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Presented at Computer Science Department, Sharif University of Technology (Advanced Numerical Methods).
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1
Gaussian Gaussian IntegrationIntegration
M. Reza Rahimi,M. Reza Rahimi,Sharif University of Technology,Sharif University of Technology,
Tehran, Iran.Tehran, Iran.
2
OutlineOutline• Introduction• Gaussian Integration• Legendre Polynomials• N-Point Gaussian Formula• Error Analysis for Gaussian Integration• Gaussian Integration for Improper Integrals• Legendre-Gaussian Integration Algorithms• Chebyshev-Gaussian Integration Algorithms• Examples, MATLAB Implementation and Results• Conclusion
3
IntroductionIntroduction• Newton-Cotes and Romberg Integration usually
use table of the values of function.• These methods are exact for polynomials less
than N degrees.• General formula of these methods are as bellow:
• In Newton-Cotes method the subintervals has the same length.
b
a
n
iii xfwdxxf
1
)()(
4
• But in Gaussian Integration we have the exact formula of function.
• The points and weights are distinct for specific number N.
5
Gaussian Gaussian IntegrationIntegration
• For Newton-Cotes methods we have:
• And in general form:
.)()2
(4)(6
)( .2
.)()(2
)( .1
b
bfba
fafab
dxxf
bfafab
dxxf
b
b
a
dtji
jt
n
abw
nihiaxxfwdxxf
n n
ijji
b
a
n
iiii
0 ,1
1
1
,...,3,2,1 )1( )()(
6
• But suppose that the distance among points are not equal, and for every w and x we want the integration to be exact for polynomial of degree less than 2n-1.
1
1
12
1
12
1
1
1
1
1
1
.2
.............
.2
.1
dxxwxn
xdxwx
dxw
ni
n
i
n
n
iii
n
ii
7
• Lets look at an example:
• So 2-point Gaussian formula is:
.1 ,3
1
.3
134,2
0.4
3
2.3
0.2
2.1
.,,, 2
2121
12
22
12
223
113
222
112
2211
21
2121
wwxx
xxx
wxwx
wxwx
wxwx
ww
xxwwn
.)3
1()
3
1()(
1
1
ffdxxf
8
Legendre Legendre PolynomialsPolynomials
• Fortunately each x is the roots of Legendre Polynomial.
• We have the following properties for Legendre Polynomials.
.0,1,2,....n .)1(!2
1)( 2 n
nN xdx
d
nxP
1
1-
21
1
1
1
1
11
)!12(
)!(2)(5.
1-n.,0,1,2.....k 0)(.4
12
2)()(.3
).()()12()()1(2.
1,1).interval(-in ZerosN Has )(.1
n
ndxxPx
dxxPx
mndxxPxP
xnPxxPnxPn
xP
n
nn
nk
mnmn
nnn
n
9
• Legendre Polynomials make orthogonal bases in (-1,1) interval.
• So for finding Ws we must solve the following equations:
))1(1(1
.
....................
....................
0.2
2.1
1
1
1
1
1
n
1i
1
1
2
1
11
nnn
i
in
i
ii
n
ii
ndxxxwn
xdxxw
dxw
10
• We have the following equation which has unique answer:
• Theorem: if Xs are the roots of legendre polynomials and we got W from above equation then is exact for .
1
1
)( dxxP
12 nP
))1(1(1
.
0
2
.
...1
............
...1
...1
2
1
1
122
111
n
n
T
nnn
n
n
nw
w
w
xx
xx
xx
11
• Proof:
.2)()()(
)())()()(()(
.2)()()()(
))()()(())()()(()(
).()( ; )()(
).()()()(
0
1
1
1
01
1
01
1
0
1 1 1
0
1
1
0
1
0
1
1
1
0
1
0
1
1
1
0
1
1-
1
1
1
0
1
0
12
rdxxPrxPwrxPrw
xrwxrxPxqwxpw
rdxxPxPrdxxPxPq
dxxPrxPqxPdxxrxPxqdxxp
xPrxrxPqxq
xrxPxqxpp
j
n
jj
n
iji
n
jj
n
i
n
jijji
n
i
n
i
n
iiiiNiiii
j
n
jjnj
n
jj
n
jjj
n
jjjnn
n
jjj
n
jjj
nn
12
Theorem:
1
1 ,1
2
)(
)()( )(
n
ijj ji
jiii xx
xxxLdxxLw
Proof:
.)()()(21
1 1
222
2i
n
jjijini wxLwxLxL
13
Error Analysis for Error Analysis for Gaussian IntegrationGaussian Integration
• Error analysis for Gaussian integrals can be derived according to Hermite Interpolation.
