simpson rule composite simpson rule
DESCRIPTION
Simpson rule Composite Simpson rule. Simpson rule for numerical integration. y=sin(x) for x within [1.2 2]. a=1.2;b=2; x= linspace (a,b);plot(x,sin(x));hold on. Quadratic polynomial. Approximate sin using a quadratic polynomial . c=0.5*(a+b); x=[a b c]; y=sin(x); p=polyfit(x,y,2); - PowerPoint PPT PresentationTRANSCRIPT
數值方法2008, Applied Mathematics NDHU 1
Simpson ruleComposite Simpson rule
數值方法2008, Applied Mathematics NDHU 2
Simpson rule for numerical integration
b
aabfbfbafafabdxxf
baCf
5)4(
4
)(2880
)()()2
(4)(6
)(
withexists b)(a,in number a then ],,[ If
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]
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
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
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
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
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]
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))
數值方法2008, Applied Mathematics NDHU 10
))2
()((23a-b
Area(I)abfaf
3ab
))2
()2
((23a-b
Area(II)abfabf
))()2
((23a-b
Area(III)
bfabf
2ab
• The area under a quadratic polynomial is a sum of area I, II and III
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))
數值方法2008, Applied Mathematics NDHU 12
))2
()((23a-b
Area(I)abfaf
3ab
))2
()2
((23a-b
Area(II)abfabf
))()2
((23a-b
Area(III)
bfabf
))()2
(4)((23a-b
Area(III)Area(II)Area(I)
bfabfaf
2ab
數值方法2008, Applied Mathematics NDHU 13
passes )()( 2
xpPxp
))(,( bfb ))2
(,2
( bafba
b
adxxp )(
that Show
)()2
(4)(6
bfbafafab
))(,( afa
數值方法2008, Applied Mathematics NDHU 14
)()()( :))(())(()(
))(())(()(
))(())(()( )(
cf,p(c)bf,p(b)afp(a)proofbcacbxaxcf
cbabcxaxbf
cabacxbxafxp
))(,( afa
))(,( bfb
2)),(,( baccfc
數值方法2008, Applied Mathematics NDHU 15
31
)02)(12()1)(0(
))(())((
34
)01)(21()2)(0(
))(())((
31
)10)(20()1)(2(
))(())(2(
))(())((
2))(())(()(
))(())(()(
))(())(()(
))(())(()(
))(())(()(
))(())(()( )(
2
0
2
0
2
0
2
hdttthdxcbabcxax
hdttthdxbcacbxax
hdttthdxcaba
haxhaxdxcabacxbx
hachab
dxbcacbxaxcfdx
cbabcxaxbfdx
cabacxbxaf
dxbcacbxaxcf
cbabcxaxbf
cabacxbxafdxxp
b
a
b
a
ha
a
b
a
b
a
b
a
b
a
b
a
b
a
))()(4)((
3bfcfafh
))()(4)((6
)(
))()(4)((3
bfcfafab
bfcfafh
數值方法2008, Applied Mathematics NDHU 16
dttthdthhhtht
dxcaba
haxhaxdxcabacxbx
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
數值方法2008, Applied Mathematics NDHU 17
Shortcut to Simpson 1/3 rule
))()2
(4)((6
)()()()(
)(
2
2
bfbafafab
IIIQIIQIQdxxp
PpCBxAxxp
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
數值方法2008, Applied Mathematics NDHU 18
Proof
))()2
(4)((6
23
)()(
))()((4))()((2))()((2)(
1)(
23
2
2
2
2
bfbafafab
CxxBxA
dxCBxAxdxxp
PpCBxAx
bxaxcfcxaxbfcxbxafba
xp
b
a
b
a
b
a
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
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
)(
bfbafafab
dxxpb
a
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
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
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
xfhxfhbfafh
xfxfxfxfxfh
hiafhiafihafh
dxxf
njjhax h=(b-a)/2n
數值方法2008, Applied Mathematics NDHU 24
Composite Simpson rule
1
112
1
12 )(
34)(
32))()((
3
)(
2,...,1,0,
n
kk
n
kk
b
a
k
xfhxfhbfafh
dxxf
njjhax h=(b-a)/2n
Simpson's Rule for Numerical Integration