1. Some smallest eigenvalues of a large matrix
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Go to the SuiteSparse Matrix Collection and get out of the 
DIMACS10 matrix set the "G_n_pin_pout" graph

https://sparse.tamu.edu/DIMACS10/G_n_pin_pout

https://sparse.tamu.edu/mat/DIMACS10/G_n_pin_pout.mat

In this case it is a 100000 x 100000 sparse matrix with
1002396 non zero elements. 
The rows and columns stands for the nodes and A_ij is set to one
if there is and edge from node i to node j.

Load this in matlab, you will find a struct "Problem"
holding some informations and in Problem.A 
the adjacency matrix of the graphas a sparse matrix.
Use it as 

  A=Problem.A

The degree matrix D of a graph is defined as 
d_i := row sum of row i in A
D=diag(d_1 .. d_n)

The Laplacian matrix L of a graph is defined as L=D-A

Create the matrix L. Have repect for the sparsity of the matrix A. 

Now compute the 10 smallest eigenvalues of L and 
determine the number of strongly connected components of the
graph by counting the zero eigenvalues.
Hint: it should be 6 ;-) 


2. QR algorithm without/with shift
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Create random full matrices with known eigenvalues,
for example lambda_i=i , i=1...n

Now perform the QR algorithm with and without shift,
compute in every iteration the maximum error of the iterated 
eigenvalues and plot the convergence about the iterations.