數值方法 2008, Applied Mathematics NDHU 1 Numerical Integration.
數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.
-
Upload
ethelbert-poole -
Category
Documents
-
view
217 -
download
1
Transcript of 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.
![Page 1: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/1.jpg)
數值方法2008, Applied Mathematics NDHU 1
Simpson ruleComposite Simpson rule
![Page 2: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/2.jpg)
數值方法2008, Applied Mathematics NDHU 2
Simpson rule for numerical integration
b
aab
fbf
bafaf
abdxxf
baCf
5)4(
4
)(2880
)()()
2(4)(
6)(
withexists b)(a,in number a then ],,[ If
![Page 3: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/3.jpg)
數值方法2008, Applied Mathematics NDHU 3
a=1.2;b=2;x=linspace(a,b);plot(x,sin(x));hold on
y=sin(x) for x within [1.2 2]
![Page 4: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/4.jpg)
數值方法2008, Applied Mathematics NDHU 4
Quadratic polynomial
c=0.5*(a+b);x=[a b c]; y=sin(x);p=polyfit(x,y,2);z=linspace(a,b);plot(z,polyval(p,z) ,'r')
• Approximate sin using a quadratic polynomial
![Page 5: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/5.jpg)
數值方法2008, Applied Mathematics NDHU 5
a=1;b=3;x=linspace(a,b);plot(x,sin(x))hold on;c=0.5*(a+b);x=[a b c]; y=sin(x);p=polyfit(x,y,2);z=linspace(a,b);plot(z,polyval(p,z) ,'r')
Approximate sin(x) within [1 3] by a quadratic polynomial
![Page 6: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/6.jpg)
數值方法2008, Applied Mathematics NDHU 6
a=1;b=3;x=linspace(a,b);plot(x,sin(x))hold on;c=0.5*(a+b);x=[a b ]; y=sin(x);p=polyfit(x,y,1);z=linspace(a,b);plot(z,polyval(p,z) ,'r')
Approximate sin(x) within [1 3] by a line
1 1.5 2 2.5 30.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
![Page 7: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/7.jpg)
數值方法2008, Applied Mathematics NDHU 7
1 1.5 2 2.5 30.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Strategy I : Apply Trapezoid rule to calculate area under a lineStrategy II : Apply Simpson rule to calculate area under a second order polynomial
Strategy II is more accurate and general than strategy I, since a line is a special case of quadratic polynomial
![Page 8: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/8.jpg)
數值方法2008, Applied Mathematics NDHU 8
Simpson rule
Original task: integration of f(x) within [a,b]
c=(a+b)/2 Numerical task
Approximate f(x) within [a,b] by a quadratic polynomial, p(x)
Integration of p(x) within [a,b]
![Page 9: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/9.jpg)
數值方法2008, Applied Mathematics NDHU 9
Blue: f(x)=sin(x) within [1,3] Red: a quadratic polynomial that pass (1,sin(1)), (2,sin(2)),(3,sin(3))
![Page 10: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/10.jpg)
數值方法2008, Applied Mathematics NDHU 10
))2
()((23
a-b
Area(I)
abfaf
3
ab
))2
()2
((23
a-b
Area(II)
abf
abf
))()2
((23
a-b
Area(III)
bfab
f
2
ab
• The area under a quadratic polynomial is a sum of area I, II and III
![Page 11: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/11.jpg)
數值方法2008, Applied Mathematics NDHU 11
• The area under a quadratic polynomial is a sum of area I, II and III
• Partition [a,b] to three equal-size intervals, and use the high of the middle point c to produce three Trapezoids
• Use the composite Trapezoid rule to determine area I, II and III h/2*(f(a)+f(c) +f(c)+f(c)+ f(c)+f(b)) • substitute h=(b-a)/3 and c=(a+b)/2• area I + II + III = (b-a)/6 *(f(a)+4*f((a+b)/2) +f(b))
![Page 12: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/12.jpg)
數值方法2008, Applied Mathematics NDHU 12
))2
()((23
a-b
Area(I)
abfaf
3
ab
))2
()2
((23
a-b
Area(II)
abf
abf
))()2
((23
a-b
Area(III)
bfab
f
))()2
(4)((23
a-b
Area(III)Area(II)Area(I)
