Lecture 14 - Eigen-analysis

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Lecture 14 - Eigen-analysis

• CVEN 302
• July 10, 2002

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Lectures Goals

• QR Factorization
• Householder
• Hessenberg Method

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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
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QR Factorization
Another form of factorization
A QR
Produces an orthogonal matrix (Q) and a right
upper triangular matrix (R) Orthogonal matrix
- inverse is transpose
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QR Factorization
Why do we care? We can use Q and R to find
eigenvalues 1. Get Q and R (A QR) 2. Let A
RQ 3. Diagonal elements of A are eigenvalue
approximations 4. Iterate until converged
Note QR eigenvalue method gives all eigenvalues
simultaneously, not just the
dominant ?
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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
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QR Factorization

• Construction of QR Factorization

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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.

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Jordan Canonical Form

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

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Similarity Transformation

• Transformation of the matrix A of the form H-1AH
  is 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.

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Upper Triangular Matrix

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

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Householder Reflector

• Householder reflector is a matrix of the form
• It is straightforward to verify that Q is
  symmetric and orthogonal

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Householder Matrix

• Householder matrix reduces zk1 ,,zn to zero
• To achieve the above operation, v must be a
  linear combination of x and ek

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Householder Transformation
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Householder Matrix

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

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Householder matrix

• Define the value ? so that
• The vector w is found by
• Choose ? sign(xk)g to reduce round-off error

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Householder Matrices
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Example Householder Matrix
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Example Householder Matrix
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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)

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Example QR Factorization
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QR Factorization
QR A

• Similarity transformation B QTAQ preserve the
  eigenvalues

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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!
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QR Eigenvalue Method

• QR factorization A QR
• Similarity transformation A(new) RQ

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Example QR Eigenvalue
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Example QR Eigenvalue
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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
A1 2 -1 2 2 -1 2 -1 2 A 1 2
-1 2 2 -1 2 -1 2
Q,RQR_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 eQR_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
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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 1n-2 H Householder(A(,k),k1)
A HAH end
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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
A1 2 -1 2 2 -1 2 -1 2 A 1 2
-1 2 2 -1 2 -1 2
Q,RQR_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 eQR_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
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Summary

• QR Factorization
• Householder matrix
• Hessenberg matrix

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Interpolation
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Lectures Goals

• Interpolation methods
• Lagranges Interpolation
• Newtons Interpolation
• Hermites Interpolation
• Rational Function Interpolation
• Spline (Linear,Quadratic, Cubic)
• Interpolation of 2-D data

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Interpolation Methods
Why would we be interested in interpolation
methods?

• Interpolation method are the basis for other
  procedures that we will deal with

• Numerical differentiation
• Numerical integration
• Solution of ODE (ordinary differential equations)
  and PDE (partial differential equations)

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Interpolation Methods
Why would we be interested in interpolation
methods?

• These methods demonstrate some important theory
  about polynomials and the accuracy of numerical
  methods.
• The interpolation of polynomials serve as an
  excellent introduction to some techniques for
  drawing smooth curves.

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Interpolation Methods
Interpolation uses the data to approximate a
function, which will fit all of the data points.
All of the data is used to approximate the values
of the function inside the bounds of the data.
We will look at polynomial and rational function
interpolation of the data and piece-wise
interpolation of the data.
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Polynomial Interpolation Methods

• Lagrange Interpolation Polynomial - a
  straightforward, but computational awkward way to
  construct an interpolating polynomial.
• Newton Interpolation Polynomial - there is no
  difference between the Newton and Lagrange
  results. The difference between the two is the
  approach to obtaining the coefficients.

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Lagrange Interpolation
This method is generally the easiest to work.
The data does not have to be equally spaced and
is useful for finding the points between
quadratic and cubic methods. However, it does
not provide an accurate model for large sets of
terms.
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Lagrange Interpolation
The function can be defined as
where,
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Lagrange Interpolation
The function can be defined as
where, the coefficients are defined as
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Lagrange Interpolation
The method works for quadratic and cubic
polynomials. As you add additional points in the
degree of the polynomial increases. So if you
have n points it will fit a (n-1)th degree
polynomial.
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Example of Lagrange Interpolation
What are the coefficients of the polynomial and
what is the value of P2(2.3)?
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Example of Lagrange Interpolation

• The values are evaluated
• P(x) 9.2983(x-1.7)(x-3.0)
• - 19.4872(x-1.1)(x-3.0)
• 8.2186(x-1.1)(x-1.7)
• P(2.3) 9.2983(2.3-1.7)(2.3-3.0)
• - 19.4872(2.3-1.1)(2.3-3.0)
• 8.2186(2.3-1.1)(2.3-1.7)
• 18.3813

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Lagrange Interpolation Program
The Lagrange interpolation is broken up into two
programs to evaluate the new polynomial.

• C Lagrange_coef(x,y), which evaluates the
  coefficients of the Lagrange technique
• P(x) Lagrange_eval(t,x,c), which uses the
  coefficients and x values to evaluate the
  polynomial
• Plottest(x,y),which will plot the Lagrange
  polynomial

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Example of Lagrange Interpolation
What happens if we increase the number of data
points?
Coefficient for 2 is
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Example of Lagrange Interpolation
Note that the coefficient creates a P4(x)
polynomial and comparison between the two curves.
The original value P2(x) is given. The problem
with adding additional points will create
bulges in the graph.
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Newton Interpolation
The Newton interpolation uses a divided
difference method. The technique allows one to
add additional points easily.
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Newton Interpolation
For given set of data points (x1,y1), (x2,y2),
(x3,y3)
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Newton Interpolation
The function can be defined as
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Newton Interpolation
The method works for quadratic and cubic
polynomials. As you add additional points in the
degree of the polynomial increases. So if you
have n points it will fit a (n-1)th degree
polynomial. The method is setup to show the a
pattern for combining a table computation and
provide additional points.
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Example of Newton Interpolation
What are the coefficients of the polynomial and
what is the value of P2(2.3)?
The true function of the points is f(x) 2x
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Example of Newton Interpolation
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Example of Newton Interpolation
The coefficients are the top row of the chart
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Example of Newton Interpolation

• The values are evaluated
• P(x) 1 (x-0)
• 0.5(x-0)(x-1)
• 0.1667(x-0)(x-1)(x-2)
• 0.04167(x)(x-1)(x-2)(x-3)
• P(2.3) 1 (2.3)
• 0.5(2.3)(1.3)
• 0.1667(2.3)(1.3)(0.3)
• 0.04167(2.3)(1.3)(0.3)(-0.7)
• 4.9183 (4.9246)

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Newton Interpolation Program
The Newton interpolation is broken up into two
programs to evaluate the new polynomial.

• C Newton_coef(x,y), which evaluates the
  coefficients of the Newton technique
• P(x) Newton_eval(t,x,c), which uses the
  coefficients and x values to evaluate the
  polynomial
• Plottest_new(x,y),which will plot the Newton
  polynomial

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Summary

• The Lagrange and Newton Interpolation are
  basically the same methods, but use different
  coefficients. The polynomials depend on the
  entire set of data points.

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Homework

• Check the Homework webpage