added more description for error bounds for bisection method

02_newton.md

@@ -12,12 +12,17 @@ Iterative techniques for solving $f(x) = 0$ for $x$.
 
 *Bisection*: start with an interval $[a, b]$ bracketing the root.
 Evaluate the midpoint.  Replace one end, maintaining a root bracket.
-Linear convergence.  Slow but **robust**.
+Linear convergence. Error is roughly halved at every iteration as the interval is halved: 
+
+$$ \frac{|e_{k+1}|}{|e_k|} \le \frac{1}{2} $$
+ Slow but **robust**.
 
 *Newton's Method*: $x_{k+1} = x_k - f(x_k) / f'(x_k)$.  Faster,
 quadratic convergence (number of correct decimals places doubles each
 iteration).
 
+$$ \frac{|e_{k+1}|}{|e_k|^2} \le c $$
+
 Downsides of Newton's Method: need derivative info, and additional
 smoothness.  Convergence usually not guaranteed unless "sufficiently
 close": not **robust**.