Numerical Analysis EE NCKU TienHao Chang Darby Chang

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

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

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

Chapter 4 Eigenvalues and eigenvectors 4

Eigenvalues and eigenvectors n 5

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

4. 1 The power method 7

The power method n Different problems have different requirements – a single, several or all of the eigenvalues – the corresponding eigenvectors may or may not also be required n n 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

The power method Basics 9

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The power method Approximated eigenvalue 12

Any Questions? 13

The power method A common practice n question 14

The power method A common practice n 15

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The power method Complete procedure 17

Any Questions? 18

http: //thomashawk. com/hello/209/1017/1024/Jackson%20 Running. jpg In action 19

what is the first estimate? 20

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Any Questions? The power method for generic matrices 24

The power method for symmetric matrices 25

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The power method variation 27

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The power method Approximated eigenvalue Recall that http: //www. dianadepasquale. com/Thinking. Monkey. jpg 29

Any Questions? The power method for symmetric matrices 30

Why to Require the matrix to be symmetric? 31

Any Questions? 4. 1 The power method 32

An application of eigenvalue 33

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

Undirected graph The dominant eigenvalue 37

Undirected graph The corresponding eigenvector 38

Any Questions? 39

4. 2 The inverse power method 40

The inverse power method n later 41

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The inverse power method n 45

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Any Questions? 47

How to Find the eigenvalue smallest in magnitude 48

Any Questions? 4. 2 The inverse power method 49

4. 3 Deflation 50

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