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

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:

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).

Equilibrium positions Mass-Spring System Equilibrium positions

Mass-Spring System Homogeneous system Find the eigenvalues  from det[ ] = 0

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

Principal Modes of Vibration Tp = 1.62 s X1 = X2 Tp = 2.81 s X1 = X2

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

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

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

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

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

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

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

Example: Power Method

Script file: Power_eig.m

MATLAB Example: Power Method 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 » x=eig(A) x = -7.7013 0.6686 19.0327 MATLAB Example: Power Method eigenvector eigenvalue MATLAB function

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

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

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.