%%
clear
clc
rng(2)

%%
% The sprandsym command generates a random sparse matrix with
% prescribed eigenvalues.
n = 3000;
density = 1.23e-3;
lambda = 1./(1:n);
A = sprandsym(n, density, lambda);
spy(A)
title('Sparse symmetric matrix')

%%
eigs(A, 5)    % largest magnitude

%%
eigs(A, 5, 0)  % closest to zero

%%
% The scaling of time to solve a sparse linear system is not easy to
% predict unless you have some more information about the matrix (such as
% bandedness). But it will typically be a great deal faster than the dense
% or full matrix case.
x = 1./(1:n)';  
b = A*x;
tic()
sparse_err = norm(x - A\b)
sparse_time = toc()

%%
A = full(A);
tic()
dense_err = norm(x - A\b)
dense_time = toc()