11.4 gaussian quadrature
TRANSCRIPT
11. Numerical Differentiation and Integration11.3 Better Numerical Integration, 11.4 Gaussian Quadrature, 11.5 MATLAB’s Methods
Natural Language Processing LabDept. of Computer Science and Engineering, Korea Univertity
CHOI Won-Jong ([email protected])Woo Yeon-Moon([email protected])Kang Nam-Hee([email protected])
2
Contents 11.3 BETTER NUMERICAL INTEGRATION
11.3.1 Composite Trapezoid Rule 11.3.2 Composite Simpson’s Rule 11.3.3 Extrapolation Methods for Quadrature
11.4 GAUSSIAN QUADRATURE 11.4.1 Gaussian Quadrature on [-1, 1] 11.4.2 Gaussian Quadrature on [a, b]
11.5 MATLAB’s Methods
5
11.3 BETTER NUMERICAL INTEGRATION
Composite integration(복합적분 ) : Applying one of the lower order methods presented in the previous section repeatedly on several sub intervals.
6
11.3.1 Composite Trapezoid Rule
If we divide the interval of integration, [a, b], into two or more subintervals and use the trapezoid rule on each subintervals, we obtain the composite trapezoid rule.
1
11 1
1 1
( ) ( ) ( ) [ ( ) ( )] [ ( ) ( )]2 2
[ ( ) 2 ( ) ( )] [ ( ) 2 ( ) ( )]2 4
b x b
a a x
h hf x dx f x dx f x dx f a f x f x f b
h b af a f x f b f a f x f b
2b ah
7
11.3.1 Composite Trapezoid Rule
If we divide the interval into n subintervals, we get
1
1
1 1
1 1
( ) ( ) ( )
[ ( ) ( )] [ ( ) ( )]2 2
[ ( ) 2 ( ) 2 ( ) ( )]2
n
b x b
a a x
n
n
f x dx f x dx f x dx
h hf a f x f x f b
b a f a f x f x f bn
b ahn
MATLAB CODE
8
11.3.1 Composite Trapezoid Rule
Example 11.9
n=1 n=2 n=3
n=4 n=20 n=100
9
11.3.1 Composite Trapezoid Rule
Example 11.9
2
1
1 [log | | ]
1 2[log | 2 | ] [log |1| ] log 0.693147180559951
b baa
dx x Cx
dx C Cx
11
11.3.2 Composite Simpson’s Rule
Example 11.10
12
11.3.2 Composite Simpson’s Rule
Applying the same idea of subdivision of intervals to Simpson’s rule and requiring that n be even gives the composite Simpson rule.
[a,b] 를 two subintervals [a,x2], [x2, b] 로 나눈다면 ,
2 ,2 4
b a b ax h
2
2
1 2 2 3
( ) ( ) ( )
[ ( ) 4 ( ) ( )] [ ( ) 4 ( ) ( )]3 3
b x b
a a xf x dx f x dx f x dx
h hf a f x f x f x f x f b
13
11.3.2 Composite Simpson’s Rule
In general, for n even, we have h=(b-a)/n, and Simpson’s rule is
b ahn
1 2 3 4 2 1( ) [ ( ) 4 ( ) 2 ( ) 4 ( ) 2 ( ) 2 ( ) 4 ( ) ( )]3
b
n na
hf x dx f a f x f x f x f x f x f x f b
14
11.3.2 Composite Simpson’s Rule
Example 11.10
15
11.3.2 Composite Simpson’s Rule
Example 11.11 Length of Elliptical Orbit
2 2 2 2
3( ) cos( ), ( ) sin( )4
( ') ( ') 0.25 16sin ( ) 9cos ( )b b
a a
x r r y r r
L x y dr r r dr
16
11.3.2 Composite Simpson’s Rule
Example 11.11 Length of Elliptical Orbit
2 2 2 2
3( ) cos( ), ( ) sin( )4
( ') ( ') 0.25 16sin ( ) 9cos ( )b b
a a
x r r y r r
L x y dr r r dr
days 0 10 20 30 40 50 60 70 80 90 100r = [0.00 1.07 1.75 2.27 2.72 3.14 3.56 4.01 4.53 5.22 6.28]
Using Composite Simpson’s Rule and the length between day 0 and 10 (n=20) is 0.88952. (Trapezoid=0.889567, Text=0.8556)Using Composite Simpson’s Rule and the length between day 60 and 70 (n=20) is 0.382108. (Trapezoid=0.382109, Text=0.3702)The former is 2.3279 times faster than the latter.
18
Richardson Expolation
Truncation error(절단 오차 )• ( , ) ( , )I I f h E f h
21 1
1
( ) [ ( ) 2 ( ) ... 2 ( ) ( )]2
bj
n jja
hf x dx f a f x f x f b c h
2
1 1[ ( ) 2 ( ) ... 2 ( ) ( )]2 nh f a f x f x f b ch
41 2 2 3[ ( ) 4 ( ) 2 ( )] [ ( ) 4 ( ) ( )]
3 3h hf a f x f x f x f x f b ch
사다리꼴
simpson
19
Richardson Expolation
To obtain an estimate that is more accurate• using two or more subintervals (h를 줄임 )
- 그러나 , 세부구간의 수가 일정한 범위를 넘어서면 round-off error가 커지게 된다 .
