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HW1 Q8: bisection update #2

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Update 02_newton.md
munchichii Sep 27, 2015
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Merge pull request #1 from munchichii/munchichii-patch-1
munchichii Sep 27, 2015
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6 changes: 4 additions & 2 deletions 02_newton.md
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Expand Up @@ -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,
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