1. CPE 332 Computer Engineering Mathematics II
Part III, Chapter 10
Numerical Differentiation and Integration

2. Today Topics Chapter 10 Numerical Differentiation and Integration
Derivative Approximation
Forward, Backward, Centered Difference
High Order Derivative
High Accuracy Approximation
Integral Approximation
Polynomial
Zero Order
First Order (Trapezoidal)
Second Order (Simpson 1/3)
Third Order (Simpson 3/8)
More Accurate Method
Richardson Extrapolation

5. Slope of Line m=slope = tan  = (y2-y1)/ (x2-x1) (x2,y2) (y2-y1)  

6. Definition of Derivative
Derivative of f(x) at any point x is the slope of the tangent line at that point (x,f(x))
Mathematically
For function y=f(x), derivative of function is written in many forms

7. Approximation of slope at point x using secant line
Slope at ‘x’  [f(x+x) - f(x)] / x = y / x
As x approaches 0,
we have y/x
approach a true slope
of f at x.
(x+x, f(x+x))
f(x+x) - f(x)
=y
(x, f(x)) x

8. Derivative(Forward) y

9. Derivative(Backward)y

10. Derivative(Central) y

11. Introduction h h
f(xi+1)
f(xi)
f(xi-1)
xi-1 xi
xi+1

12. Finite Divided-Difference

14. Finite Divided-Difference

16. Finite Divided-Difference

18. Second Derivative

19. Second Derivative

20. High Accuracy Finite Divided-Difference

21. High Accuracy Finite Divided-Difference

22. Summary

23. Summary

24. Summary

25. Numerical Integration
Newton-Cotes Integration Formula
Zero-Order Approximation
First-Order Approximation
Trapezoidal Rule
Second-Order Approximation
Simpson 1/3 rule
Third-Order Approximation
Simpson 3/8 rule
Romberg Integration
Richardson Extrapolation
Romberg Integration Algorithm

28. Integral Approximation

29. Integral Approximation
xn-1 xn

30. Zero-Order Approximation
xn-1 xn

31. Zero-Order Approximation
a0=f(a) x0 x1 x2 x3 x4
xn-1 xn

32. Zero-Order Approximation
a0=f((a+b)/2) x0 x1 x2 x3 x4
xn-1 xn

33. Zero-Order Approximation

34. First-Order Approximation
xn-1 xn

35. First-Order Approximation

36. First-Order Approximation

37. First-Order Approximation

39. Trapezoidal Rule

40. Trapezoidal Rule

44. Second-Order Approximation
xn-1 xn
Simpson’s 1/3 Rule

45. Second-Order Approximation

46. Second-Order Approximation

47. Second-Order Approximation

48. Second-Order Approximation

50. Third-Order Approximation
xn-1 xn
Simpson’s 3/8 Rule

51. Third Degree Approximation

52. Simson’s 3/8 Rule

54. Romberg Integration

55. Romberg Integration

56. Romberg Integration

57. Romberg Integration

58. Romberg Integration

59. Romberg Integration

60. Romberg Integration

61. Romberg Integration

62. Romberg Integration

63. Romberg Integration

64. Romberg Integration

65. Romberg Integration

66. Summary

67. Homework 10: Ch 10
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68. End of Week 13 Week 14 Week 15: End of Part III
Chapter 11 Solutions of ODE + HW 11
Week 15:
Ch 12 Curve Fitting + HW 12
End of Part III
Prepare for Final Exam