Lecture 14 - Eigen-analysis

1. Lecture 14 - Eigen-analysis CVEN 302 July 10, 2002

2. Lecture�s Goals QR Factorization Householder Hessenberg Method

3. QR Factorization

4. QR Factorization

5. QR Factorization

6. QR Eigenvalue Method

7. QR Factorization Construction of QR Factorization

8. QR Factorization Use Householder reflections and given rotations to reduce certain elements of a vector to zero. Use QR factorization that preserve the eigenvalues. The eigenvalues of the transformed matrix are much easier to obtain.

9. Jordan Canonical Form Any square matrix is orthogonally similar to a triangular matrix with the eigenvalues on the diagonal

10. Similarity Transformation Transformation of the matrix A of the form H-1AH is known as similarity transformation. A real matrix Q is orthogonal if QTQ = I. If Q is orthogonal, then A and Q -1AQ are said to be orthogonally similar The eigenvalues are preserved under the similarity transformation.

11. Upper Triangular Matrix The diagonal elements Rii of the upper triangular matrix R are the eigenvalues

12. Householder Reflector Householder reflector is a matrix of the form It is straightforward to verify that Q is symmetric and orthogonal

13. Householder Matrix Householder matrix reduces zk+1 ,�,zn to zero To achieve the above operation, v must be a linear combination of x and ek

14. Householder Transformation

15. Householder Matrix Corollary (kth Householder matrix): Let A be an nxn matrix and x any vector. If k is an integer with 1< k<n-1 we can construct a vector w(k) and matrix H(k) = I - 2w(k)w�(k) so that

16. Householder matrix Define the value ? so that The vector w is found by Choose ? = sign(xk)g to reduce round-off error

17. Householder Matrices

18. Example: Householder Matrix

19. Example: Householder Matrix

20. Basic QR Factorization [A] = [Q] [R] [Q] is orthogonal, QTQ = I [R] is upper triangular QR factorization using Householder matrices Q = H(1)H(2)�.H(n-1)

21. Example: QR Factorization

22. Similarity transformation B = QTAQ preserve the eigenvalues QR Factorization

23. Finding Eigenvalues Using QR Factorization Generate a sequence A(m) that are orthogonally similar to A Use Householder transformation H-1AH the iterates converge to an upper triangular matrix with the eigenvalues on the diagonal

24. QR Eigenvalue Method QR factorization: A = QR Similarity transformation: A(new) = RQ

25. Example: QR Eigenvalue

26. Example: QR Eigenvalue

28. Improved QR Method Using similarity transformation to form an upper Hessenberg Matrix (upper triangular matrix & one nonzero band below diagonal) . More efficient to form Hessenberg matrix without explicitly forming the Householder matrices (not given in textbook).

30. Summary QR Factorization Householder matrix Hessenberg matrix

31. Interpolation

32. Lecture�s Goals Interpolation methods Lagrange�s Interpolation Newton�s Interpolation Hermite�s Interpolation Rational Function Interpolation Spline (Linear,Quadratic, & Cubic) Interpolation of 2-D data

33. Interpolation Methods Interpolation method are the basis for other procedures that we will deal with:

34. Interpolation Methods These methods demonstrate some important theory about polynomials and the accuracy of numerical methods. The interpolation of polynomials serve as an excellent introduction to some techniques for drawing smooth curves.

35. Interpolation Methods

36. Polynomial Interpolation Methods Lagrange Interpolation Polynomial - a straightforward, but computational awkward way to construct an interpolating polynomial. Newton Interpolation Polynomial - there is no difference between the Newton and Lagrange results. The difference between the two is the approach to obtaining the coefficients.

37. Lagrange Interpolation

38. Lagrange Interpolation

39. Lagrange Interpolation

40. Lagrange Interpolation

41. Example of Lagrange Interpolation

42. Example of Lagrange Interpolation The values are evaluated P(x) = 9.2983*(x-1.7)(x-3.0) - 19.4872*(x-1.1)(x-3.0) + 8.2186*(x-1.1)(x-1.7) P(2.3) = 9.2983*(2.3-1.7)(2.3-3.0) - 19.4872*(2.3-1.1)(2.3-3.0) + 8.2186*(2.3-1.1)(2.3-1.7) = 18.3813

43. Lagrange Interpolation Program C = Lagrange_coef(x,y), which evaluates the coefficients of the Lagrange technique P(x) = Lagrange_eval(t,x,c), which uses the coefficients and x values to evaluate the polynomial Plottest(x,y),which will plot the Lagrange polynomial

44. Example of Lagrange Interpolation

45. Example of Lagrange Interpolation

46. Newton Interpolation

47. Newton Interpolation

48. Newton Interpolation

49. Newton Interpolation

50. Example of Newton Interpolation

51. Example of Newton Interpolation

52. Example of Newton Interpolation

53. Example of Newton Interpolation The values are evaluated P(x) = 1 + (x-0) + 0.5*(x-0)(x-1) 0.1667*(x-0)(x-1)(x-2) + 0.04167*(x)(x-1)(x-2)(x-3) P(2.3) = 1 + (2.3) + 0.5*(2.3)(1.3) + 0.1667*(2.3)(1.3)(0.3) + 0.04167*(2.3)(1.3)(0.3)(-0.7) = 4.9183 (4.9246)

54. Newton Interpolation Program C = Newton_coef(x,y), which evaluates the coefficients of the Newton technique P(x) = Newton_eval(t,x,c), which uses the coefficients and x values to evaluate the polynomial Plottest_new(x,y),which will plot the Newton polynomial

55. Summary The Lagrange and Newton Interpolation are basically the same methods, but use different coefficients. The polynomials depend on the entire set of data points.

56. Homework Check the Homework webpage