Numerical Analysis

1. Numerical Analysis EE, NCKU Tien-Hao Chang (Darby Chang)

2. Summary 1 exam, 1project and some exercises http://zoro.ee.ncku.edu.tw/na/

3. Target Solve problems with numerical methods

4. In this slide • Why numerical methods? • differences between human and computer • a very simple numerical method • What is algorithm? • definition and components • three problems and three algorithms • Convergence • compare rate of convergence

5. Why such methods? Computer is stupid

6. x-2=0 Human says, “x=2, easy!”

7. { x-2=0; } Computer says, “compilation error!”

8. What is the difference?

9. http://www.wallcoo.com/paint/Donald_Zolan_Early_Childhood_02/wallpapers/1280x1024/painting_children_kjb_DonaldZolan_68TheThinker_sm.jpghttp://www.wallcoo.com/paint/Donald_Zolan_Early_Childhood_02/wallpapers/1280x1024/painting_children_kjb_DonaldZolan_68TheThinker_sm.jpg Human is logical (thinking)

10. http://files.myopera.com/conansakura/albums/31567/thumbs/2.jpg_thumb.jpghttp://files.myopera.com/conansakura/albums/31567/thumbs/2.jpg_thumb.jpg Can do inference

11. http://www.aclibrary.org/eventkeeper/Graphics/SLZ/computer.jpghttp://www.aclibrary.org/eventkeeper/Graphics/SLZ/computer.jpg Computer is procedural (executing)

12. An example • (((x+3)-2)+6)=0 • Human requires only the rules (in this case, arithmetic), • and can inference the steps for the solution

13. Computer • (((x+3)-2)+6)=0 • Requires the exact procedure (steps) • { x0=0–6; } • { x1=x0+2; } • { x=x1–3; } • These steps is numerical method

14. Does computer have any advantage?

15. http://www.masternewmedia.org/images/fast_snail_id86636_size350.jpghttp://www.masternewmedia.org/images/fast_snail_id86636_size350.jpg It is fast

16. So, why numerical methods? • Computer is stupid • Computer is fast (and works hard) • Sometimes, stupid methods can solve difficult problems

19. is the time of death,which cannot be solved explicitly

20. We know that is no earlier than PM 7:15, and is no later than PM 8:00. So…

21. could be PM 7:38 rubbish =.=

22. A systematic procedure • Let as PM 7:38 • Evaluate the above formula • If the result exceeds 98.6, we use PM 7:27, otherwise, we use PM 7:49 instead • Repeat step 2 & 3 until the result is close to 98.6 enough

23. http://www.leda-tutorial.org/en/unofficial/Pictures/BisectionMethod.pnghttp://www.leda-tutorial.org/en/unofficial/Pictures/BisectionMethod.png Bisection method

24. Bisection method • The concept is • 1) find the mid-point, 2) evaluate it, and 3) shrink the solution range • It is stupid: just trial and error • But it works, because is ascending • And…

25. And very accurate Actually, it is getting accurate after every trial

26. When #trails Computer works hard, so it could happen

28. Algorithm The heart of numerical analysis

29. Algorithm • Definition • A precisely defined sequence of steps • In this course • design; • implement; and • examine the performance

30. How to implement?

31. By hand too painful (but you might need to)

32. With computer in other words, do programming

33. Programming Even scared!

34. Algorithm could be simple

35. An example from statistics • Mean and standard deviation on nvalues

37. In action input is

39. It is also an algorithm (a precisely defined sequence of steps)

40. Not A difficult sequence of steps

42. Another example Definite integral using trapezoidal rule

43. A partition

44. where , and

46. In action

48. Error • The analytic solution is • The absolute error is

50. Observations of the errors • , the absolute error, is a decreasing function of • When is doubled, is reduced by a factor (roughly 1/4) • From the numerical evidencewhere is independent of