09 Power Method

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Numerical Analysis

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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)

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In this slide Eigenvalues and eigenvectors

The power method

locate the dominant eigenvalue Inverse power method

Deflation

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Chapter 4

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Eigenvalues and eigenvectors


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Eigenvalues and eigenvectors Eigenvalue

= = 0 det = 0

characteristic polynomial

Eigenvector the nonzero vector for which =

associated with the eigenvalue

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In Chapter 4 Determine the dominant eigenvalue

Determine a specific eigenvalue

Remove a eigenvalue Determine all eigenvalues

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4.1

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The power method


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The power method Different problems have different requirements

a single, several or all of the eigenvalues

the corresponding eigenvectors may or may notalso 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

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The power method

Basics

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The power method

Approximated eigenvalue

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Any Questions?

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The power method

A common practice

Make the vector () have a unit

length

Why we need this step?

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question


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The power method

A common practice


length

Why we need this step?

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length

to avoid overflow and underflow

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The power method

Complete procedure

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Any Questions?

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In action

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


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what is the first estimate?


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Any Questions?

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The power method for generic matrices


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The power method for symmetric matrices


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When is symmetric

more rapid convergence

still linear, but smaller asymptotic error different scaling scheme (norm)

based on the theorem

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The power method variation

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The power method

Approximated eigenvalue

29http://www.dianadepasquale.com/ThinkingMonkey.jpgRecall that


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Any Questions?

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The power method for symmetric

matrices


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Why to

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Require the matrix to be symmetric?


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Any Questions?

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4.1 The power method


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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 canbe used in a proper coloring of a graph

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Undirected graph

The dominant eigenvalue

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Undirected graph

The corresponding eigenvector

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Any Questions?

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4.2

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The inverse power method


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The inverse power method An eigenvalue other than the dominant one To derive the inverse power method, we will

need

the relationship between the eigenvalues of amatrix to a class of matrices constructed

from With that, we can

transform an eigenvalue of the dominant

eigenvalue of =

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later


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is a polynomial of

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The inverse power method An eigenvalue other than the dominant one To derive the inverse power method, we will

need

the relationship between the eigenvalues of amatrix to a class of matrices constructed

from With that, we can

transform an eigenvalue of the dominanteigenvalue of

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How to

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Find the eigenvalue smallest in

magnitude


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Any Questions?

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4.2 The inverse power method


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4.3

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Deflation


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Deflation 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 thelargest/smallest eigenvalues?

Deflation

to remove an already determined solution,while leaving the remainder solutionsunchanged

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Within the context of polynomial rootfinding

remove each root by dividing out themonomial

6 + 11 6 = 1 5 + 6 = (

1)( 2)( 3) 5 + 6 = ( 2)( 3) is a deflation of

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+ 11 6 For the matrix eigenvalue problem

shift the previously determined eigenvalue tozero (while leaving the remainder eigenvalues

unchanged) to do this, we need the relationship among the

eigenvalues of a matrix and 52


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Deflation

Shift an eigenvalue to zero

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While leaving the remaining eigenvalues unchanged


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Deflation

Summary

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Any Questions?

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Do we

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Miss something?


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How to choose for the formula = ?


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Wielandt deflation

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l d d fl


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Wielandt deflation

Bonus

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In action

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


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Hotelling deflation

Recall that we choose , based oninfinity norm

Like the power method, there is

another deflation variation for

symmetric matrices

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Any Questions?4.3 Deflation

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09 Power Method