1. The Power Method for Finding
Scientific Computing
The Power Method for Finding
Dominant Eigenvalue

2. Eigenvalues-Eigenvectors
The eigenvectors of a matrix are vectors that satisfy
Ax = λx
Or, (A – λI)x = 0
So, λ is an eigenvalue iff det(A – λI) = 0
Example:

3. Eigenvalues-Eigenvectors
Eigenvalues are used in the solution of Engineering Problems involving vibrations, elasticity, oscillating systems, etc.
Eigenvalues are also important for the analysis of Markov Chains in statistics.
The next set of slides are from the course “Computer Applications in Engineering and Construction” at Texas A&M (Fall 2008).

4. Equilibrium positions
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 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. 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

9. k is the dominant eigenvalue
Power Method
Start with initial guess z = x0
rescaling
k is the dominant eigenvalue

10. Power Method
For large number of iterations,  should converge to the largest eigenvalue
The normalization make the right hand side converge to  , rather than n

11. Example: Power Method Consider Start with
Assume all eigenvalues are equally important, since we don’t know which one is dominant
Start with
Eigenvalue estimate
Eigenvector

12. Example
Current estimate for largest eigenvalue is 21 Rescale w by eigenvalue to get new x
Check Convergence (Norm < tol?)
Norm

13. Update the estimated eigenvector and repeat
New estimate for largest eigenvalue is Rescale w by eigenvalue to get new x
Norm

14. Convergence criterion -- Norm (or relative error) < tol
Example
One more iteration
Norm
Convergence criterion -- Norm (or relative error) < tol

15. Example: Power Method

16. Script file: Power_eig.m

17. MATLAB Example: Power Method
» [z,m] = Power_eig(A,100,0.001);
it m z(1) z(2) z(3) z(4) z(5)
error =
8.3175e-004
» z
z =
0.9199
0.7085
1.0000
» m
m =
» x=eig(A)
x =
0.6686
MATLAB Example:
Power Method
eigenvector
eigenvalue
MATLAB function

18. MATLAB’s 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

19. Theorem: If A has a complete set of eigenvectors, then the Power method converges to the dominate eigenvalue of the matrix A.
Proof: A has n eigenvalues 1,2,3,…,n with 1>2>3>…>n with a corresponding basis of eigenvectors w1,w2,w3,…,wn. Let the initial vector w0 be a linear combination of the vectors w1,w2,w3,…,wn.
w0 = a1w1+a2w2+a3w3+…+anwn
Aw0 = A(a1w1+a2w2+a3w3+…+anwn)
=a1Aw1+a2Aw2+a3Aw3+…+anAwn
=a11w1+a22w2+a33w3+…+annwn
Akw0 =a1(1)kw1+a2(2)kw2+…+an(n)kwn
Akw0/(1)k-1 =a1(1)k /(1)k-1 w1 +…+an(n)k /(1)k-1 wn

20. For large values of k (as k goes to infinity) we get the following:
At each stage of the process we divide by the dominant term of the vector. If we write w1 as shown to the right and consider what happens between two consecutive estimates we get the following.
Dividing by the dominant term gives something that is approximately 1.