SE301: Numerical Methods Topic 7 Numerical Integration Lecture 24-27

1. SE301: Numerical MethodsTopic 7Numerical Integration Lecture 24-27 KFUPM (Term 101) Section 04 Read Chapter 21, Section 1 Read Chapter 22, Sections 2-3

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

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

4. Fundamental Theorem of Calculus

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

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

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

8. Example

9. Example

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.

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.

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

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

14. Trapezoid Method f(x)

15. Trapezoid MethodDerivation-One Interval

16. Trapezoid Method f(x)

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

18. Trapezoid MethodGeneral Formula and Special Case

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

20. Example 1

21. Error in estimating the integralTheorem

22. Estimating the Error For Trapezoid Method

23. Example

24. Example

25. Example

26. Recursive Trapezoid Method f(x)

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

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

29. Recursive Trapezoid MethodFormulas

30. Recursive Trapezoid Method

31. Example on Recursive Trapezoid Estimated Error = |R(3,0) – R(2,0)| = 0.009669

32. 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.

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

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

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

36. First Column Recursive Trapezoid Method

37. Derivation of Romberg Method

38. Romberg Method

39. Property of Romberg Method Error Level

40. Example

41. Example (cont.)

42. When do we stop?

43. Lecture 27Gauss Quadrature Motivation General integration formula Read 22.3

44. Motivation

45. General Integration Formula

46. Lagrange Interpolation

47. Example • Determine the Gauss Quadrature Formula of If the nodes are given as (-1, 0 , 1) • Solution: First we need to find l0(x), l1(x), l2(x) • Then compute:

48. Solution

49. Using the Gauss Quadrature Formula

50. Using the Gauss Quadrature Formula