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

2. In the previous slide Fixed point iteration scheme –what is a fixed point? –iteration function –convergence Newton’s method –tangent line approximation –convergence Secant method 2

3. In this slide Accelerating convergence –linearly convergent –Newton’s method on a root of multiplicity >1 –(exercises) Proceed to systems of equations –linear algebra review –pivoting strategies 3

4. 2.6 4 Accelerating convergence

5. 5

6. Accelerating convergence Linearly convergence Thus far, the only truly linearly convergent sequence –false position –fixed point iteration Bisection method is not according to the definition 6

7. 7

8. Aitken’s Δ 2 -method 8

9. 9

10. Aitken’s Δ 2 -method Accelerated? 10 which implies super- linearly convergence later answer

11. Aitken’s Δ 2 -method Accelerated? 11 which implies super- linearly convergence later

12. Aitken’s Δ 2 -method Accelerated? 12 which implies super- linearly convergence

13. 13

14. Any Questions? 14 About Aitken’s Δ 2 -method

15. Accelerating convergence Anything to further enhance? 15

16. 16

17. Steffensen’s method 17

18. 18 Restoring quadratic convergence to Newton’s method

19. 19

20. 20

21. Any Questions? 21

22. Two disadvantages 22 answer

23. Two disadvantages 23

24. Any Questions? 24 Chapter 2 Rootfinding (2.7 is skipped)

25. Exercise 25 Due at 2011/4/25 2:00pm Email to darby@ee.ncku.edu.tw or hand over in class. Note that the last problem includes a programming work.darby@ee.ncku.edu.tw

26. 26

27. 27

28. 28

29. 29

30. 30 (Programming)

31. Chapter 3 31 Systems of equations

32. Systems of equations Definition 32

33. 3.0 33 Linear algebra review (vectors and matrices)

34. Matrix Definitions 34

35. Any Questions? 35 m, n, i, j, E QUAL, S UM, S CALAR M ULTIPLICATION, P RODUCT …

36. The inverse matrix 36 cannot be skipped

37. 37

38. Any questions? 38 answerquestion

39. Any questions? 39 answer

40. Any questions? 40

41. The determinant 41 cannot be skipped, either

42. 42 cofactor

43. 43

44. Link the concepts All these theorems will be extremely important throughout this chapter Nonsingular matrices Determinants Solutions of linear systems of equations 44

45. 45

46. 46 (Hard to prove)

47. Any Questions? 47 3.0 Linear algebra review

48. 3.1 48 Gaussian elimination (I suppose you have already known it)

49. An application problem 49

50. 50

51. Following Gaussian elimination 51

52. Any Questions? 52 Gaussian elimination

53. Gaussian elimination Operation counts 53

54. Operation counts Comparison 54

55. 3.2 55 Pivoting strategy

56. 56

57. 57

58. 58 Compare to x 1 =1, x 2 =7, x 3 =1

59. Pivoting strategy To avoid small pivot elements A scheme for interchanging the rows (interchanging the pivot element) Partial pivoting 59

60. 60 In action http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg

61. 61

62. 62

63. Any Questions? 63

64. From the algorithm view How to implement the interchanging operation? –change implicitly Introduce a row vector r –each time a row interchange is required, we need only swap the corresponding elements of the vector –number of operations from 3n to 3 64 hint answer

65. From the algorithm view How to implement the interchanging operation? –change implicitly Introduce a row vector r –each time a row interchange is required, we need only swap the corresponding elements of the vector –number of operations from 3n to 3 65 answer

66. From the algorithm view 66

67. 67 In action http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg

68. 68 Without pivoting

69. 69

70. 70

71. 71

72. 72 Scaled partial pivoting

73. Scaled partial pivoting An example 73

74. Any Questions? 74

75. Scaled partial pivoting A blind spot of partial pivoting 75 answer

76. Scaled partial pivoting A blind spot of partial pivoting 76

77. Scaled partial pivoting 77

78. 78

79. 79 In action http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg

80. 80

81. 81

82. 82

83. Any Questions? 83 3.2 Pivoting strategy