Natural Language Processing Lab Dept. of Computer Science and Engineering, Korea Univertity

1. 11. Numerical Differentiation and Integration11.3 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 (wjchoi@nlp.korea.ac.kr) Woo Yeon-Moon(wooym@nlp.korea.ac.kr) Kang Nam-Hee(nhkang@nlp.korea.ac.kr)

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

3. 11.3.1 Composite Trapezoid Rule • 11.3.2 Composite Simpson’s Rule CHOI WonJong (wjchoi@nlp.korea.ac.kr)

4. 11.3.1 Composite Trapezoid Rule CHOI WonJong (wjchoi@nlp.korea.ac.kr)

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=1 n=2 n=3 n=4 n=20 n=100

9. 11.3.1 Composite Trapezoid Rule • Example 11.9

10. 11.3.2 Composite Simpson’s Rule CHOI WonJong (wjchoi@nlp.korea.ac.kr)

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 11.11 Length of Elliptical Orbit

16. 11.3.2 Composite Simpson’s Rule • Example 11.11 Length of Elliptical Orbit • days 0 10 20 30 40 50 60 70 80 90 100 • r = [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.

17. 11.3.3 Extrapolation Methods for Quadrature Woo Yeon-Moon(wooym@nlp.korea.ac.kr)

18. 사다리꼴 simpson Richardson Expolation • Truncation error(절단 오차)

19. 계산오차 trapezoid simpson 세부 구간의 수 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

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)=0.693147..

22. Example 11.12 Integral of 1/x • Form a table of the approximations • 0.6944 ≠0.693147

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 = 0.7083 • For k=2, I_2 = 0.6941 • To apply Richardson extrapolation

26. Example 11.12 Integral of 1/x using Romberg Integration • second level of extrapolation

27. Example 11.12 Integral of 1/x using Romberg Integration • five levels of extrapolation to find values for

28. Matlab function for Romberg Integration

29. 11.4 Gaussian Quadrature Kang Nam-Hee (nhkang@nlp.korea.ac.kr)

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 xkand 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 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 11.14 (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 Yeon-Moon(wooym@nlp.korea.ac.kr)

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 = [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)