Lecture04 - Eigenvalues and Eigenvectors

8/9/2019 Lecture04 - Eigenvalues and Eigenvectors

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Solution of Linear System of Equations

Lecture 4:

Eigenvalues and Eigenvectors

MTH2212 Computational Methods and Statistics


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Dr. M. HrairiDr. M. Hrairi MTH2212MTH2212 -- Computational Methods and StatisticsComputational Methods and Statistics 22

Objectives

Introduction

Mathematical background

Physical background

Polynomial Method

Power Method


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Introduction

Eigenvalue problems are a special class of problems that

are common in engineering contexts involving vibrations

and elasticity.

Many systems of ODEs can be reduced to eigenvalue

problems.


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Mathematical Background

So far we learned to solve [A]{x}={b}

Such systems are called nonhomogeneous because of the presence

of{b}.

Ifdet[A] 0 unique solution of{x}

Homogeneous systems has the general form [A]{x}=0

Nontrivial solutions of such systems are possible but

generally they are not unique.


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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 55


Eigenvalue problems are of the general form:

Pis the unknown parameter called the eigenvalue or

characteristic value.

A solution {x1,x2, ,xn} for such a system is referred to as an

eigenvector.

0)(

0)(

0)(

2211

2222121

1212111

!

!

!

nnnnn

nn

nn

xaxaxa

xaxaxa

xaxaxa

P

P

P

.

////

.

.


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The set of equations may also be expressed as:

The determinant of the matrix [[A]-P[I]] must equal to zero for

nontrivial solutions to be possible.

Expanding the determinant yields a polynomial in P.

The roots of this polynomial are the solutions to the eigenvalues.

? A ? A? A_ a 0!P xIA


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Physical Background

The following mass-spring system is a simple illustration of how

eigenvalues occur in engineering context.

)( 12121

2

1 xxkkxdt

xdm !

2122

22

2 )( kxxxkdt

xd!

Force balance for each mass is

developed using Newtonssecond law:


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Physical Background

From vibration theory, the solutions to these equations

where

Xi is the amplitude of the vibration of mi

is the frequency of the vibration given by

Tp is the period.

)sin( tXx ii [!

pT

T![

2


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Physical Background

This system of equations can be converted to an eigenvalue

problem of vibrations.

02

211

2

1

!

[ X

m

kX

m

k

02

2

2

2

1

2

!

[ X

m

kX

m

k


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The PolynomialMethod

When dealing with complicated systems or systems withheterogeneous properties, analytical solutions are often difficult orimpossible to obtain.

Numerical solutions to such equations may be the only practicalalternatives.

These equations can be solved by substituting a central finite-

divided difference approximation for the derivatives.

Writing this equation for a series of nodes yields a homogeneoussystem of equations.


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The PolynomialMethod Procedure

Convert the system to an eigenvalue problem

[[A]- [I]]{x}= 0

Expand determinant det[[A]- [I]]= 0.This will yield a

polynomial in .

Solve for

For each value of, establish the relationship between the

unknowns xs called an eigenvector (note no unique solution).


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Example 1

Use the polynomial method to evaluate the eigenvalues and

eigenvectors of the spring-mass example for the case where

m1 = m2 = 40 kg and k = 200 N/m.


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Example 1 - Solution


Expand determinant det[[A]- [I]]= 0.

Solve for2

2 = 15 and 2 = 5 s-2

The frequencies for the vibrations of the masses are = 3.873 and = 2.236 s-1

0510 212 ![ XX

0105 22

1!

[ XX

07520)( 222 ![[


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The periods for the vibrations

Tp= 1.62 s and Tp= 2.81 s

For each value of2, plug into matrix equation to solve for

eigenvectors Xs.

- For the first mode (2 = 15)

X1 = - X2

- Similarly, for the second mode (2 = 5) X1 =X2

051510 21 ! XX

015105 21 ! XX


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What does this mean

physically?

Valuable information

about: Period

Amplitude

1st mode

X1 = -

X2

2nd mode

X1=X2


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The PowerMethod

An iterative approach that can be employed to determine the

largest eigenvalue.

With slight modification, it can also be used to determine thesmallest eigenvalue.

To determine the largest eigenvalue the system must be expressed

in the form:? A_ a _ aXXA P!


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The PowerMethod Procedure

Rearrange equations so that we have:

Plug in an initial guess for LHS X .Assume all the Xs on the LHS

of the equations are equal to 1.

Solve forRHS.

Pull scalar out of RHS so maximum value in vector is equal

to 1.

Plug eigenvector back into LHS and repeat until eigenvalue

converges with a < s

? A_ a _ aXXA P!


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Example 2

Employ the power method to determine the highest eigenvalue

and its associated eigenvector of a three mass-four spring

system for the case where m1 = m2 = m3 = 1 kg and k1 = k2 =

k3 = k4 = k = 20 N/m.


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07520)( 222 ![[

02

2

1

1

2

1

!

[ X

m

kX

m

k

02

3

2

22

2

1

2

!

[ X

m

kX

m

kX

m

k

02 323

2

3!

[ XmkX

mk


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Substitute the values ofms and ks and express the system

in the matrix form

P!

-

3

2

1

3

2

1

40200

204020

02040

X

X

X

X

X

X

? A_ a _ aXXA P!


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Eventually converges

- Eigenvalue = 68.28427

- Eigenvector =

707107.0

1

707107.0


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Assignment #2

Computational Methods

27.11, 28.25, 28.27

Statistics

2.83, 2.86, 2.88, 2.108, 2.120


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Quiz #2

Solve the following system of equations using Gauss-Seidel

method

20 x1 + 2 x2 5 x3 = 13 (1)

5 x1 20 x2 + 2 x3 = 27 (2)

4 x1 + 5 x2 + 20 x3 = 19 (3)

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Lecture04 - Eigenvalues and Eigenvectors