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

2. In the previous slide 2

3. In this slide Special matrices –strictly diagonally dominant matrix –symmetric positive definite matrix Cholesky decomposition –tridiagonal matrix Iterative techniques –Jacobi, Gauss-Seidel and SOR methods –conjugate gradient method Nonlinear systems of equations (Exercise 3) 3

4. 3.7 4 Special matrices

5. Linear systems –which arise in practice and/or in numerical methods –the coefficient matrices often have special properties or structure Strictly diagonally dominant matrix Symmetric positive definite matrix Tridiagonal matrix 5

6. Strictly diagonally dominant 6

7. 7

8. Symmetric positive definite 8

9. Symmetric positive definite Theorems for verification 9

10. 10

11. Symmetric positive definite Relations to Eigenvalues Leading principal sub-matrix 11

12. Cholesky decomposition 12

13. 13

14. 14

15. Tridiagonal 15

16. 16

17. Any Questions? 17 3.7 Special matrices

18. Before entering 3.8 So far, we have learnt three methods algorithms in Chapter 3 –Gaussian elimination –LU decomposition –direct factorization Are they algorithms? What’s the differences to those algorithms in Chapter 2? –they report exact solutions rather than approximate solutions 18 question further question answer

19. Before entering 3.8 So far, we have learnt three methods algorithms in Chapter 3 –Gaussian elimination –LU decomposition –direct factorization Are they algorithms? What’s the differences to those algorithms in Chapter 2? –they report exact solutions rather than approximate solutions 19 further question answer

20. Before entering 3.8 So far, we have learnt three methods algorithms in Chapter 3 –Gaussian elimination –LU decomposition –direct factorization Are they algorithms? What’s the differences to those algorithms in Chapter 2? –they report exact solutions rather than approximate solutions 20 answer

21. Before entering 3.8 So far, we have learnt three methods algorithms in Chapter 3 –Gaussian elimination –LU decomposition –direct factorization Are they algorithms? What’s the differences to those algorithms in Chapter 2? –they report exact solutions rather than approximate solutions 21

22. 3.8 22 Iterative techniques for linear systems

23. Iterative techniques 23

24. Iterative techniques Basic idea 24

25. Iteration matrix Immediate questions 25

26. 26

27. 27

28. 28

29. (in section 2.3 with proof) 29 http://www.dianadepasquale.com/ThinkingMonkey.jpg Recall that

30. 30 http://www.dianadepasquale.com/ThinkingMonkey.jpg Recall that

31. Iteration matrix For these questions 31 question hint answer

32. Iteration matrix For these questions 32 hint answer

33. Iteration matrix For these questions 33 answer

34. Iteration matrix For these questions 34

35. Splitting methods 35

36. Splitting methods 36

37. 37

38. 38

39. Gauss-Seidel method 39

40. Gauss-Seidel method Iteration matrix 40

41. 41 The SOR method (successive overrelaxatoin)

42. Any Questions? 42 Iterative techniques for linear systems

43. 3.9 Conjugate gradient method 43

44. Conjugate gradient method Not all iterative methods are based on the splitting concept The minimization of an associated quadratic functional 44

45. Conjugate gradient method Quadratic functional 45

46. 46 http://fuzzy.cs.uni-magdeburg.de/~borgelt/doc/somd/parabola.gif

47. 47

48. Minimizing quadratic functional 48

49. 49

50. 50 Global optimization problem http://www.mathworks.com/cmsimages/op_main_wl_3250.jpg

51. Any Questions? 51 Conjugate gradient method

52. 3.10 52 Nonlinear systems of equations

53. 53

54. Generalization of root-finding 54

55. Generalization Newton’s method 55

56. Generalization of Newton’s method Jacobian matrix 56

57. 57

58. 58 A lots of equations bypassed… http://www.math.ucdavis.edu/~tuffley/sammy/LinAlgDEs1.jpg

59. 59 And this is a friendly textbook :)

60. Any Questions? 60 Nonlinear systems of equations

61. Exercise 3 61 2011/5/2 2:00pm Email to darby@ee.ncku.edu.tw or hand over in class. Note that the fourth problem is a programming work.darby@ee.ncku.edu.tw

62. 62

63. 63

64. 64

65. Implement LU decomposition 65

66. 66