Bisection method, Newton method, Falsi method and Secant method with examples.

1. By
Ahmed Haider Ahmed
Pre-M.Sc. Physics – ASU

2. To My father ,
May Allah
enter you his
paradise

3. 1 – Bisection Method
 If F(wi+1) is negative we put wi+1 instead of ai or bi
2
1
ii
i
ba
w

4. 2 – Falsie Method
)()(
)()(
1
ii
iiii
i
bfaf
afbbfa
x

5. 3 – Newton Method
)(
)(
1
i
i
ii
xf
xf
xx

6. 4 – Secant Method
)()(
))((
1
1
1
ii
iii
ii
xfxf
xxxf
xx

7. Examples

8. Find the positive root of x – cos x = 0 using
bisection method.
 We will take points that give positive value
with the negative one i.e 0 and 1
)(1)2(
)(00015.0)1(
)(1)0(
cos)(
vef
vef
vef
xxxf

9. )(249.0)75.0(
75.0
2
15.0
150
)(5.0)5.0(
)(5.0
2
10
10
2
1
vef
x
and.enroot betweHence the
vef
vex
andenroot betweHence the

10. 062.0)9375.0(
9375.0
2
1875.0
18750
)(12488.0)875.0(
875.0
2
75.01
1750
4
3
f
x
and.enroot betweHence the
vef
x
and.enroot betweHence the

11. Solve equation x3 – x – 1 = 0 using
falsie method.
5128)2(
1111)1(
1)( 3
f
f
xxxf
2,1
)()(
)()(
00
00
0000
1
ba
bfaf
afbbfa
x

12. )(279351.0)2547.1(
2547.1
5684.05
5684.02517.1
217.1from
)(5684.0)17.1(
17.1
6
25
51
1251
2
2
1
vef
x
tox
vef
x

13. )(0578.0)311.1(
311.1
127.05
127.025294.1
2294.1from
)(127.0)294.1(
294.1
279351.05
279351.0252547.1
22547.1from
4
4
3
3
vef
x
tox
vef
x
tox

14. 3238.1
0109.05
0109.0253223.1
23223.1from
0109.00)3223.1(
3223.1
024255.05
024255.025319.1
2319.1from
)(024255.0)319.1(
319.1
0578.05
0578.025311.1
2311.1from
7
7
6
6
5
5
x
tox
f
x
tox
vef
x
tox

15. Solve equation x3 – x – 1 = 0 using
Newton method at x0 = 1
213)(
1111)(
13)(
1)(
)(
)(
0
0
2
3
1
xf
xf
xxf
xxxf
xf
xf
xx
i
i
ii

16. 4497.4)3478.1(
1006.0)3478.1(
3478.1
75.5
875.0
5.1
)(
)(
75.5)5.1(
875.0)5.1(
5.1
2
1
1
)(
)(
1
1
12
0
0
01
f
f
xf
xf
xx
f
f
xf
xf
xx

17. 000077.0)3247.1(
3247.1
269.4
00206.0
3252.1
)(
)(
269.4)3252.1(
00206.0)3252.1(
3252.1
4497.4
1006.0
3478.1
)(
)(
3
3
34
2
2
23
f
xf
xf
xx
f
f
xf
xf
xx

18. REFERANCES
 Lecture notes on numerical methods ,
Dr. Shemi 2011, Minia university written
by Aya Hassan.
 Lecture notes on computational physics
for Pre-M.Sc. Students , Prof. Dr.
S.Hendawi 2013, Ain Shams University.