eigen-analysis and the power method. module goals –power method –shift technique (optional)...
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Eigen-analysis and the
Power Method
Module Goals
– Power Method
– Shift technique (optional)
– Inverse Method
– Accelerated Power Method
Power method
The special advantage of the power method is that the eigenvector corresponds to the dominant eigenvalue and is generated at the same time. The inverse power method solves for the minimal eigenvalue/vector pair.
The disadvantage is that the method only supplies obtains one eigenvalue
Power MethodReaders Digest Version
Eigenvalues can be ordered in magnitude and the largest is called the dominant eigenvalue or spectral radius.
Think about how eigenvalues are a reflection of the nature of a matrix. Now if we multiply by that matrix over and over again..eventually the biggest eigenvalue will make everyone else have eigen-envy.
One λ to rule them all, One λ to find them, One λ to bring them all and in the darkness bind them.
Power Method
In general continue the multiplication:
AAAA k
where,
3knn3
k332
k221
k11
k xA
Power Method
Factor the large value term
As you continue to multiply the vector by [A]
3
k
1
nn2
k
1
2211
k1
k
xA
k as 11k1
k xA
Power Method
The basic computation of the power method is summarized as
k as 1
1
1k
1k
1k
1k
0k
0k
k
xA
xAu
Power Method
The basic computation of the power method is summarized as
lim and 1kk
1-k
1-kk
uAu
Auu
The equation can be written as:
1-k
1-k11-k11-k u
AuuAu
The Power Method Algorithm(algorithm 3.3.1 pg 107)
y=nonzero random vector
Initialize x = A*y vector
for k =1,2,…n
y=x/||x||
x =Ay (x is the approximate eigenvector)
approximate eigenvalue μ= (yT*x)/(yT*y)
r=μy-x
k++
Example of Power Method
Consider the follow matrix A
100
120
014
A
Assume an arbitrary vector x0 = { 1 1 1}T
Example of Power Method
Multiply the matrix by the matrix [A] by {x}
1
3
5
1
1
1
100
120
014
Normalize the result of the product
2.0
6.0
1
5
1
3
5
Example of Power Method
0435.0
4783.0
2174.4
0435.0
217.0
1
100
120
014
0435.0
217.0
1
6.4
2.0
1
6.4
2.0
1
6.4
2.0
6.0
1
100
120
014
0183.0
1134.0
1
2174.4
0435.0
4783.0
2174.4
Example of Power Method
0103.0
2165.0
1134.4
0183.0
1134.0
1
100
120
014
0025.0
0526.0
1
1134.4
0103.0
2165.0
1134.4
As you continue to multiple each successive vector = 4 and the vector uk={1 0 0}T
Shift method(optional)
It is possible to obtain another eigenvalue from the set equations by using a technique known as shifting the matrix.
xxA Subtract the a vector from each side, thereby changing the maximum eigenvalue
xsxIsxA
Shift method
The eigenvalue, s, is the maximum value of the matrix A. The matrix is rewritten in a form.
IAB max
Use the Power method to obtain the largest eigenvalue of [B].
Example of Power Method
Consider the follow matrix A
500
120
010
100
010
001
4
100
120
014
B
Assume an arbitrary vector x0 = { 1 1 1}T
Example of Power Method
Multiply the matrix by the matrix [A] by {x}
5
1
1
1
1
1
500
120
010
Normalize the result of the product
1
6.0
2.0
5-
5
1
1
Example of Power Method
1
12.0
04.0
5
5
6.0
2.0
5
6.0
2.0
1
2.0
2.0
500
120
010
Continue with the iteration and the final value is = -5. However, to get the true you need to shift back by:
145max
Inverse Power Method
The inverse method is similar to the power method, except that it finds the smallest eigenvalue. Using the following technique.
xxA xAxAA 11
xAx 11
xBx
Inverse Power Method
The algorithm is the same as the Power method and the “eigenvector” is not the eigenvector for the smallest eigenvalue. To obtain the smallest eigenvalue from the power method.
1
1
Inverse Power Method
The inverse method is similar to the power method, except that it finds the smallest eigenvalue. Using the following technique.
xxA xAxAA 11
xAx 11
xBx
Inverse Power Method
The algorithm is the same as the Power method and the “eigenvector” is not the eigenvector for the smallest eigenvalue. To obtain the smallest eigenvalue from the power method.
1
1
Inverse Power Method
The inverse algorithm use the technique avoids calculating the inverse matrix and uses a LU decomposition to find the {x} vector.
xxA xxUL 1
Example
512
131
024
A
The matrix is defined as:
82.1
2535.5
9264.4
Accelerated Power MethodThe Power method can be accelerated by using the Rayleigh Quotient instead of the largest wk value.
The Rayeigh Quotient is defined as:
11 zA
zz
wz
'
'1
Accelerated Power MethodThe values of the next z term is defined as:
The Power method is adapted to use the new value.
12
wz
Example of Accelerated Power Method
Consider the follow matrix A
100
120
014
A
Assume an arbitrary vector x0 = { 1 1 1}T
Example of Power Method
Multiply the matrix by the matrix [A] by {x}
1
3
5
1
1
1
100
120
014
333.23
7
1
1
1
111
1
3
5
111
1
4286.0
2857.1
1429.2
1
12
wz
Example of Accelerated Power Method
Multiply the matrix by the matrix [A] by {x}
4286.0
1429.2
8571.9
4286.0
2857.1
1429.2
100
120
014
6857.3
429.0
2857.1
143.2
429.0286.1143.2
429.0
143.2
857.9
429.02857.1142.2
2
1163.0
5814.0
6744.2
2
23
wz
Example of Accelerated Power Method
1163.0
2791.1
2791.11
1163.0
5814.0
6744.2
100
120
014
1171.43
0282.0
3107.0
7396.2
3
34
wz
Example of Accelerated Power Method
0282.0
5931.0
2689.11
0282.0
3107.0
7396.2
100
120
014
0849.44
0069.0
1452.0
7587.2
4
45
wz
And so on ...