mp2: done 1-5

mp2/Project.2.Maggioni.Claudio/ex4.m

@@ -0,0 +1,8 @@
+clear;
+clc;
+load('householder/housegraph.mat')
+names = split(strtrim(convertCharsToStrings(name')));
+
+common1  = names((A(:,Golub) .* A(:, Moler)) == 1)
+common2 = names((A(:,Golub) .* A(:, Saunders)) == 1)
+common3 = names((A(:,TChan) .* A(:, Demmel)) == 1)

mp2/project.2.Maggioni.Claudio.tex

@@ -91,11 +91,11 @@ blanks):
 
 \[
 A :=  \begin{bmatrix}
-	1 & 1 & 1 & \hdots & 1 \\
+	n & 1 & 1 & \hdots & 1 \\
-	1 & 1 &           && 1 \\
+	1 & n + 1 &           && 1 \\
-	1 &   & 1         && 1 \\
+	1 &   & n + 2         && 1 \\
 	\vdots & & & \ddots & \vdots \\
-	1 & 1 & 1 & \hdots & 1 \\
+	1 & 1 & 1 & \hdots & 2n - 1 \\
     \end{bmatrix}
 \]
 
@@ -103,7 +103,14 @@ A :=  \begin{bmatrix}
 to solve $Ax = b$
 for a given righthand-side vector would be problematic.}
 
-\textbf{A IS NOT POSITIVE DEFINITE: CHECK WITH EDOARDO}
+Solving $Ax = b$ would be a costly operation since the a Cholesky
+decomposition of matrix $A$ (performed using MATLAB's \texttt{chol(\ldots)})
+would drastically reduce the number of zero elements in the matrix in the very
+first iteration. This is due to the fact that the first row, by definition, is
+made of of only nonzero elements (namely 1s) and by subtracting the first row to
+every other row (as what would effectively happen in the first iteration of the
+Cholesky decomposition of A) the zero elements would become (negative) nonzero
+elements, thus making all columns but the first almost empty of 0s.
 
 \section{Degree Centrality [10 points]}
 
@@ -233,6 +240,21 @@ Varah 1: Golub
 
 \section{The Connectivity of the Coauthors [10 points]}
 
+The author indexes of the common authors between the author at index $i$ and the
+author at index $j$ can be computed by listing the indexes of the nonzero
+elements in the Schur product (or element-wise product) between $A_{:,i}$ and
+$A_{:,j}$ (respectively the i-th and j-th column vector of $A$). Therefore the set $C$ of common coauthor's indexes can be defined
+as:
+
+\[C = \{i \in N_0 \;|\; (A_{:,i} \odot A_{:,j})_i = 1\}\]
+
+The common Co-authors between Golub and Moler are Wilkinson and Van Loan.
+
+The common Co-authors between Golub and Saunders are Golub, Saunders and Gill.
+
+The common Co-authors between TChan and Demmel are Schreiber, Arioli, Duff and
+Heath.
+
 \section{PageRank of the Coauthor Graph [10 points]}
 
 \section{Zachary's karate club: social network of friendships between 34 members [50 points]}