Numerical methods
18
NUMERICAL METHODS By Ahmed Haider Ahmed Pre-M.Sc. Physics – ASU
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Bisection method, Newton method, Falsi method and Secant method with examples.
Transcript of Numerical methods
NUMERICAL METHODS
ByAhmed Haider Ahmed Pre-M.Sc. Physics – ASU
To My father , May Allah
enter you his paradise
1 – Bisection Method
If F(wi+1) is negative we put wi+1 instead of ai or bi
21ii
i
baw
2 – Falsie Method
)()(
)()(1
ii
iiiii bfaf
afbbfax
3 – Newton Method
)(
)(1
i
iii xf
xfxx
4 – Secant Method
)()(
))((
1
11
ii
iiiii xfxf
xxxfxx
Examples
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
)(249.0)75.0(
75.02
15.0
150
)( 5.0)5.0(
)( 5.02
10
10
2
1
vef
x
and .en root betweHence the
vef
vex
and en root betweHence the
062.0)9375.0(
9375.02
1875.0
18750
)( 12488.0)875.0(
875.02
75.01
1750
4
3
f
x
and .en root betweHence the
vef
x
and .en root betweHence the
Solve equation x3 – x – 1 = 0 using falsie method.
5128)2(
1 111)1(
1)( 3
f
f
xxxf
2 , 1
)()(
)()(
00
00
00001
ba
bfaf
afbbfax
)(279351.0)2547.1(
2547.15684.05
5684.02517.1
217.1 from
)(5684.0)17.1(
17.16
25
51
1251
2
2
1
vef
x
tox
vef
x
)(0578.0)311.1(
311.1127.05
127.025294.1
2 294.1 from
)(127.0)294.1(
294.1279351.05
279351.0252547.1
2 2547.1 from
4
4
3
3
vef
x
tox
vef
x
tox
3238.10109.05
0109.0253223.1
2 3223.1 from
0109.00)3223.1(
3223.1024255.05
024255.025319.1
2 319.1 from
)(024255.0)319.1(
319.10578.05
0578.025311.1
2 311.1 from
7
7
6
6
5
5
x
tox
f
x
tox
vef
x
tox
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
xfxx
i
iii
4497.4)3478.1(
1006.0)3478.1(
3478.175.5
875.05.1
)(
)(
75.5)5.1(
875.0)5.1(
5.12
11
)(
)(
1
112
0
001
f
f
xf
xfxx
f
f
xf
xfxx
000077.0)3247.1(
3247.1269.4
00206.03252.1
)(
)(
269.4)3252.1(
00206.0)3252.1(
3252.14497.4
1006.03478.1
)(
)(
3
334
2
223
f
xf
xfxx
f
f
xf
xfxx
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.
