1. SE301_Topic 6Al-Amer20051 SE301:Numerical Methods Topic 6 Numerical Integration Dr. Samir Al-Amer Term 053

2. SE301_Topic 6Al-Amer20052 Lecture 17 Introduction to Numerical Integration  Definitions  Upper and Lower Sums  Trapezoid Method  Examples

3. SE301_Topic 6Al-Amer20053 Integration Indefinite Integrals Indefinite Integrals of a function are functions that differ from each other by a constant. Definite Integrals Definite Integrals are numbers.

4. SE301_Topic 6Al-Amer20054 Fundamental Theorem of Calculus

5. SE301_Topic 6Al-Amer20055 The Area Under the Curve One interpretation of the definite integral is Integral = area under the curve ab f(x)

6. SE301_Topic 6Al-Amer20056 Numerical Integration Methods Numerical integration Methods Covered in this course  Upper and Lower Sums  Newton-Cotes Methods:  Trapezoid Rule  Simpson Rules  Romberg Method  Gauss Quadrature

7. SE301_Topic 6Al-Amer20057 Upper and Lower Sums ab f(x) The interval is divided into subintervals

8. SE301_Topic 6Al-Amer20058 Upper and Lower Sums ab f(x)

9. SE301_Topic 6Al-Amer20059 Example

10. SE301_Topic 6Al-Amer200510 Example

11. SE301_Topic 6Al-Amer200511 Upper and Lower Sums Estimates based on Upper and Lower Sums are easy to obtain for monotonic functions (always increasing or always decreasing). For non-monotonic functions, finding maximum and minimum of the function can be difficult and other methods can be more attractive.

12. SE301_Topic 6Al-Amer200512 Newton-Cotes Methods  In Newton-Cote Methods, the function is approximated by a polynomial of order n  Computing the integral of a polynomial is easy.

13. SE301_Topic 6Al-Amer200513 Newton-Cotes Methods Trapezoid Method ( First Order Polynomial are used ) Simpson 1/3 Rule ( Second Order Polynomial are used ),

14. SE301_Topic 6Al-Amer200514 Trapezoid Method f(x)

15. SE301_Topic 6Al-Amer200515 Trapezoid Method Derivation-One interval

16. SE301_Topic 6Al-Amer200516 Trapezoid Method f(x)

17. SE301_Topic 6Al-Amer200517 Trapezoid Method Multiple Application Rule ab f(x) x

18. SE301_Topic 6Al-Amer200518 Trapezoid Method General Formula and special case

19. SE301_Topic 6Al-Amer200519 Example Given a tabulated values of the velocity of an object. Obtain an estimate of the distance traveled in the interval [0,3]. Time (s)0.01.02.03.0 Velocity (m/s)0.0101214 Distance = integral of the velocity

20. SE301_Topic 6Al-Amer200520 Example 1 Time (s)0.01.02.03.0 Velocity (m/s) 0.0101214

21. SE301_Topic 6Al-Amer200521 Estimating the Error For Trapezoid method

22. SE301_Topic 6Al-Amer200522 Error in estimating the integral Theorem

23. SE301_Topic 6Al-Amer200523 Example

24. SE301_Topic 6Al-Amer200524 Example x1.01.52.02.53.0 f(x)2.13.23.42.82.7

25. SE301_Topic 6Al-Amer200525 Example x1.01.52.02.53.0 f(x)2.13.23.42.82.7

26. SE301_Topic 6Al-Amer200526 SE301: Numerical Method Lecture 18 Recursive Trapezoid Method Recursive formula is used

27. SE301_Topic 6Al-Amer200527 Recursive Trapezoid Method f(x)

28. SE301_Topic 6Al-Amer200528 Recursive Trapezoid Method f(x) Based on previous estimate Based on new point

29. SE301_Topic 6Al-Amer200529 Recursive Trapezoid Method f(x) Based on previous estimate Based on new points

30. SE301_Topic 6Al-Amer200530 Recursive Trapezoid Method Formulas

31. SE301_Topic 6Al-Amer200531 Recursive Trapezoid Method

32. SE301_Topic 6Al-Amer200532 Advantages of Recursive Trapezoid Recursive Trapezoid:  Gives the same answer as the standard Trapezoid method.  Make use of the available information to reduce computation time.  Useful if the number of iterations is not known in advance.

33. SE301_Topic 6Al-Amer200533 SE301:Numerical Methods 19. Romberg Method Motivation Derivation of Romberg Method Romberg Method Example When to stop?

34. SE301_Topic 6Al-Amer200534 Motivation for Romberg Method  Trapezoid formula with an interval h gives error of the order O(h 2 )  We can combine two Trapezoid estimates with intervals 2h and h to get a better estimate.

35. SE301_Topic 6Al-Amer200535 Romberg Method First column is obtained using Trapezoid Method R(0,0) R(1,0)R(1,1) R(2,0)R(2,1)R(2,2) R(3,0)R(3,1)R(3,2)R(3,3) The other elements are obtained using the Romberg Method

36. SE301_Topic 6Al-Amer200536 First Column Recursive Trapezoid Method

37. SE301_Topic 6Al-Amer200537 Derivation of Romberg Method

38. SE301_Topic 6Al-Amer200538 Romberg Method R(0,0) R(1,0)R(1,1) R(2,0)R(2,1)R(2,2) R(3,0)R(3,1)R(3,2)R(3,3)

39. SE301_Topic 6Al-Amer200539 Property of Romberg Method R(0,0) R(1,0)R(1,1) R(2,0)R(2,1)R(2,2) R(3,0)R(3,1)R(3,2)R(3,3) Error Level

40. SE301_Topic 6Al-Amer200540 Example 1 0.5 3/81/3

41. SE301_Topic 6Al-Amer200541 Example 1 cont. 0.5 3/81/3 11/321/3

42. SE301_Topic 6Al-Amer200542 When do we stop?

43. SE301_Topic 6Al-Amer200543 SE301:Numerical Methods 20. Gauss Quadrature Motivation General integration formula Read 22.3-22.3

44. SE301_Topic 6Al-Amer200544 Motivation

45. SE301_Topic 6Al-Amer200545 General Integration Formula

46. SE301_Topic 6Al-Amer200546 Lagrange Interpolation

47. SE301_Topic 6Al-Amer200547 Question What is the best way to choose the nodes and the weights?

48. SE301_Topic 6Al-Amer200548 Theorem

49. SE301_Topic 6Al-Amer200549 Weighted Gaussian Quadrature Theorem

50. SE301_Topic 6Al-Amer200550 Determining The Weights and Nodes

51. SE301_Topic 6Al-Amer200551 Determining The Weights and Nodes Solution

52. SE301_Topic 6Al-Amer200552 Theorem

53. SE301_Topic 6Al-Amer200553 Determining The Weights and Nodes Solution

54. SE301_Topic 6Al-Amer200554 Determining The Weights and Nodes Solution

55. SE301_Topic 6Al-Amer200555 Gaussian Quadrature See more in Table 22.1 (page 626) 626

56. SE301_Topic 6Al-Amer200556 Error Analysis for Gauss Quadrature 627

57. SE301_Topic 6Al-Amer200557 Example

58. SE301_Topic 6Al-Amer200558 Example

59. SE301_Topic 6Al-Amer200559 Improper Integrals 627

60. SE301_Topic 6Al-Amer200560 Quiz x1.01.52.02.53.0 f(x)2.13.23.42.82.7