Richardson Extrapolation간격이 다른 2개의 식을 구한 결과를 대수적으로 정리함으로써보다 정확한 값을 산출
계산오차
세부 구간의 수
simpson
trapezoid
20
Richardson Extrapolation Richardson Extrapolation using the trapezoid rule
(if h_2 = ½ h_1)
2 21 1 2 2( ) ( )T TI I h ch I h ch
2 14 ( ) ( )3
I h I hI
2 12 2
1
2
( ) ( )( )
1
I h I hI I hhh
Simpson rules
21
Example 11.12 Integral of 1/x start with one subinterval (h=1)
two subintervals (h=1/2)
to apply Richardson extrapolation
exact value of the integral is ln(2)=0.693147..
2
01
1 1 1 1 3[ (1) (2)] [ ] 0.752 2 1 2 4
dx I f fx
11 1 1 2 1 17[ (1) 2 (1.5) (2)] [ ] 0.70834 4 1 1.5 2 24
I f f f
1[4 ( ) ( )]3 2
hA A A h
1 01[4 ] [4(0.7083) 0.7500] / 3 0.69443
I I I
22
Example 11.12 Integral of 1/x Form a table of the approximations
0.6944 ≠0.693147
Ⅰ Ⅱ
h=1 0.75000.6944
h=1/2 0.7083
0
1
20 1
0.75 0.6944 0.05560.7083 0.6944 0.0139
(2)
EE
E E
2( ) ( )E h O h
23
Romberg Integration Approximate an Error
Trapezoid rules : Richardson extrapolation :
continued ( using simpson rules)4 4
1 1 2 2( ) ( )S SI I h ch I h ch
2 116 ( ) ( )15
I h I hI
2 12 4
1
2
( ) ( )( )
1
I h I hI I hhh
2( )O h4( )O h
24
Romberg Integration Improving the result by Richardson extrapolation
Romberg integration : iterative procedure using Richardson extrapolation
k means the improving level(= )
2 4 6 8 101 2 3 4 5E c h c h c h c h c h
4 ( / 2) ( )( )4 1
k
k
I h I hI h
2degree of the error
1st 2nd 3rd
25
Example 11.12 Integral of 1/x using Romberg Integration
Trapezoid rule
For k=0, I_0 = 0.75 For k=1, I_1 = 0.7083 For k=2, I_2 = 0.6941
To apply Richardson extrapolation
2
1 11
1( ) [ ( ) 2 ( ) ... 2 ( ) ( )]2
b
na
hf x dx dx f a f x f x f bx
Ⅰ Ⅱ
h=1 0.75000.69440.69330.6943
h=1/2 0.7083h=1/4 0.6970h=1/8 0.6941
1( ) 4 ( ) ( )3 2
hA h A A h
26
Example 11.12 Integral of 1/x using Romberg Integration
second level of extrapolation
1( ) 16 ( ) ( )15 2
hC h B B h
Ⅰ Ⅱ Ⅲ
h=1 0.75000.69440.6933
h=1/2 0.7083 [16(0.6933)-0.6944]/15
h=1/4 0.6970
27
Example 11.12 Integral of 1/x using Romberg Integration
five levels of extrapolation to find values for 2
1
1 dxx
0.7500
0.6944
0.6932
0.6931
0.6931
0.6931
0.7083
0.6933
0.6931
0.6931
0.6931
0.6970
0.6932
0.6931
0.6931
0.6941
0.6931
0.6931
0.6934
0.6931
0.6932
28
Matlab function for Romberg Integration
30
11.4.1 Gaussian Quadrature on [-1,1]
Gaussian Quadrature Formular
Get the definite integration of f(x) on [-1,1] using linear combinations of coefficient ck and evaluated function value f(xk) at the point xk
Appropriate values of the points xk and ck depend on the choice of n
By choosing the quadrature point x1 ,… xn as the n zeros of the nth-degree Gauss-Legendre polynomial, and by using the appropriate coefficients, the integration formular is exact for polynomials of degree up to 2n-1
31
11.4.1 Gaussian Quadrature on [-1,1]
Gaussian Quadrature Formular (cont.)
n=2
n=3
32
11.4.1 Gaussian Quadrature on [-1,1]
Example 11.13 integral of exp(-x2) Using G.Q
n Xi ci
23
4
±0.557753 0±0.77459±0.861136±0.339981
18/95/90.347850.652145
Table 11.2 parameters of Gaussian quadrature
33
Gaussian-Legendre Polynomials
11.4.1 Gaussian Quadrature on [-1,1]
34
Extends Gaussian Quadrature for f(t) on [a, b] by Transformation f(t) on [a, b] to f(x) on [-1,1]
For the given integral
change interval of t by using next formular
so the interval
11.4.2 Gaussian Quadrature on [a,b]
35
Extends Gaussian Quadrature for f(t) on [a, b] (cont.) f(t) rewrite for variable x
remark the factor (b-a)/2 (∵td convert to dx)
Apply f(x) to the integral
11.4.2 Gaussian Quadrature on [a,b]
36
Example 11.14 integral of exp(-x2) on [0,2] using G.Q with n = 2
Consider again the integral
Transform f(t) on [0,2] to f(x) on [-1,1] using next formular
11.4.2 Gaussian Quadrature on [a,b]
37
Example 11.14 (cont) So we can get
Apply Gaussian Quadrature to the integral with n = 2
11.4.2 Gaussian Quadrature on [a,b]
38
Matlab function for Gaussian Quadrature
11.4.2 Gaussian Quadrature on [a,b]
40
11.5 MATLAB’s Methods p=polyfit(x,y,n) – find the coefficients of the p
olynomial of degree n polyder(p) - calculates the derivative of polynom
ials diff(x) - x = [1 2 3 4 5];
y = diff(x)y = 1 1 1 1
traps(x,y) Q=quad(‘f’,xmin,xmax) (simpson rules) Q=quad8(‘f’,xmin,xmax) (Newton-Cotes eight-panel
rule)