CISE301 : Numerical Methods Topic 7 Numerical Integration Lecture 24-27

1. CISE301: Numerical MethodsTopic 7Numerical Integration Lecture 24-27 KFUPM Read Chapter 21, Section 1 Read Chapter 22, Sections 2-3 KFUPM

2. Lecture 24Introduction to Numerical Integration Definitions Upper and Lower Sums Trapezoid Method (Newton-Cotes Methods) Romberg Method Gauss Quadrature Examples KFUPM

3. Integration Indefinite Integrals Indefinite Integrals of a function are functionsthat differ from each other by a constant. Definite Integrals Definite Integrals are numbers. KFUPM

4. Fundamental Theorem of Calculus KFUPM

5. The Area Under the Curve One interpretation of the definite integral is: Integral = area under the curve f(x) a b KFUPM

6. Upper and Lower Sums The interval is divided into subintervals. f(x) a b KFUPM

7. Upper and Lower Sums f(x) a b KFUPM

8. Example KFUPM

9. Example KFUPM

10. 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. KFUPM

11. 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. KFUPM

12. Newton-Cotes Methods • Trapezoid Method (First Order Polynomials are used) • Simpson 1/3 Rule (Second Order Polynomials are used) KFUPM

13. Lecture 25Trapezoid Method Derivation-One Interval Multiple Application Rule Estimating the Error Recursive Trapezoid Method Read 21.1 KFUPM

14. Trapezoid Method f(x) KFUPM

15. Trapezoid MethodDerivation-One Interval KFUPM

16. Trapezoid Method f(x) KFUPM

17. Trapezoid MethodMultiple Application Rule f(x) x a b KFUPM

18. Trapezoid MethodGeneral Formula and Special Case KFUPM

19. Example Given a tabulated values of the velocity of an object. Obtain an estimate of the distance traveled in the interval [0,3]. Distance = integral of the velocity KFUPM

20. Example 1 KFUPM

21. Estimating the Error For Trapezoid Method KFUPM

22. Error in estimating the integralTheorem KFUPM

23. Example KFUPM

24. Example KFUPM

25. Example KFUPM

26. Recursive Trapezoid Method f(x) KFUPM

27. Recursive Trapezoid Method f(x) Based on previous estimate Based on new point KFUPM

28. Recursive Trapezoid Method f(x) Based on previous estimate Based on new points KFUPM

29. Recursive Trapezoid MethodFormulas KFUPM

30. Recursive Trapezoid Method KFUPM

31. Advantages of Recursive Trapezoid Recursive Trapezoid: • Gives the same answer as the standard Trapezoid method. • Makes use of the available information to reduce the computation time. • Useful if the number of iterations is not known in advance. KFUPM

32. Lecture 26Romberg Method Motivation Derivation of Romberg Method Romberg Method Example When to stop? Read 22.2 KFUPM

33. Motivation for Romberg Method • Trapezoid formula with an interval h gives an error of the order O(h2). • We can combine two Trapezoid estimates with intervals 2h and h to get a better estimate. KFUPM

34. Romberg Method First column is obtained using Trapezoid Method The other elements are obtained using the Romberg Method KFUPM

35. First Column Recursive Trapezoid Method KFUPM

36. Derivation of Romberg Method KFUPM

37. Romberg Method KFUPM

38. Property of Romberg Method Error Level KFUPM

39. Example 1 KFUPM

40. Example 1 (cont.) KFUPM

41. When do we stop? KFUPM

42. Lecture 27Gauss Quadrature Motivation General integration formula Read 22.3 KFUPM

43. Motivation KFUPM

44. General Integration Formula KFUPM

45. Lagrange Interpolation KFUPM

46. Question What is the best way to choose the nodes and the weights? KFUPM

47. Theorem KFUPM

48. Weighted Gaussian QuadratureTheorem KFUPM

49. Determining The Weights and Nodes KFUPM

50. Determining The Weights and NodesSolution KFUPM