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Lecture 14 - Eigen-analysis

Lecture's Goals . QR FactorizationHouseholder Hessenberg Method. QR Factorization. The technique can be used to find the eigenvalue using a successive iteration using Householder transformation to find an equivalent matrix to [A] having an eigenvalues on the diagonal. QR Factorization. Another f

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Lecture 14 - Eigen-analysis

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    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

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