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  Numerical Methods for Eigenvalues

Numerical Methods for Eigenvalues:

As mentioned over the ew’s as well as ev’s of an n×n matrix where n ≥ 4 should be found numerically instead of by hand. The numerical methods that are utilized in practice depend on the geometric meaning of ew’s as well as ev’s which is equation. The core of all these methods is captured in the Power method which we now introduce.

The Power Method:

In the command window of Mat lab enter the following:

> A = hilb(5)
> x = ones(5,1)
> x = A*x
>el = max(x)
> x = x/el

Evaluate the new value of x with the original. Repeat the last three lines (you can utilize the scroll up button). Evaluate the newest value of x with the previous one and the original. Notice that there is less change among the second two. Replicate the last three commands over and over until the values stop changing. You have finished what is known as the Power Method. Now try the command

> [v e] = eig(A)

The last entry in e must be the final el we computed. The previous column in v is x/norm(x). Below we describe why our commands gave this eigenvalue and eigenvector.

For example consider a 2×2 matrix whose ew’s are 1/3 and 2 and whose corresponding ev’s are v1 and v2. Let x0 be any vector which is a combination of v1 and v2 example:

x0 = v1 + v2.

Now let x1 be Atimes x0. It follows from previous equation that:

x1= Av1 + Av2
   =1/3v1 + 2v2.

Therefore the v1 part is shrunk while the v2 is stretched. If we replicate this process k times then:

893_replication of process.jpg

Evidently xk grows in the direction of v2 as well as shrinks in the direction of v1. This is the code of the Power Method vectors multiplied by A are stretched most in the direction of the ev whose ew has the largest absolute value.

The ew with the largest complete value is called the dominant ew. In several applications this quantity will necessarily be positive for physical reasons. When this is the case the Mat lab code above will work since max(v) will be the element with the largest total value. In applications where the dominant ew perhaps negative the program should use flow control to determine the correct number.

Summarizing the Power Method:

• Repeatedly multiply x by A as well as divide by the element with the largest absolute value.
• The element of largest complete value converges to largest absolute ew.
• The vector converges to the corresponding ev.

Note that this logic merely works when the eigenvalue largest in magnitude is real. If the matrix as well as starting vector are real then the power method can never give a result with an imaginary part.

Eigenvalues with imaginary part signify the matrix has a rotational component, so the eigenvector would not settle down either.

Try:

> A = rand(15,11) ;
> e = eig(A)

You be able to see that for a random square matrix many of the ew’s are complex.

Nevertheless matrices in applications are not just random. They have structure as well as this can lead to real eigenvalues as seen in the next section.

Symmetric Positive-Definite Matrices:

As noted in the previous paragraph the power method be able to fail if A has complex eigenvalues. One class of matrices that appear frequently in applications and for which the eigenvalues are always real are called the symmetric matrices. A matrix is symmetric if:

A′ = A,

That is- A is symmetric with respect to reflections about its diagonal.

Try

> A = rand(5,5)
> C = A’*A
> e = eig(C)

You are able to see that the eigenvalues of these symmetric matrices are real.

Next we consider an even additionally specialized class for which the eigenvalues aren’t only real however positive. A symmetric matrix is called the positive definite if for all vectors v 6= 0 the following holds

Av • v > 0.

Geometrically A doesn’t rotate any vector by more than 180 degrees. In summary

• If A is symmetric afterwards its eigenvalues are real.
• If A is symmetric positive definite afterwards its eigenvalues are positive numbers.

Notice that the B matrices in the preceding section were symmetric and the ew’s were all real. Notice that the Hilbert as well as Pascal matrices are symmetric.

The Inverse Power Method:

In the application of vibration analysis the mode (ev) with the least frequency (ew) is the most dangerous for the machine or structure. The Power Method gives us in its place the largest ew which is the least important frequency. In this section we initiate a method the Inverse Power Method which produces exactly what is needed.

The subsequent facts are at the heart of the Inverse Power Method

• If λ is an ew of A then 1/λ is an ew for A−1.
• The ev’s for A and A−1 are the same.

Therefore if we apply the Power Method to A−1 we will obtain the largest absolute ew of A−1 which is precisely the reciprocal of the smallest absolute ew of A. We will as well obtain the corresponding ev, which is an ev for both A−1 and A. Recollect that in the application of vibration mode analysis the smallest ew and its ev correspond exactly to the frequency and mode that we are most interested in that is the one that can do the most damage.

Here as always we don’t really want to calculate the inverse of A directly if we can help it.

Opportunely multiplying xi by A−1 to get xi+1 is equal to solving the system

Axi+1 = xi,

This can be done efficiently and accurately. Because iterating this process involves solving a linear system with the same A but several different right hand sides, it is a perfect time to use the LU decomposition to save computations. The subsequent function program does n steps of the Inverse Power Method.

function [v e] = myipm(A,n)
[L U P] = lu(A); % LU decomposition of A with pivoting
m = size(A,1);
v = ones(m,1); % make an preliminary vector with ones
for i = 1:n
pv = P*v;
y = L\pv;
v = U\y;
M = max(v);
m = min(v);
if abs(M) >= abs(m)
el = M;
else
el = m;
end
v = v/el;
end
e = 1/el;

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