1. CPE 332 Computer Engineering Mathematics II Part III, Chapter 11 Numerical Differentiation and Integration Numerical Differentiation and Integration

2. Today Topics Chapter 11 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 (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

18. Second Derivative

20. High Accuracy Finite Divided-Difference

22. 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. 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

39. Trapezoidal Rule

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

45. Second-Order Approximation

47. 3n

48. Second-Order Approximation

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

54. Romberg Integration

66. Summary

67. Homework 10: Ch 11 Download HW Next Week – ส่งการบ้าน HW 10 –Chapter 11 Solutions of ODE –HW 11 (Option)