From ddd68bf84692f58ec02182884fdc864036b62351 Mon Sep 17 00:00:00 2001 From: Gary Yan Date: Sun, 27 Sep 2015 23:14:16 -0700 Subject: [PATCH] added more description for error bounds for bisection method and Newton's method. --- 02_newton.md | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/02_newton.md b/02_newton.md index 4b01352..450bdac 100644 --- a/02_newton.md +++ b/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**.