From 455a331912818a39cb5924f83b1ec9721e275a42 Mon Sep 17 00:00:00 2001 From: munchichii Date: Sun, 27 Sep 2015 06:26:30 -0700 Subject: [PATCH] Update 02_newton.md --- 02_newton.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/02_newton.md b/02_newton.md index 4b01352..caa4428 100644 --- a/02_newton.md +++ b/02_newton.md @@ -10,8 +10,10 @@ Rooting Finding 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. +*Bisection*: start with an interval $[a, b]$ bracketing the root such that $f(a)f(b)<0$ +From the Intermediate Value Theorem, there must be at least one root in $[a, b]$. +Let $x_1=\frac{a+b}{2}$. Replace one of the endpoints (a or b) with $x_1$,maintaing a root bracket. +Repeat this until it satisfies some error tolerance. Linear convergence. Slow but **robust**. *Newton's Method*: $x_{k+1} = x_k - f(x_k) / f'(x_k)$. Faster,