0.75in,0.653in)},
 scale only axis,
-point meta min=0.0282583971491394,
+point meta min=-0.0235938830592792,
-point meta max=0.921555841989536,
+point meta max=0.99393991750956,
 axis on top,
 xmin=0.5,
 xmax=250.5,

mp5/submit.sh

@@ -0,0 +1,8 @@
+#!/bin/sh
+
+PID="5"
+dname="Project_${PID}_Maggioni_Claudio"
+zname="Project_${PID}_Maggioni_Claudio"
+
+rm -v $zname.zip
+zip $zname.zip *.tex *.sty usi_inf.pdf $zname.{pdf,tex} $dname/blur_data/*.mat $dname/test_data/*.mat $dname/*.m

mp5/test_semilogy.tex

@@ -12,7 +12,7 @@ scale only axis,
 xmin=0,
 xmax=180,
 ymode=log,
-ymin=1e-12,
+ymin=1e-08,
 ymax=1000000,
 yminorticks=true,
 axis background/.style={fill=white},
@@ -183,23 +183,6 @@ title={Residual vector squared 2-norm (log) over iterations}
 160	1.65115695649721e-06\\
 161	2.09460062061473e-06\\
 162	5.05915189772462e-07\\
-163	1.32841240174248e-07\\
-164	1.28360359603615e-06\\
-165	1.43607180361307e-05\\
-166	5.04094734598852e-06\\
-167	7.41197749055341e-06\\
-168	5.04948498688418e-07\\
-169	1.54446803267932e-08\\
-170	1.81444161747323e-07\\
-171	8.78831793802036e-08\\
-172	1.53613965535421e-06\\
-173	2.81415884074516e-06\\
-174	4.43530815074645e-07\\
-175	2.7224336876613e-08\\
-176	1.43297134609307e-06\\
-177	1.62527199539552e-09\\
-178	2.50972491576067e-10\\
-179	1.74075823612926e-11\\
 };
 \end{axis}
 \end{tikzpicture}%