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Lecture 13 - Eigen-analysis. CVEN 302 July 1, 2002. Lecture’s Goals. Shift Method Inverse Power Method Accelerated Power Method QR Factorization Householder Hessenberg Method. Shift method.

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Lecture 13 eigen analysis

Lecture 13 - Eigen-analysis

CVEN 302

July 1, 2002


Lecture s goals
Lecture’s Goals

  • Shift Method

  • Inverse Power Method

  • Accelerated Power Method

  • QR Factorization

  • Householder

  • Hessenberg Method


Shift method
Shift method

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


Shift method1
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].


Example of shift method
Example of Shift Method

Consider the follow matrix A

Assume an arbitrary vector x0 = { 1 1 1}T


Example of shift method1
Example of Shift Method

Multiply the matrix by the matrix [A] by {x}

Normalize the result of the product


Example of shift method2
Example of Shift Method

Continue with the iteration and the final value is l = -5. However, to get the true you need to shift back by:


Inverse power method
Inverse Power Method

The inverse method is similar to the power method, except that it finds the smallest eigenvalue. Using the following technique.


Inverse power method1
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.


Inverse power method2
Inverse Power Method

The inverse algorithm use the technique avoids calculating the inverse matrix and uses a LU decomposition to find the {x} vector.


Example
Example

The matrix is defined as:


Matlab program
Matlab Program

  • There are set of programs Power and InversePower.

    • The InversePower(A, x0,iter,tol) does the inverse method.


Accelerated power method
Accelerated Power Method

The Power method can be accelerated by using the Rayleigh Quotient instead of the largest wk value.

The Rayleigh Quotient is defined as:


Accelerated power method1
Accelerated Power Method

The values of the next z term is defined as:

The Power method is adapted to use the new value.


Example of accelerated power method
Example of Accelerated Power Method

Consider the follow matrix A

Assume an arbitrary vector x0 = { 1 1 1}T


Example of accelerated power method1
Example of Accelerated Power Method

Multiply the matrix by the matrix [A] by {x}


Example of accelerated power method2
Example of Accelerated Power Method

Multiply the matrix by the matrix [A] by {x}




Qr factorization
QR Factorization

The technique can be used to find the eigenvalue using a successive iteration using Householder transformation to find an equivalent matrix to [A] having an eigenvalues on the diagonal


Qr factorization1
QR Factorization

Another form of factorization

A = Q*R

Produces an orthogonal matrix (“Q”) and a right upper triangular matrix (“R”)

Orthogonal matrix - inverse is transpose


Qr factorization2
QR Factorization

Why do we care?

We can use Q and R to find eigenvalues

1. Get Q and R (A = Q*R)

2. Let A = R*Q

3. Diagonal elements of A are eigenvalue

approximations

4. Iterate until converged

Note:QR eigenvalue method gives all eigenvalues

simultaneously, not just the dominant 


Qr eigenvalue method
QR Eigenvalue Method

In practice, QR factorization on any given matrix requires a number of steps

First transform A into Hessenberg form

Hessenberg matrix - upper triangular plus first sub-diagonal

Special properties of Hessenberg matrix make it easier to find Q, R, and eigenvalues


Qr factorization3
QR Factorization

  • Construction of QR Factorization


Qr factorization4
QR Factorization

  • Use Householder reflections and given rotations to reduce certain elements of a vector to zero.

  • Use QR factorization that preserve the eigenvalues.

  • The eigenvalues of the transformed matrix are much easier to obtain.


Jordan canonical form
Jordan Canonical Form

  • Any square matrix is orthogonally similar to a triangular matrix with the eigenvalues on the diagonal


Similarity transformation
Similarity Transformation

  • Transformation of the matrix A of the form H-1AHis known as similarity transformation.

  • A real matrix Q is orthogonal if QTQ = I.

  • If Q is orthogonal, then A and Q -1AQ are said to be orthogonally similar

  • The eigenvalues are preserved under the similarity transformation.


