Numerical Analysis 1 EE, NCKU Tien-Hao Chang (Darby Chang)
In the previous slide Eigenvalues and eigenvectors The power method –locate the dominant eigenvalue Inverse power method Deflation 2
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
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
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
4.4 Reduction to Symmetric Tridiagonal Form 6
Before going into 4.4 Similarity transformations Orthogonal matrices 7
8 Similarity transformation
9 Recall that
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
Householder matrix 11
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In practice 13
n-2 similarity transformations 14
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Generating appropriate Householder matrices 16
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Any Questions? 19 About generating Householder matrices
20 Is there any possible cancellation errors?
21 In action
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Any Questions? Reduction to Symmetric Tridiagonal Form
4.5 Eigenvalues of Symmetric Tridiagonal Matrices 26
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28 The off-diagonal elements zero while the diagonal elements the eigenvalues (in decreasing order)
QR Factorization The heart of the QR algorithm 29
Rotation matrix 30
QR factorization 31
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36 In action
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The Product R (i) Q (i) 39
40 Recall that
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The algorithm for R (i) Q (i) 43
44 In action
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Any Questions? 46 Eigenvalues and Eigenvectors
Exercise /5/26 9:00am to or hand over in class. You may arbitrarily pick one problem among the first three, which means this exercise contains only five
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Write a program to obtain R given a symmetric tridiagonal matrix A 51
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