✔ 最佳答案
draw the graph
http://in.geocities.com/myisland8132/Book1.pub
we see that
x=0 is a solution by inspection
there is another root about [-2,-1] and [1,2] use netwon's method to find the roots
let f(x)=sinx-0.5x
df(x)/dx=cosx-0.5
x(n+1)=x(n)-f(x(n))/f'(x(n))
x(n+1)=x(n)-[sinx(n)-0.5x(n)]/[cosx(n)-0.5]
let x(0)=-2
interation
x(n)
x(n+1)
0
-2
-1.901
1
-1.901
-1.89551
2
-1.89551
-1.89549
3
-1.89549
-1.89549
4
-1.89549
-1.89549
5
-1.89549
-1.89549
6
-1.89549
-1.89549
7
-1.89549
-1.89549
8
-1.89549
-1.89549
9
-1.89549
-1.89549
10
-1.89549
-1.89549
let x(0)=1
then
interation
x(n)
x(n+1)
0
1
-7.47274
1
-7.47274
14.47852
2
14.47852
6.935115
3
6.935115
16.63568
4
16.63568
8.343938
5
8.343938
4.954633
6
4.954633
-8.30132
7
-8.30132
-4.81732
8
-4.81732
3.792574
9
3.792574
1.861061
10
1.861061
1.896214
11
1.896214
1.895495
12
1.895495
1.895494
13
1.895494
1.895494
14
1.895494
1.895494
15
1.895494
1.895494
16
1.895494
1.895494
17
1.895494
1.895494
18
1.895494
1.895494
19
1.895494
1.895494
20
1.895494
1.895494
21
1.895494
1.895494
22
1.895494
1.895494
23
1.895494
1.895494
24
1.895494
1.895494
25
1.895494
1.895494
so the roots of sinx=0.5x are -1.895494 0 and 1.895494