bfab
faf
2
ab
![Page 13: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/13.jpg)
數值方法2008, Applied Mathematics NDHU 13
passes )(
)( 2
xp
Pxp ))(,( bfb ))
2(,
2(
baf
ba
b
adxxp )(
that Show
)()
2(4)(
6bf
bafaf
ab
))(,( afa
![Page 14: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/14.jpg)
數值方法2008, Applied Mathematics NDHU 14
)()()( :
))((
))(()(
))((
))(()(
))((
))(()( )(
cf,p(c)bf,p(b)afp(a)proof
bcac
bxaxcf
cbab
cxaxbf
caba
cxbxafxp
))(,( afa
))(,( bfb
2)),(,(
baccfc
![Page 15: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/15.jpg)
數值方法2008, Applied Mathematics NDHU 15
3
1
)02)(12(
)1)(0(
))((
))((
3
4
)01)(21(
)2)(0(
))((
))((
3
1
)10)(20(
)1)(2(
))((
))(2(
))((
))((
2
))((
))(()(
))((
))(()(
))((
))(()(
))((
))(()(
))((
))(()(
))((
))(()( )(
2
0
2
0
2
0
2
hdttt
hdxcbab
cxax
hdttt
hdxbcac
bxax
hdttt
hdxcaba
haxhaxdx
caba
cxbx
hac
hab
dxbcac
bxaxcfdx
cbab
cxaxbfdx
caba
cxbxaf
dxbcac
bxaxcf
cbab
cxaxbf
caba
cxbxafdxxp
b
a
b
a
ha
a
b
a
b
a
b
a
b
a
b
a
b
a
))()(4)((
3bfcfaf
h
))()(4)((6
)(
))()(4)((3
bfcfafab
bfcfafh
![Page 16: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/16.jpg)
數值方法2008, Applied Mathematics NDHU 16
dttt
hdthh
htht
dxcaba
haxhaxdx
caba
cxbx
h
ha
a
b
a
2
0
2
0
2
)10)(20(
)0)(2(
)0)(20(
))(2(
))((
))(2(
))((
))((
-1 -0.5 0 0.5 1 1.5 2 2.5 3-0.5
0
0.5
1
1.5
2
2.5
3
Shift Lagrange polynomialRescale Lagrange polynomial
![Page 17: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/17.jpg)
數值方法2008, Applied Mathematics NDHU 17
Shortcut to Simpson 1/3 rule
))()2
(4)((6
)()()()(
)(
2
2
bfba
fafab
IIIQIIQIQdxxp
Pp
CBxAxxp
b
a
1. Partition [a,b] to three equal intervals2. Draw three trapezoids, Q(I), Q(II) and Q(III)3. Calculate the sum of areas of the three trapezoids
![Page 18: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/18.jpg)
數值方法2008, Applied Mathematics NDHU 18
Proof
))()2
(4)((6
23
)()(
))()((4))()((2))()((2)(
1)(
23
2
2
2
2
bfba
fafab
CxxB
xA
dxCBxAxdxxp
Pp
CBxAx
bxaxcfcxaxbfcxbxafba
xp
b
a
b
a
b
a
![Page 19: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/19.jpg)
數值方法2008, Applied Mathematics NDHU
19
Composite Simpson rule
Partition [a,b] into n interval Integrate each interval by Simpson
rule h=(b-a)/2n
1
0
)22(
2)(
)(
n
i
hia
iha
b
a
dxxf
dxxf
![Page 20: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/20.jpg)
數值方法2008, Applied Mathematics NDHU 20
Apply Simpson sule to each interval
)))22(())12((4)2((3
)()22(
2
hiafhiafihafh
dxxfhia
iha
))()2
(4)((6
)(
bfba
fafab
dxxpb
a
![Page 21: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/21.jpg)
數值方法2008, Applied Mathematics NDHU 21
Composite Simpson rule
1
0
1
0
)22(
2
)))22(())12((4)2((3
)()(
n
i
n
i
hia
iha
b
a
hiafhiafihafh
dxxfdxxf
![Page 22: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/22.jpg)
數值方法2008, Applied Mathematics NDHU 22
1
0
1
0
)22(
2
)))22(())12((4)2((3
)()(
n
i
n
i
hia
iha
b
a
hiafhiafihafh
dxxfdxxf
Set nSet fx
h=(b-a)/(2*n)ans=0;
for i=0:n-1
aa=a+2*i*h;cc=aa+h;bb=cc+h;
ans=ans+h/3*(fx(aa)+4*fx(cc)+fx(bb))
exit
![Page 23: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/23.jpg)
數值方法2008, Applied Mathematics NDHU 23
Composite Simpson rule
1
112
1
12
23210
1
0
)(3
4)(
3
2))()((
3
)(...)(4)(2)(4)(3
)))22(())12((4)2((3
)(
2,...,1,0,
n
kk
n
kk
n
n
i
b
a
k
xfh
xfh
bfafh
xfxfxfxfxfh
hiafhiafihafh
dxxf
njjhax h=(b-a)/2n
![Page 24: 數值方法 2008, Applied Mathematics NDHU 1 Simpson rule Composite Simpson rule.](https://reader037.fdocuments.net/reader037/viewer/2022110401/56649e185503460f94b04482/html5/thumbnails/24.jpg)
數值方法2008, Applied Mathematics NDHU 24
Composite Simpson rule
1
112
1
12 )(
3
4)(
3
2))()((
3
)(
2,...,1,0,
n
kk
n
kk
b
a
k
xfh
xfh
bfafh
dxxf
njjhax h=(b-a)/2n
Simpson's Rule for Numerical Integration