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8/9/2019 Lecture04 - Eigenvalues and Eigenvectors
1/24
Solution of Linear System of Equations
Lecture 4:
Eigenvalues and Eigenvectors
MTH2212 Computational Methods and Statistics
8/9/2019 Lecture04 - Eigenvalues and Eigenvectors
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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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Dr. M. HrairiDr. M. Hrairi MTH2212MTH2212 -- Computational Methods and StatisticsComputational Methods and Statistics 33
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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Dr. M. HrairiDr. M. Hrairi MTH2212MTH2212 -- Computational Methods and StatisticsComputational Methods and Statistics 44
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
Mathematical Background
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 66
Mathematical Background
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 77
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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Dr. M. HrairiDr. M. Hrairi MTH2212MTH2212 -- Computational Methods and StatisticsComputational Methods and Statistics 88
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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Dr. M. HrairiDr. M. Hrairi MTH2212MTH2212 -- Computational Methods and StatisticsComputational Methods and Statistics 99
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1010
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1111
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1212
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1313
Example 1 - Solution
Convert the system to an eigenvalue problem
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1414
Example 1 - Solution
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1515
Example 1 - Solution
What does this mean
physically?
Valuable information
about: Period
Amplitude
1st mode
X1 = -
X2
2nd mode
X1=X2
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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1616
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1717
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1818
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 1919
Example 2 - Solution
Convert the system to an eigenvalue problem
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 2020
Example 2 - Solution
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 2121
Example 2 - Solution
Eventually converges
- Eigenvalue = 68.28427
- Eigenvector =
707107.0
1
707107.0
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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 2323
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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Dr. M. HrairiDr. M. Hrairi MTHMTH22122212 -- Computational Methods and StatisticsComputational Methods and Statistics 2424
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)