CPE 332 Computer Engineering Mathematics II

1. CPE 332Computer 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)   (x1,y1) (x2-x1)

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 xusing 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-OrderApproximation • 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 x0 x1 x2 x3 x4 xn-1 xn

30. Zero-Order Approximation x0 x1 x2 x3 x4 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 x0 x1 x2 x3 x4 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 x0 x1 x2 x3 x4 xn-1 xn Simpson’s 1/3 Rule

45. Second-Order Approximation

46. Second-Order Approximation

47. Second-Order Approximation 3n

48. Second-Order Approximation

50. Third-Order Approximation x0 x1 x2 x3 x4 xn-1 xn Simpson’s 3/8 Rule