Eigen Values and Eigen Vectors
� Eigen values and Eigen vectors are important in many areas of nu-merical computation and engineering applications.
� Defined as the zeros of the characteristic polynomial:
f (λ) = |A− λI|
� Useful in analysis of convergence characteristics of iterative meth-ods
� Ratios of Eigen values (largest to smallest) is a measure of the con-dition of a matrix.
� Applications include solutions of differential equations relating tophysical characteristics of a structure(such as principal stress, mo-ments of inertia, vibration analysis, etc.
ITCS 4133/5133: Numerical Comp. Methods 1 Regression
Application
ITCS 4133/5133: Numerical Comp. Methods 2 Regression
Application
ITCS 4133/5133: Numerical Comp. Methods 3 Regression
Determining Eigen Values: Power Method
� An iterative procedure for determining the largest eigen value
� Begin with the definition,
λx = Ax
� Assume an initial guess for z, thus
w = Az
� If z is the eigen vector, then zk = wk, else iterate.
� Each iteration, z is scaled by its largest component.
wk ≈ λzk =⇒ λ ≈wk
zk
� Given that z is scaled, λ ≈ wk
ITCS 4133/5133: Numerical Comp. Methods 4 Regression
Power Method: Algorithm
ITCS 4133/5133: Numerical Comp. Methods 5 Regression
Power Method: Example
ITCS 4133/5133: Numerical Comp. Methods 6 Regression
Accelerated Power Method
� For symmetric matrices, we can use the Rayleigh coefficient for thelambda estimate to accelerate the convergence,
λ =zTw
zTz
ITCS 4133/5133: Numerical Comp. Methods 7 Regression
Shifted Power Method
� How can we determine the other Eigen values?
� Property: If λ1, · · · , λn are the eigen values of A, then the eigenvalues of A−bI are µ1 = λ1− b, · · · , λn− b. Eigen vectors are thesame.
� If we know the eigen value λ of a A, then a second eigen value canbe found by applying the power method to the shifted matrix A asfollows:
B = A− bI
ITCS 4133/5133: Numerical Comp. Methods 8 Regression
Shifted Power Method:Example
ITCS 4133/5133: Numerical Comp. Methods 9 Regression
Determining Eigen Values: Inverse PowerMethod
� To compute the smallest Eigen value of a matrix.
� Apply power method to A−1.
� Compute the reciprocals of the Eigen values of A−1; the dominantEigen value is the smallest Eigen value of A.
� In practice, the inverse of A is not computed.
A−1z = w =⇒ Aw = z
ITCS 4133/5133: Numerical Comp. Methods 10 Regression
Inverse Power Method:Algorithm
ITCS 4133/5133: Numerical Comp. Methods 11 Regression
Inverse Power Method:Example
ITCS 4133/5133: Numerical Comp. Methods 12 Regression
ITCS 4133/5133: Numerical Comp. Methods 13 Regression
Top Related