5.Newton's Method
(a) Draw a picture and deduce from it the Newton's iteration
process from Xn to Xn+1(n+1是下標) to approximate roots
of f(x)=0
3 2
(b) Show that for f(x)= 2x -3x -1,thereis exactly one root
c屬於(1,2) for f(x)=0.Let X1=2,compute x2,x3 and f(x3)
(c) Let f be a continuous function on [a,b] such that f''(x)
exists for all x屬於(a,b) and moreover |f''(x)| < M,
|f'(x)| >= m > 0 on (a,b).Let c屬於(a,b) be a root of
f ( f(c) = 0 )
Show that
M 2
| Xn+1 - c | < --- | Xn - c |
m
in Newton's iteration process.
(Hint:Apply the mean value theorem twice.)
--
挑你是讓你成長
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.114.216.88