.ba, )())!2)((12(
)!()()(
::is )( integral ion theapproximatin n integratiogaussian by madeerror The :Theorem
)2(3
412
b
a
nn
N fnn
NabfE
dxxf
14
Gaussian Integration for Improper Gaussian Integration for Improper IntegralsIntegrals
• Suppose we want to compute the following integral:
• Using Newton-Cotes methods are not useful in here because they need the end points results.
• We must use the following:
1
121
)(dx
x
xf
dxx
xfdx
x
xf
1
12
1
12 1
)(
1
)(
15
• But we can use the Gaussian formula because it does not need the value at the endpoints.
• But according to the error of Gaussian integration, Gaussian integration is also not proper in this case.
• We need better approach.
12
2
b
a 1
en exactly wh integral thecompute will
)()(
and for roots theare where
)()()(
:ionapproximat following thehave then we
jifor 0)()()(
:if w(x)respect to with b)(a,in orthogonal is Pset Polynomial The :Definition
n
b
a
ii
ni
n
iii
j
b
a
i
i
f
dxxLxww
Px
xfwdxxfxw
dxxPxPxw
16
• So we have following approximation:
1
12
101
222
0
j.i if 0)()(1
1
.2
)12(cos roots ).arccoscos()(
: then11- If
.)(,1)( ,1 ),()(2)(
)1(2
)(
:as defined is )( sPolynomial Chebyshev :Definition
dxxTxTx
n
ixxnxT
x
xxTxTnxTxxTxT
xxk
nxT
xT
ji
in
nnn
kkn
n
kn
n
.,...,3,2,1 2
)12(cos ,)()(
1
1
1
1
12
nin
ixxf
ndxxf
x
n
iii
17Legendre-Gaussian Integration Legendre-Gaussian Integration
AlgorithmsAlgorithmsa,b: Integration Interval,
N: Number of Points,f(x):Function Formula.
Initialize W(n,i),X(n,i).Ans=0;
).22
(2
)(ba
xab
fab
xA
For i=1 to N do:Ans=Ans+W(N,i)*A(X(N,i));
Return Ans;
End
FigureFigure 1: Legendre-Gaussian Integration Algorithm1: Legendre-Gaussian Integration Algorithm
18
a,b: Integration Interval,tol=Error Tolerance.
f(x):Function Formula.
Initialize W(n,i),X(n,i).Ans=0;
).22
(2
)(ba
xab
fab
xA
For i=1 to N do:If |Ans-Gaussian(a,b,i,A)|<tol then return Ans;
ElseAns=Gaussian(a,b,i,A);
Return Ans;
End
Figure 2: Adaptive Legendre-Gaussian Integration Algorithm.Figure 2: Adaptive Legendre-Gaussian Integration Algorithm.(I didn’t use only even points as stated in the book.)(I didn’t use only even points as stated in the book.)
19
Chebychev-Gaussian Integration Chebychev-Gaussian Integration AlgorithmsAlgorithms
a,b: Integration Interval,N: Number of Points,
f(x):Function Formula.
For i=1 to N do:Ans=Ans+ A(xi); //xi chebyshev
roots
Return Ans*pi/n;
End
FigureFigure 3: Chebyshev-Gaussian Integration Algorithm3: Chebyshev-Gaussian Integration Algorithm
)22
(2
)(1)( 2 x
babaf
abxxA
20
a,b: Integration Interval,tol=Error Tolerance.
f(x):Function Formula.
For i=1 to N do:If |Ans-Chebyshev(a,b,I,A)|<tol then return Ans;
ElseAns=Chebyshev(a,b,I,A);
Return Ans;
End
Figure 4: Adaptive Chebyshev-Gaussian Integration AlgorithmFigure 4: Adaptive Chebyshev-Gaussian Integration Algorithm
)22
(2
)(1)( 2 x
babaf
abxxA
21
Example and MATLAB Example and MATLAB Implementation and ResultsImplementation and Results
Figure 5:Legendre-Gaussian IntegrationFigure 5:Legendre-Gaussian Integration
22
Figure 6: AdaptiveFigure 6: Adaptive Legendre-Gaussian IntegrationLegendre-Gaussian Integration
23
Figure 7:Chebyshev-Gaussian IntegrationFigure 7:Chebyshev-Gaussian Integration
24
Figure 8:Adaptive Chebyshev-Gaussian IntegrationFigure 8:Adaptive Chebyshev-Gaussian Integration
25
Testing Strategies:Testing Strategies:
• The software has been tested for polynomials less or equal than 2N-1 degrees.
• It has been tested for some random inputs.• Its Result has been compared with MATLAB
Trapz function.
