We first construct a polynomial of degree six with known integer roots.
r = [-2 -1 1 1 3 6]'; % column vector of exact roots
p = poly(r) % polynomial having those roots
Now we use MATLAB's built in function roots for finding them.
r_computed = sort( roots(p) ) % algorithmically computed roots
These are the relative errors in each computed root.
format short e
abs(r - r_computed) ./ r
It seems that the forward error is acceptably small in all cases except the double root at 
. This is not a surprise, though, given the poor conditioning at such roots.
. This is not a surprise, though, given the poor conditioning at such roots. Let's consider the backward error. The data in the rootfinding problem are the polynomial coefficients. We can apply poly to find the coefficients of the polynomial (that is, the data) whose roots were actually computed.
p_computed = poly(r_computed)'
We find that in a relative sense, these coefficients are very close to those of the original, exact polynomial:
(p_computed - p') ./ p'
In summary, even though there are some computed roots relatively far from their correct values, they are nevertheless the roots of a polynomial that is very close to the original.