Eigen-analysis and the Power Method

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Eigen-analysis and the Power Method
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Module Goals

• Power Method
• Shift technique (optional)
• Inverse Method
• Accelerated Power Method

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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
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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 ? to
rule them all, One ? to find them, One ? to
bring them all and in the darkness bind them.
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Power Method
In general continue the multiplication
where,
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Power Method
Factor the large l value term
As you continue to multiply the vector by A
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Power Method
The basic computation of the power method is
summarized as
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Power Method
The basic computation of the power method is
summarized as
The equation can be written as
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The Power Method Algorithm(algorithm 3.3.1 pg
107)
ynonzero random vector Initialize x Ay
vector for k 1,2,n yx/x x Ay (x
is the approximate eigenvector) approximate
eigenvalue µ (yTx)/(yTy) rµy-x k
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Example of Power Method
Consider the follow matrix A
Assume an arbitrary vector x0 1 1 1T
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Example of Power Method
Multiply the matrix by the matrix A by x
Normalize the result of the product
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Example of Power Method
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Example of Power Method
As you continue to multiple each successive
vector l 4 and the vector uk1 0 0T
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Shift method(optional)
It is possible to obtain another eigenvalue from
the set equations by using a technique known as
shifting the matrix.
Subtract the a vector from each side, thereby
changing the maximum eigenvalue
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Shift method
The eigenvalue, s, is the maximum value of the
matrix A. The matrix is rewritten in a form.
Use the Power method to obtain the largest
eigenvalue of B.
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Example of Power Method
Consider the follow matrix A
Assume an arbitrary vector x0 1 1 1T
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Example of Power Method
Multiply the matrix by the matrix A by x
Normalize the result of the product
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Example of Power Method
Continue with the iteration and the final value
is l -5. However, to get the true you need to
shift back by
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Inverse Power Method
The inverse method is similar to the power
method, except that it finds the smallest
eigenvalue. Using the following technique.
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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.
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Inverse Power Method
The inverse method is similar to the power
method, except that it finds the smallest
eigenvalue. Using the following technique.
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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.
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Inverse Power Method
The inverse algorithm use the technique avoids
calculating the inverse matrix and uses a LU
decomposition to find the x vector.
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Example
The matrix is defined as
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Accelerated Power Method
The Power method can be accelerated by using the
Rayleigh Quotient instead of the largest wk
value. The Rayeigh Quotient is defined as
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Accelerated Power Method
The values of the next z term is defined
as The Power method is adapted to use the new
value.
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Example of Accelerated Power Method
Consider the follow matrix A
Assume an arbitrary vector x0 1 1 1T
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Example of Power Method
Multiply the matrix by the matrix A by x
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Example of Accelerated Power Method
Multiply the matrix by the matrix A by x
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Example of Accelerated Power Method
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Example of Accelerated Power Method
And so on ...