Final

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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