Upper triangular matrix
Upper Triangular Matrix

  • The diagonal elements Rii of the upper triangular matrix R are the eigenvalues


Householder reflector
Householder Reflector

  • Householder reflector is a matrix of the form

  • It is straightforward to verify that Qis symmetric and orthogonal


Householder matrix
Householder Matrix

  • Householder matrix reduces zk+1 ,…,zn to zero

  • To achieve the above operation, v must be a linear combination of x and ek



Householder matrix1
Householder matrix

  • Corollary (kth Householder matrix):Let A be an nxn matrix and x any vector. If k is an integer with 1< k<n-1 we can construct a vector w(k) and matrix H(k) = I - 2w(k)w’(k) so that


Householder matrix2
Householder matrix

  • Define the value  so that

  • The vector w is found by

  • Choose  = sign(xk)g to reduce round-off error





Basic qr factorization
Basic QR Factorization

  • [A] = [Q] [R]

  • [Q] is orthogonal, QTQ = I

  • [R] is upper triangular

  • QR factorization using Householder matrices

  • Q = H(1)H(2)….H(n-1)



Qr factorization5
QR Factorization

  • Similarity transformation B = QTAQ preserve the eigenvalues

QR = A


Finding eigenvalues using qr factorization
Finding Eigenvalues Using QR Factorization

  • Generate a sequence A(m) that are orthogonally similar to A

  • Use Householder transformation H-1AH

  • the iterates converge to an upper triangular matrix with the eigenvalues on the diagonal

Find all eigenvalues simultaneously!


Qr eigenvalue method1
QR Eigenvalue Method

  • QR factorization: A = QR

  • Similarity transformation: A(new) = RQ




MATLAB Example

A =

2.4634 1.8104 -1.3865

-0.0310 3.0527 1.7694

0.0616 -0.1047 -0.5161

A =

2.4056 1.8691 1.3930

0.0056 2.9892 -1.9203

0.0099 -0.0191 -0.3948

A =

2.4157 1.8579 -1.3937

-0.0010 3.0021 1.8930

0.0017 -0.0038 -0.4178

A =

2.4140 1.8600 1.3933

0.0002 2.9996 -1.8982

0.0003 -0.0007 -0.4136

A =

2.4143 1.8596 -1.3934

0.0000 3.0001 1.8972

0.0001 -0.0001 -0.4143

e =

2.4143

3.0001

-0.4143

» A=[1 2 -1; 2 2 -1; 2 -1 2]

A =

1 2 -1

2 2 -1

2 -1 2

» [Q,R]=QR_factor(A)

Q =

-0.3333 -0.5788 -0.7442

-0.6667 -0.4134 0.6202

-0.6667 0.7029 -0.2481

R =

-3.0000 -1.3333 -0.3333

0.0000 -2.6874 2.3980

0.0000 0.0000 -0.3721

» e=QR_eig(A,6);

A =

2.1111 2.0535 1.4884

0.1929 2.7966 -2.2615

0.2481 -0.2615 0.0923

QR factorization

eigenvalue


Improved qr method
Improved QR Method

  • Using similarity transformation to form an upper Hessenberg Matrix (upper triangular matrix & one nonzero band below diagonal) .

  • More efficient to form Hessenberg matrix without explicitly forming the Householder matrices (not given in textbook).

function A = Hessenberg(A)

[n,nn] = size(A);

for k = 1:n-2

H = Householder(A(:,k),k+1);

A = H*A*H;

end


Improved QR Method

A =

2.4056 -2.1327 0.9410

-0.0114 -0.4056 -1.9012

0.0000 0.0000 3.0000

A =

2.4157 2.1194 -0.9500

-0.0020 -0.4157 -1.8967

0.0000 0.0000 3.0000

A =

2.4140 -2.1217 0.9485

-0.0003 -0.4140 -1.8975

0.0000 0.0000 3.0000

A =

2.4143 2.1213 -0.9487

-0.0001 -0.4143 -1.8973

0.0000 0.0000 3.0000

e =

2.4143

-0.4143

3.0000

» eig(A)

ans =

2.4142

-0.4142

3.0000

» A=[1 2 -1; 2 2 -1; 2 -1 2]

A =

1 2 -1

2 2 -1

2 -1 2

» [Q,R]=QR_factor_g(A)

Q =

0.4472 0.5963 -0.6667

0.8944 -0.2981 0.3333

0 -0.7454 -0.6667

R =

2.2361 2.6833 -1.3416

-1.4907 1.3416 -1.7889

-1.3333 0 -1.0000

» e=QR_eig_g(A,6);

A =

2.1111 -2.4356 0.7071

-0.3143 -0.1111 -2.0000

0 0.0000 3.0000

A =

2.4634 2.0523 -0.9939

-0.0690 -0.4634 -1.8741

0.0000 0.0000 3.0000

Hessenberg matrix

eigenvalue

MATLAB function


Summary
Summary

  • Single value eigen-analysis

    • Power Method

    • Shifting technique

    • Inverse Power Method

  • QR Factorization

    • Householder matrix

    • Hessenberg matrix


Homework
Homework

  • Check the Homework webpage


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