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

งานนำเสนอเรื่อง: "CPE 332 Computer Engineering Mathematics II Part III, Chapter 10 Numerical Differentiation and Integration Numerical Differentiation and Integration."— ใบสำเนางานนำเสนอ:

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

3

4

5 Slope of Line (x 1,y 1 ) (x 2,y 2 ) (y 2 -y 1 ) (x 2 -x 1 )  m=slope = tan  = (y 2 -y 1 )/ (x 2 -x 1 ) 

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 (x, f(x)) (x+  x, f(x+  x)) xx f(x+  x) - f(x) =  y As  x approaches 0, we have  y/  x approach a true slope of f at x.

8 Derivative(Forward) yy

9 Derivative(Backward) yy

10 Derivative(Central) yy

11 Introduction ในทางปฎิบัติ เราได้ Sample ของ Data เป็นจุด การหา Derivative ก็คือการลบค่าของจุด Data ที่อยู่ติดกันและ หารด้วยระยะห่างระหว่างจุด นี่คือวิธีการของ Finite Divided-Difference xixi x i+1 x i-1 f(x i ) f(x i-1 ) f(x i+1 ) h h

12 Finite Divided-Difference

13

14

15

16

17

18 Second Derivative

19

20 High Accuracy Finite Divided-Difference

21

22 Summary

23

24

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

26

27

28 Integral Approximation

29 x0x0 x1x1 x2x2 x3x3 x4x4 x n-1 xnxn

30 Zero-Order Approximation x0x0 x1x1 x2x2 x3x3 x4x4 x n-1 xnxn

31 Zero-Order Approximation a 0 =f(a) x0x0 x1x1 x2x2 x3x3 x4x4 x n-1 xnxn

32 Zero-Order Approximation a 0 =f((a+b)/2) x0x0 x1x1 x2x2 x3x3 x4x4 x n-1 xnxn

33 Zero-Order Approximation

34 First-Order Approximation x0x0 x1x1 x2x2 x3x3 x4x4 x n-1 xnxn

35 First-Order Approximation

36

37

38

39 Trapezoidal Rule

40

41

42

43

44 Second-Order Approximation x0x0 x1x1 x2x2 x3x3 x4x4 x n-1 xnxn Simpson’s 1/3 Rule

45 Second-Order Approximation

46

47 3n

48 Second-Order Approximation

49

50 Third-Order Approximation x0x0 x1x1 x2x2 x3x3 x4x4 x n-1 xnxn Simpson’s 3/8 Rule

51 Third Degree Approximation

52 Simson’s 3/8 Rule

53

54 Romberg Integration

55

56

57

58

59

60

61

62

63

64

65

66 Summary

67 Homework 10: Ch 10 Download Due Wed 22 April –No Class Next Week (Song-Kran)

68 End of Week 13 Week 14 วันที่ 15 เมษายน หยุดสงกรานต์ Week 15 วันที่ 22 เมษายน – ส่งการบ้าน HW 10 –Chapter 11 Solutions of ODE + HW 11 Week 16: ( ถ้ามีเวลา ขึ้นอยู่กับวันที่ 29 เมษายน นี้ จะมีการเสด็จหรือไม่ ) –Ch 12 Curve Fitting + HW 12 –End of Part III –Prepare for Final Exam