1. 11. Numerical Differentiation and Integration 11
11. Numerical Differentiation and Integration Better Numerical Integration, 11.4 Gaussian Quadrature, 11.5 MATLAB’s Methods
Natural Language Processing Lab
Dept. of Computer Science and Engineering, Korea Univertity
CHOI Won-Jong
Woo
Kang

2. Contents 11.3 BETTER NUMERICAL INTEGRATION
Composite Trapezoid Rule
Composite Simpson’s Rule
Extrapolation Methods for Quadrature
11.4 GAUSSIAN QUADRATURE
Gaussian Quadrature on [-1, 1]
Gaussian Quadrature on [a, b]
11.5 MATLAB’s Methods

3. 11.3.1 Composite Trapezoid Rule 11.3.2 Composite Simpson’s Rule
CHOI WonJong

4. 11.3.1 Composite Trapezoid Rule
CHOI WonJong

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.

7. 11.3.1 Composite Trapezoid Rule
If we divide the interval into n subintervals, we get
MATLAB CODE

8. 11.3.1 Composite Trapezoid Rule
Example 11.9
n= n= n=3
n= n= n=100

9. 11.3.1 Composite Trapezoid Rule
Example 11.9

10. 11.3.2 Composite Simpson’s Rule
CHOI WonJong

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,

13. 11.3.2 Composite Simpson’s Rule
In general, for n even, we have h=(b-a)/n, and Simpson’s rule is

14. 11.3.2 Composite Simpson’s Rule
Example 11.10

15. 11.3.2 Composite Simpson’s Rule
Example Length of Elliptical Orbit

16. 11.3.2 Composite Simpson’s Rule
Example Length of Elliptical Orbit
days
r = [ ]
Using Composite Simpson’s Rule and the length between day 0 and 10 (n=20) is (Trapezoid= , Text=0.8556)
Using Composite Simpson’s Rule and the length between day 60 and 70 (n=20) is (Trapezoid= , Text=0.3702)
The former is times faster than the latter.

17. 11.3.3 Extrapolation Methods for Quadrature
Woo

18. Richardson Expolation
Truncation error(절단 오차)
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 Extrapolation
간격이 다른 2개의 식을 구한 결과를 대수적으로 정리함으로써
보다 정확한 값을 산출
계산오차
세부 구간의 수
simpson
trapezoid

20. Richardson Extrapolation
Richardson Extrapolation using the trapezoid rule
(if h_2 = ½ h_1)
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)=

22. Example 11.12 Integral of 1/x Form a table of the approximations≠ Ⅰ Ⅱ
h=1
0.7500
0.6944
h=1/2
0.7083

23. Romberg Integration Approximate an Error
Trapezoid rules :
Richardson extrapolation :
continued ( using simpson rules)

24. Romberg Integration Improving the result by Richardson extrapolation
Romberg integration : iterative procedure using Richardson extrapolation
k means the improving level(= )
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 =
For k=2, I_2 =
To apply Richardson extrapolation Ⅰ Ⅱ
h=1
0.7500
0.6944
0.6933
0.6943
h=1/2
0.7083
h=1/4
0.6970
h=1/8
0.6941

26. Example 11.12 Integral of 1/x using Romberg Integration
second level of extrapolation Ⅰ Ⅱ Ⅲ
h=1
0.7500
0.6944
0.6933
h=1/2
0.7083
[16(0.6933) ]/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
0.7500
0.6944
0.6932
0.6931
0.7083
0.6933
0.6970
0.6941
0.6934

28. Matlab function for Romberg Integration

29. 11.4 Gaussian Quadrature
Kang Nam-Hee

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 integral of exp(-x2) Using G.Q n Xi ci 2 3 4 ± ± ± ± 1
8/9
5/9
Table 11.2 parameters of Gaussian quadrature

33. 11.4.1 Gaussian Quadrature on [-1,1]
Gaussian-Legendre Polynomials

34. 11.4.2 Gaussian Quadrature on [a,b]
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

35. 11.4.2 Gaussian Quadrature on [a,b]
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

36. 11.4.2 Gaussian Quadrature on [a,b]
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

37. 11.4.2 Gaussian Quadrature on [a,b]
Example (cont)
So we can get
Apply Gaussian Quadrature to the integral with n = 2

38. 11.4.2 Gaussian Quadrature on [a,b]
Matlab function for Gaussian Quadrature

39. 11.5 MATLAB’s Methods
Woo

40. 11.5 MATLAB’s Methods
p=polyfit(x,y,n) – find the coefficients of the polynomial of degree n
polyder(p) - calculates the derivative of polynomials
diff(x) - x = [ ];
y = diff(x)
y =
traps(x,y)
Q=quad(‘f’,xmin,xmax) (simpson rules)
Q=quad8(‘f’,xmin,xmax) (Newton-Cotes eight-panel rule)