Lecture 15 LU Decomposition and Eigenanalysis

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Lecture 15 - LU Decomposition and Eigen-analysis

• CVEN 302
• September 28, 2001

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Lectures Goals

• LU Decomposition with Pivoting
• What is an eigenvalue and eigenvector?
• Direct computation of the ls and eigenvectors.
• Power Method
• Shift technique

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Pivoting of the LU Decomposition

• Still need pivoting in LU decomposition
• Messes up order of L
• What to do?
• Need to pivot both L and a permutation matrix
  P
• Initialize P as identity matrix and pivot when
  A is pivoted ? Also pivot L

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Pivoting of the LU Decomposition

• Permutation matrix P
• - permutation of identity matrix I
• Permutation matrix performs bookkeeping
  associated with the row exchanges
• Permuted matrix P A
• LU factorization of the permuted matrix
• P A L U

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Permutation Matrix

• Bookkeeping for row exchanges
• Example P1 interchanges row 1 and 3
• Multiple permutations P

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LU Decomposition with Pivoting
Start with
No need to consider b in decomposition
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Forward Elimination
Interchange rows 1 4
Gaussian elimination of first column
Save -mi1 in the first column of L
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Forward Elimination
No interchange required
Gaussian elimination of second column
Save -mi2 in the second column of L
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Forward Elimination
Interchange rows 3 4
Partial pivoting for L and P
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Forward Elimination
Gaussian elimination of third column
Save -mi3 in third column of L
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Doolittle LU with Pivoting
Gauss elimination with partial pivoting
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Doolittle LU with Pivoting
Forward substitution
Back substitution
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LU Pivoting Program

• Test program
• L U PLU_pivot(A) - the program does a
  decomposition of a matrix and returns the L and
  U matrices and P matrix which represents the
  pivoting problem.

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Chapter 7 Eigen-Analysis
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Eigenvalue Problems

• In solving homogeneous linear differential
  equations, an acceptable form of solution is
  determined by an eigenvalue problem.
• To solve ODE, one must find an eigenvalue/eigenfun
  ction for which in general are infinite in
  numbers. These are different than eigen problems.

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Eigen-Analysis

• Matrix eigenvalues arise from discrete models of
  physical systems
• Discrete models
• Finite number of degrees of freedom result in a
  finite number of eigenvalues and eigenvectors.

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Eigenvalues

• Computing eigenvalues of a matrix is important in
  numerous applications
• In numerical analysis, the convergence of an
  iterative sequence involving matrices is
  determined by the size of the eigenvalues of the
  iterative matrix.
• In dynamic systems, the eigenvalues indicate
  whether a system is oscillatory, stable (decaying
  oscillations) or unstable(growing oscillation)

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Eigenvalues

• Oscillator system, the eigenvalues of
  differential equations or the coefficient matrix
  of a finite element model are directly related to
  natural frequencies of the system
• Regression analysis, eigenvectors of correlation
  matrix are used to select new predictor variables
  that are linear combinations of the original
  predictor variables.

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Physical Examples

• Natural vibration of systems of mass springs
• Flutter of the airplane wings
• vibration of membranes
• oscillation of a suspension bridge
• torsional vibration of multi-cylindrical engine
• structural response of earthquakes

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General form of the equations

• The general form of the equations

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Example

• For a set of equations

Rewrite the equations
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Example

• The equations can be

The determinant of the matrix is
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Example
Determinant

• Expand the equation

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Examples

• The equation can be factored

Eigenvalues are
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Example

• The eigenvector for l 3, can be determined by
  plug-in the equation

The matrix is singular so there are infinite
number of results.
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Example
Assume that one value of the x values is 1.
Therefore, x2 is 1. So the eigenvector for l 3
is 1, 1T.
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Example
For second eigenvalue, l -1, the equation
becomes
Assume x11 therefore x2 is -1. So the
eigenvector for l -1 is 1, -1T.
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Eigen-analysis
Unfortunately, we can not find the eigenvalues of
A general matrix by simply reducing it to a
triangular form by Gaussian elimination as we
might hope.
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Eigen-analysis
We can find the largest eigenvalue by using an
iterative procedure called the power method. Any
x vector can be represented by a combination of
the systems eigenvectors.
Multiply the equation by A for each Af is
equal to lf.
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Summary

• Eigen-analysis of the set of equations
• Finding an eigenvalue.
• Power Method
• Shifting technique
• Inverse Power Method

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Homework

• Check the Homework webpage