Eigenvalues

1
Chapter 22

• Eigenvalues
• Appendix A

2
Eigenvalue Problems

• Engineering Problems involving vibrations,
  elasticity, oscillating systems, etc.,
• Determine the eigenvalues for n homogenous linear
  equations in n unknowns

Non-homogeneous system
homogeneous system
3
Mathematical Background

• For nontrivial solutions gt
• Characteristic polynomial det( ) fn( )

The root of fn( ) 0 are the solutions for the
eigenvalues
4
Mass-Spring System
Equilibrium positions
5
Mass-Spring System

• Homogeneous system
• Find the eigenvalues ? from det 0

6
Polynomial Method

• m1 m2 40 kg, k 200 N/m
• Characteristic equation det 0
• Two eigenvalues ?? 3.873s?1 or 2.236 s ?1
• Period Tp 2?/? 1.62 s or 2.81 s

7
Principal Modes of Vibration
Tp 1.62 s X1 ?X2
Tp 2.81 s X1 X2
8
Buckling of Column

• Axially loaded column
• Buckling modes
• M bending moment
• E modulus of elasticity
• I moment of inertia

Curvature
9
Buckling of Axially Loaded Column

• Eigenvalue problem
• Buckling loads
• Fundamental mode n 1

Euler formula
10
Buckling Modes
11
Polynomial Method

• ODE
• Finite-difference method
• Characteristic equation (2n)th-order polynomial

Which Scheme? Order of Errors?
12
Polynomial Method

• One interior node (h L/2)
• Two interior nodes (h L/3)

13
Polynomial Method

• Three interior nodes (h L/4)

14
Power Method

• Power method for finding eigenvalues
• Start with an initial guess for x
• Calculate w Ax
• Largest value (magnitude) in w is the estimate of
  eigenvalue
• Get next x by rescaling w (to avoid the
  computation of very large matrix An )
• Continue until converged

Power method also gives you eigenvectors
15
Power Method

• Start with initial guess z x0

rescaling
?k is the dominant eigenvalue
16
Power Method

• For large number of iterations, ? should converge
  to the largest eigenvalue
• The normalization make the right hand side
  converges to ? , rather than ?n

17
Example Power Method
Consider
Assume all eigenvalues are equally important,
since we dont know which one is dominant
Start with
eigenvalue eigenvector
18
Example
Current estimate for largest eigenvalue is 21
Rescale w by eigenvalue to get new x
Check Convergence (Norm lt tol?)
Norm
19

• Update the estimated eigenvector and repeat
• New estimate for largest eigenvalue is 19.381
  Rescale w by eigenvalue to get new x

Norm
20
Example
One more iteration
Norm
Convergence criterion -- Norm (or relative
error) lt tol
21
Example Power Method
22
Script file Power_eig.m
23
A2 8 10 8 3 4 10 4 7 A 2 8
10 8 3 4 10 4 7 z,m
Power_eig(A,100,0.001) it m z(1)
z(2) z(3) z(4) z(5) 1.0000 21.0000
0.9524 0.7143 1.0000 2.0000
19.3810 0.9091 0.7101 1.0000 3.0000
18.9312 0.9243 0.7080 1.0000 4.0000
19.0753 0.9181 0.7087 1.0000
5.0000 19.0155 0.9206 0.7084 1.0000
6.0000 19.0396 0.9196 0.7085 1.0000
7.0000 19.0299 0.9200 0.7085 1.0000
8.0000 19.0338 0.9198 0.7085
1.0000 9.0000 19.0322 0.9199 0.7085
1.0000 error 8.3175e-004 z z 0.9199
0.7085 1.0000 m m 19.0322
xeig(A) x -7.7013 0.6686 19.0327
MATLAB Example Power Method
eigenvector eigenvalue MATLAB function
24
MATLABs Methods

• e eig(A)
• gives eigenvalues of A
• V, D eig(A)
• eigenvectors in V(,k)
• eigenvalues Dii (diagonal matrix D)
• V, D eig(A, B) (more general eigenvalue
  problems) (Ax ?Bx)
• AV BVD

25
Inverse Power Method

• Power method give the largest eigenvalue
• Inverse Power method gives the smallest
• Eigenvalues of B A-1 are inverse of
  eigenvalues of A (i.e., ? 1/?)
• So one could use power method on w Bx to get
  largest eigenvalue of B - smallest of A
• Calculating B is wasteful - instead use

26
Inverse Power Method

• Basic power method gives the dominant eigenvalue
• Inverse power method gives the smallest
  eigenvalue

27
Script file for Inverse Power Method Use
LU_factor and LU_solve
28
A2 8 10 8 3 4 10 4 7 A 2 8
10 8 3 4 10 4 7
max_it100 tol0.001 z,m
InvPower(A,max_it,tol) L 1.0000 0
0 4.0000 1.0000 0
5.0000 1.2414 1.0000 U 2.0000
8.0000 10.0000 0 -29.0000 -36.0000
0 0 1.6897 B 2 8
10 8 3 4 10 4 7 A
2 8 10 8 3 4 10 4
7 1.0000 12.7826 0.3000 1.0000 -0.5333 it
1 2.0000 0.7123 0.1205 1.0000
-0.8013 it 3.0000 0.6687 0.1167
1.0000 -0.8152 it 4.0000 0.6686
0.1163 1.0000 -0.8155 it 4
z z 0.1163 1.0000 -0.8155 m m
0.6686 xeig(A) x -7.7013 0.6686
19.0327
eigenvector eigenvalue MATLAB function
L U
B LU
Smallest eigenvalue
29
CVEN 302-501Homework No. 15

• Finish the HW but do not hand in. I will post the
  solution on the net.