1. Today’s class Numerical Integration Newton-Cotes Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

2. Discuss Mid-term Exam 1 Average = 74.5, Standard dev = 17.6
Histogram of scores
Average = 74.5, Standard dev = 17.6
Max = 98 Min = 35
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

3. Discuss Mid-term Exam 1
1 (30) Use Taylor series to approximate at x=0.5 until t  1%, using a base point x=0. (Use 4 significant digits with rounding, and the true value is )
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

4. Discuss Mid-term Exam 1
2 (30) Use the false-position method with initial guesses of 50 and 70 to determine x to a level of a  0.5%. (Use 4 significant digits with rounding)
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

5. Discuss Mid-term Exam 1
3 (20) Use the Newton-Raphson method with an initial guess of 6 to determine a root of the following function to a level of a  0.01% (use 4 significant digits with rounding)
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

6. Discuss Mid-term Exam 1
4 (20) Use the Gauss-Seidel method to solve the following linear equation system to a tolerance of s = 5% with a starting point (0, 0, 0).(Use 4 significant digits with rounding)
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

7. Numerical Integration
Numerical technique to solve definite integrals
Find the area under the curve f(x) from a to b
Another way to put it is that we need to solve the following differential equation
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

8. Numerical Integration
Where do we use numerical techniques?
When you can’t integrate directly
When you have a sampling of points representing f(x) as with the experiments results
Graphical techniques
Grid approximation
Strip approximation
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

9. Graphical Techniques Grid approximation Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

10. Graphical Techniques Strip approximation Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

11. Numerical Integration
Numerical techniques
Treat integral as a summation
Basic idea is to convert a continuous function into discrete function
The smaller the interval, the more accurate the solution but also at more computational expense
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

12. Numerical Integration
Newton-Cotes formulas
Approximate function with a series of polynomials that are easy to integrate
Zero-order approximation is equivalent to strip approximation
First order approximation is equivalent to trapezoidal approximation
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

13. Newton-Cotes Formulas
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

14. Newton-Cotes Formulas
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

15. Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12
CSE, UConn

16. Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12
CSE, UConn

17. Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12
CSE, UConn

18. Trapezoidal Rule Evaluate integral Analytical solution
Trapezoidal solution
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

19. Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12
CSE, UConn

20. Trapezoidal Rule Trapezoidal solution
To increase accuracy, shorten up the interval
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

21. Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12
CSE, UConn

22. Trapezoidal Rule
If you split the interval into n sub-intervals, each sub-interval has a width of h = (b – a)/n
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

23. Trapezoidal Rule More accurate as you increase n
Double n and reduce the error by a factor of four
However, if n is too large, you will start encountering round-off error and the integral solution can diverge
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

24. Simpson’s Rules
Second and third-order polynomial forms of the Newton-Cotes formulas
Simpson’s 1/3 Rule
Use three points on the curve to form a second order polynomial
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

25. Simpson’s Rules Approximate the function by a parabola
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

26. Simpson’s Rules
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

27. Simpson’s Rules
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

28. Simpson’s 1/3 Rule Error Multiple Application n must be even
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

29. Simpson’s 3/8-Rule Approximate with a cubic polynomial
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

30. Simpson’s 3/8-Rule Numerical Methods Prof. Jinbo Bi Lecture 12
CSE, UConn

31. Simpson’s 3/8-Rule Error Multiple Application
n must be a multiple of 3
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

32. Simpson’s 3/8-Rule
Both the cubic and quadratic approximations are of the same order
The cubic approximation is slightly more accurate then the quadratic approximation
Usually not worth the extra work required
Only use the 3/8 rule if you need odd number of segments
Can combine with 1/3 rule on some segments
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn 32

33. Example
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

34. Integration with unequal segments
We have until now assumed that each segment is equal
If we are integrating based on experimental or tabular data, this assumption may not be true
Newton-Cotes formulas can easily be adapted to accommodate unequal segments
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

35. Trapezoid Rule with Unequal Segments
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

36. Trapezoidal Rule Numerical Methods Prof. Jinbo Bi Lecture 12
CSE, UConn

37. Simpson’s Rule with Unequal Segments
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn

38. Next class Numerical Integration HW5 due Tuesday Oct 21
Numerical Methods
Lecture 12
Prof. Jinbo Bi
CSE, UConn