2. 2
Newton - Rapson Method
Supposewe know the function f and its derivative f’
at any point xc.
The tangent line drawn at xc is defined by:
can be thoughtof as a linear model
'
c c c c
L x f x x x f x
Newton’s Method cont.
Thezero of the linear model Lc is given by:
If Lc is a good approximationto f over a wide interval
then x+ should be a good approximation to a root of f
'
c
c
c
f x
x x
f x
Newton’s Method cont.
Repeat the formulato create an algorithm:
If at each step the linear model is a good
approximationto f then xn should get closer to a root
of f as n increases.
1
'
n
n n
n
f x
x x
f x
3. 3
(x1,f(x1))
(x2=x1-f(x1)/f’(x1),0)
First stage:
(x2,f(x2))
(x3=x2-f(x2)/f’(x2),0)
Notice: we are getting closer
Zoom in for second stage
Convergence of Newton’s Method
Wecan show that the rate of convergenceis much faster
than the bisection method.
However– as always, there is a catch. The method uses
a local linear approximation,which clearly breaks down
neara turning point.
Small f’(xn) makes the linear model very flat and will
send the search far away …
