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May 16, 2013 at 4:59 comment added arbitUser1401 we do know A has $\beta << nnz(A) < n^{2}$ non zero elements
May 16, 2013 at 3:08 comment added Vectornaut If $A$ has $\beta$ or fewer nonzero entries, it can happen that $\Delta A = A$, in which case $\|\Delta Ap\|/\|Ap\| = 1$. Thus, without knowing more about how $A$, $\Delta A$ and $p$ are lined up, you'll never find any bound whose right-hand side is less than one.
May 15, 2013 at 22:09 history edited arbitUser1401 CC BY-SA 3.0
corrected equations that were not printed correctly with markdown
May 15, 2013 at 21:17 history edited arbitUser1401 CC BY-SA 3.0
some additions in the original question
May 15, 2013 at 6:15 history asked arbitUser1401 CC BY-SA 3.0