I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate. 

Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i,j}$, $2a_{i,j}$}. However, which values are changed is not known. In fact the distribution of changed elements are uniform over elements of the matrix.We do have bound over number of elements changes $\beta\ll n^{2}$.

Let's call the perturbation in the matrix $\Delta A$. So the matrix vector product calculated is $(A+\Delta A)p$. What I am intersted  is finding bounds for following quantity
   
   $\eta =\left\Vert \frac{(\Delta A)p}{p^{T}(A+\Delta A)p}\right\Vert $ 

I would like this bound based on all known quantities like norms of $A$ and $p$ and conditions numbers, $\beta$ etc. It can be assumed that most of the parameters about $A$ and $p$ are known. 

When it is just the numerator I am able to come up with something; however when included denominator, I am left with no clues. Any help appreciated