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

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In the previous slide Special matrices –strictly diagonally dominant matrix –symmetric positive definite matrix Cholesky decomposition –tridiagonal matrix Iterative techniques –Jacobi, Gauss-Seidel and SOR methods –conjugate gradient method Nonlinear systems of equations (Exercise 3) 2

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In this slide Eigenvalues and eigenvectors The power method –locate the dominant eigenvalue Inverse power method Deflation 3

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Chapter 4 4 Eigenvalues and eigenvectors

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In Chapter 4 Determine the dominant eigenvalue Determine a specific eigenvalue Remove a eigenvalue Determine all eigenvalues 6

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4.1 7 The power method

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Different problems have different requirements –a single, several or all of the eigenvalues –the corresponding eigenvectors may or may not also be required To handle each of these situations efficiently, different strategies are required The power method –an iterative technique –locate the dominant eigenvalue –also computes an associated eigenvector –can be extended to compute eigenvalues 8

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The power method Basics 9

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The power method Approximated eigenvalue 12

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Any Questions? 13

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The power method A common practice 14 question

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The power method A common practice 15

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The power method Complete procedure 17

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Any Questions? 18

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19 In action

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20 what is the first estimate?

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Any Questions? 24 The power method for generic matrices

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25 The power method for symmetric matrices

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The power method variation 27

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The power method Approximated eigenvalue 29 Recall that

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Any Questions? 30 The power method for symmetric matrices

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Why to 31 Require the matrix to be symmetric?

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Any Questions? The power method

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33 An application of eigenvalue

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Undirected graph Relation to eigenvalue Proper coloring –how to color the geographic regions on a map regions that share a common border receive different colors Chromatic number –the minimum number of colors that can be used in a proper coloring of a graph 36

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Undirected graph The dominant eigenvalue 37

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Undirected graph The corresponding eigenvector 38

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Any Questions? 39

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The inverse power method

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The inverse power method 45

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Any Questions? 47

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How to 48 Find the eigenvalue smallest in magnitude

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Any Questions? The inverse power method

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Deflation

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So far, we can approximate –the dominant eigenvalue of a matrix –the one smallest in magnitude –the one closest to a specific value What if we need several of the largest/smallest eigenvalues? Deflation –to remove an already determined solution, while leaving the remainder solutions unchanged 51

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54 Recall that

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Deflation Shift an eigenvalue to zero 55

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56 While leaving the remaining eigenvalues unchanged

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Deflation Summary 59

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Any Questions? 60

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Do we 61 Miss something?

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62 Recall that

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Wielandt deflation 64

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Wielandt deflation Bonus 66

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67 In action

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Hotelling deflation 70

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Any Questions? Deflation

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