% construct an ill-conditioned linear system
A = eye(10)
for k = 2:10
   A(k-1,k) = 10;
end;
b = A*ones(10,1);
% Try to solve the equation A*x=b, got correct solution
format long;
x = A\b
% Try to create a nearby problem
d = b + rand(10,1)*0.000001
% solution of a nearby problem
y = A\d
% Another nearby problem A*z = g
g = b/3;
z = A\g
% Backward error
disp("backward error")
norm(A*z-g)
disp("forward error")
norm(ones(10,1)/3-z)
disp("condition number")
cond(A)
%
%  Demo for solving polynomial equations with multiple zeros
%    create a polynomial of given roots
%
p = poly([1 1 1 1 1 2 2])
%
%  try to find roots
%
z = roots(p)