26
Examples:Examples:
1.5833.Resualt Software To According
1.5000.Resualt Software To According
.6667.1))1(1
1
)0(1
14
)1(1
1)(
6
)1(1(
.0000.1))1(1
1
)1(1
1)(
2
)1(1(
.5707.12
)1tan()1tan(1
1
int3
int2
222
22
1
12
GaussianPo
GaussianPo
Simpson
Trapezoid
exact ArcArcdxx
Example 1:Gaussian-LegendreExample 1:Gaussian-Legendre
27
0.4640.
0.6494.
.0792.0)35.10)(6
03(
.0005.0)30)(2
03(
.4999.0)2
1
2()
2(
int3
int2
95.1
9
930
3
0
2
2
2
GaussianPo
GaussianPo
Simpson
Trapezoid
xexactx
ee
e
eedxxe
.3105|))!2)((12(
)!()02(|
.4105|)(sin))!2)((12(
)!()0(|)sin(
b].[a, )())!2)((12(
)!()()(
43
4122
0
423
412
0
23
412
nenn
ndxe
nnn
ndxx
fnn
nabfE
nx
nn
nn
n
Example 2:Gaussian-LegendreExample 2:Gaussian-Legendre
Example 3:Gaussian-LegendreExample 3:Gaussian-Legendre
28
.00008.0error1.151376
.00027.0error1.151565
.36227.0error1.149024
.00871.0error1.142583
.06493.0error1.21622 2
.15129.1)1ln(2
1
1
a
a
a
a
a
23
02
xdxx
x
.00005.0error0.499896
.00013.0error0.500075
.00275.0error0.502694
.03597.0error0.463973
.14943.0error0.649372
.49994.02
a
a
a
a
a
3
0
3
0
2
2
x
x edxxe
Example 4:Gaussian-LegendreExample 4:Gaussian-Legendre
Example 5:Gaussian-LegendreExample 5:Gaussian-Legendre
29
877755.1.47666903Point-3
385020.5.91940603Point2
::)sin(
019234.2.35619449Point-3
019234.2.35619449Point2
::)sin(
236915.1.60606730Point-3
797927.1.19283364Point2
::)sin(
339745.0.78539816Point-3
339745.0.78539816Point-2
::)sin(
2
0
2
2
3
0
2
0
2
2/
0
2
dxx
dxx
dxx
dxx
Example 6:Gaussian-LegendreExample 6:Gaussian-Legendre
6626731.57079632::Trapzoid
9892102.35580550::Trapzoid
3556793.14159265::Trapzoid
6903600.78460183::Trapzoid
749986.3.14131064 Point-8
162258.3.14132550Point-7
123817.3.14606122Point-6
211956.3.08922572Point-5
676239.3.53659228Point-4
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
3.5
-0.57 0.57 -0.77 0.77
30
448013.1.15137188105error::nIntegratioGaussian Adaptive)2
351486.1.15114335105error::nIntegratioGaussian Adaptive)1
::1
4-
5-
3
02
dxx
x
784837.0.49988858105error::nIntegratioGaussian 2)Adaptive
291620.0.49980229105error::nIntegratioGaussian 1)Adaptive
::
4-
5-
3
0
2
dxxe x
Example 7:Adaptive Gaussian-LegendreExample 7:Adaptive Gaussian-Legendre
31
4082711.39530571 nIntegratio ChebyshevPoint -3
428604.0.48538619nIntegratio ChebyshevPoint -2
565488.0.33089431::ChebyshevPoint -3
0::ChebyshevPoint -2
Example 8:Gaussian-ChebyshevExample 8:Gaussian-Chebyshev
32Example 9:Example 9:
0)6
5
2)5
0)4
3
2)3
0)2
2)1
533
522
511
433
422
411
333
322
311
233
222
211
332211
321
xwxwxw
xwxwxw
xwxwxw
xwxwxw
xwxwxw
www
21
23
22
322
21
23
311
22
2322
23
2111
23
522
511
322
311
23
322
311
2211
)()(),()(1
5,4
14,2
xx
xxxwxxxwxxxwxxxw
xxwxw
xwxw
xxwxw
xwxw
212
12
3122
32
111 ))(()( wwxxxwxxxw
214
112
11
332121
5
3.
5
22,
3
225,3
.0)2,
xxxwxw
xwxxww
09
81
9
5
3
22 3321
211 xwwwxw
0,5
3,
5
3,,
9
8,
9
5,
9
5,,
321
321
xxx
www
33
ConclusioConclusionn
• In this talk I focused on Gaussian Integration.• It is shown that this method has good error
bound and very useful when we have exact formula.
• Using Adaptive methods is Recommended Highly.
• General technique for this kind of integration also presented.
• The MATLAB codes has been also explained.