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Lecture's Goals . InterpolationApproximationBasic Numerical IntegrationTaylor, Euler, Runge KuttaAdam BashforthAdam MoultonSystems of ODE. Lecture 20 - Interpolation Methods. CVEN 302October 12, 2001. Interpolation. The Lagrange and Newton Interpolation are basically the same methods, but us

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1. Final CVEN 302 December 3, 2001

2. Lecture’s Goals Interpolation Approximation Basic Numerical Integration Taylor, Euler, Runge Kutta Adam Bashforth Adam Moulton Systems of ODE

3. Lecture 20 - Interpolation Methods CVEN 302 October 12, 2001

4. Interpolation The Lagrange and Newton Interpolation are basically the same methods, but use different coefficients. The polynomials depend on the entire set of data points. Hermite Interpolation is a technique to calculate the values matches the function and first derivative.

5. Interpolation The Rational function deals with fractional polynomials depend on the entire set of data points. Cubic Spline Interpolation is a piecewise technique to calculate the values matches the function and first derivative.

6. Approximation The linear least squared method is straight forward to determine the coefficients of the line.

7. Approximation The quadratic and higher order polynomial curve fits use a similar technique and involve solving a matrix of (n+1) x (n+1).

8. Approximation The higher order polynomials fit required that one selects the best fit for the data and a means of measuring the fit is the standard deviation of the results as a function of the degree of the polynomial.

9. Basic Numerical Integration

10. Trapezoid Rule Straight-line approximation

11. Simpson’s 1/3-Rule Approximate the function by a parabola

12. Simpson’s 3/8-Rule Approximate by a cubic polynomial

13. Better Numerical Integration Composite integration Composite Trapezoidal Rule Composite Simpson’s Rule Richardson Extrapolation Romberg integration

14. Composite Trapezoid Rule

15. Composite Simpson’s Rule Evaluate the integral n = 2, h = 2 n = 4, h = 1

16. Composite Simpson’s Rule with Unequal Segments Evaluate the integral h1 = 1.5, h2 = 0.5

17. Richardson Extrapolation For trapezoidal rule kth level of extrapolation

18. Romberg Integration Accelerated Trapezoid Rule

19. Gaussian Quadratures Newton-Cotes Formulae use evenly-spaced functional values Gaussian Quadratures select functional values at non-uniformly distributed points to achieve higher accuracy change of variables so that the interval of integration is [-1,1] Gauss-Legendre formulae

20. Gaussian Quadrature on [-1, 1] Choose (c1, c2, x1, x2) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3

21. Gaussian Quadrature on [-1, 1] Choose (c1, c2, c3, x1, x2, x3) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3,x4, x5

22. Gaussian Quadrature on [-1, 1] Exact integral for f = x0, x1, x2, x3, x4, x5

23. Taylor Series Method

24. Taylor Series Method

25. Euler Method

26. Euler Method

27. Euler Method

28. Euler’s Method The trouble with this method is Lack of accuracy Small step size

29. Modified Euler Method

30. Modified Euler Method

31. Modified Euler Method

32. Runge-Kutta Methods

33. Runge-Kutta Methods

34. Runge-Kutta Methods

35. Runge-Kutta Methods

36. Runge-Kutta Methods

37. Runge-Kutta Methods

38. Runge-Kutta Methods

39. Runge-Kutta Methods

40. Runge-Kutta Methods

41. The 4th order Runge-Kutta

42. The 4th order Runge-Kutta

43. 4th-order Runge-Kutta Method

44. The 4th order Runge-Kutta

45. The 4th order Runge-Kutta

46. One Step Method Euler Method Modified Euler/Midpoint Runge-Kutta Methods

47. One Step Method These methods allow us to vary the step size. Use only one initial value After each step is completed the past step is “forgotten” We do not use this information.

48. Multi-Step Methods

49. Multi-Step Methods

50. Multi-Step Methods

51. Multi-Step Methods

52. Multi-Step Methods

53. Multi-Step Methods

54. Implicit Multi-Step Methods

55. Implicit Multi-Step Methods

56. Implicit Multi-Step Methods

57. Numerical Stability Amplification or decay of numerical errors A numerical method is stable if error incurred at one stage of the process do not tend to magnify at later stages Ill-conditioned differential equation -- numerical errors will be magnified regardless of numerical method Stiff differential equation -- require extremely small step size to achieve accurate results

58. Higher order differential equations

59. Higher order differential equations

60. Higher-order differential equations

61. Higher-order differential equations

62. Higher-order differential equations

63. Higher-order differential equations

64. Higher-order differential equations Taylor Series Euler Runge-Kutta Adam Bashforth

65. System of Initial value Problems

66. System of Initial value Problems

67. Multi-Step Methods

68. Multi-Step Methods

69. Multi-Step Methods

70. Implicit Multi-Step Methods

71. Implicit Multi-Step Methods

72. Systems of ODE - Initial Value Problems

73. Ordinary differential Equations -Boundary Value Problems

74. Boundary Conditions Dirichlet boundary conditions Newman boundary conditions Mixed conditions

75. Linear ODE -Boundary Value Problems

76. Linear ODE -Boundary Value Problems

77. Linear ODE -Boundary Value Problems

78. Boundary Value Problems

79. The differential equations Taylor Series Euler / Modified Euler Runge-Kutta Adam Bashforth

80. Adapting the equations

81. The BVP - Boundary Conditions

82. The BVP - Boundary Conditions

83. The BVP - Boundary Conditions

84. The BVP - Boundary Conditions

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