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

2. In the previous slide Eigenvalues and eigenvectors The power method –locate the dominant eigenvalue Inverse power method Deflation 2

3. In this slide Find all eigenvalues of a symmetric matrix –reducing a symmetric matrix to tridiagonal form –eigenvalues of symmetric tridiagonal matrices (QR algorithm) 3

4. Obtain all eigenvalues To compute all of the eigenvalues of a symmetric matrix, we will proceed in two stages –transform to symmetric tridiagonal form this step requires a fixed, finite number of operations (not iterative) –an iterative procedure on the symmetric tridiagonal matrix that generates a sequence of matrices converged to a diagonal matrix 4

5. Who two stages? Why not apply the iterative technique directly on the original matrix? –transforming an n × n symmetric matrix to symmetric tridiagonal form requires on the order of 4/3n 3 arithmetic operations –the iterative reduction of the symmetric tridiagonal matrix to diagonal form then requires O(n 2 ) arithmetic operations –on the other hand, applying the iterative technique directly to the original matrix requires on the order of 4/3n 3 arithmetic operations per iteration 5

6. 4.4 Reduction to Symmetric Tridiagonal Form 6

7. Before going into 4.4 Similarity transformations Orthogonal matrices 7

8. 8 Similarity transformation

9. 9 http://www.dianadepasquale.com/ThinkingMonkey.jpg Recall that

10. Orthogonal matrix Similarity transformation with an orthogonal matrix maintains symmetry – A is symmetric and B=Q -1 AQ – B T =(Q -1 AQ) T =(Q T AQ) T =Q T AQ=B, that is, B is also symmetric Multiplication by an orthogonal matrix does not change the Euclidean norm – (Qx) T Qx=x T Q T Qx=x T Q -1 Qx=x T x 10

11. Householder matrix 11

12. 12

13. In practice 13

14. n-2 similarity transformations 14

15. 15

16. Generating appropriate Householder matrices 16

17. 17

18. 18 later

19. Any Questions? 19 About generating Householder matrices

20. 20 Is there any possible cancellation errors?

21. 21 In action http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg

22. 22

23. 23

24. 24

25. Any Questions? 25 4.4 Reduction to Symmetric Tridiagonal Form

26. 4.5 Eigenvalues of Symmetric Tridiagonal Matrices 26

27. 27

28. 28 The off-diagonal elements  zero while the diagonal elements  the eigenvalues (in decreasing order)

29. QR Factorization The heart of the QR algorithm 29

30. Rotation matrix 30

31. QR factorization 31

32. 32

33. 33

34. 34

35. 35

36. 36 In action http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg

37. 37

38. 38

39. The Product R (i) Q (i) 39

40. 40 http://www.dianadepasquale.com/ThinkingMonkey.jpg Recall that

41. 41

42. 42

43. The algorithm for R (i) Q (i) 43

44. 44 In action http://thomashawk.com/hello/209/1017/1024/Jackson%20Running.jpg

45. 45

46. Any Questions? 46 Eigenvalues and Eigenvectors

47. Exercise 4 47 2010/5/26 9:00am Email to darby@ee.ncku.edu.tw or hand over in class. You may arbitrarily pick one problem among the first three, which means this exercise contains only five problems.darby@ee.ncku.edu.tw

48. 48

49. 49

50. 50

51. Write a program to obtain R given a symmetric tridiagonal matrix A 51

52. 52