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

CISE301 : Numerical Methods Topic 7 Numerical Integration Lecture 24-27. KFUPM Read Chapter 21, Section 1 Read Chapter 22, Sections 2-3. L ecture 24 Introduction to Numerical Integration. Definitions Upper and Lower Sums Trapezoid Method (Newton-Cotes Methods) Romberg Method

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

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  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

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