# power method

• Feb 21st 2014, 07:38 AM
prasum
power method
use power method to find largest and smallest eigen value of matrix A=http://www.texify.com/img/%5CLARGE%5...trix%7D%5C.gif .Hence find all eigen values of matrix A.Also find eigen value that is farthest and nearest from 4 take initial vector as[1,1,1]

Here how to find the largest eigen value as so many eigen values can be calculated
also how to find eigen value that is farthest away fom 4 using power method

• Feb 21st 2014, 10:12 AM
phillip1882
Re: power method
Code:

```|2 -1 0|  |1|        |-1| |-1 2 -1|*|1| = 1*|0| |0 -1 2|  |1|      |1| |2 -1 0|  |-1|      |1| |-1 2 -1|*|0| = 1*|0| |0 -1 2|  |1|      |1|```
so its occilating between the two.
the middle term is 0 the top may either be -1 or 1 and the bottom is 1.
and the eigen value is 1.
• Feb 22nd 2014, 01:12 AM
BobP
Re: power method
With a start vector of $[1, 1, 1]^{T},$ constantly pre-multiplying by the given matrix produces the sequence

$[1,0,1]^{T}, [2,-2, 2]^{T}, [6, -8, 6]^{T}, [20, -28, 20]^{T}, [68, -96, 68]^{T}, [232, -328, 232]^{T}$

and so on.

It's usual to normalize at each step, but in this case it's convenient to leave all of the numbers as integers and you should, (after another two or three steps,) be able to calculate the dominant eigenvalue, (as the converging ratio of the normalizing factors).

Read your notes relating to the remainder of this question (regarding the use of the inverse matrix and of the technique of shifting), and post again if you need further help, but, you should show that you have made some effort, actually tried to answer this question.
• Feb 22nd 2014, 02:13 AM
prasum
Re: power method
for eigen value closest to 4 i used the concept if(eigen value-4) is smallest then 1/(eigen value-4) is largest and then i equated this with the largest eigen value of inverse of matrix
and got the eigen value closest to 4 but how to get eigen value farther away from 4
• Feb 22nd 2014, 02:12 PM
BobP
Re: power method
The wording of the question is very strange, in that you are asked to calculate all of the eigenvalues of the matrix and then also to find the eigenvalues furthest from and nearest to 4. Surely that doesn't require further calculation, you already have all of the eigenvalues and all that you need to do is to choose two of the three.
• Feb 22nd 2014, 08:16 PM
prasum
Re: power method
but then how will we know whether that eigen value is close or far enough but we stop when we see that our iteration is converging to an eigen value
• Feb 23rd 2014, 01:21 AM
BobP
Re: power method
Do you have the exact wording of this question, because as it is it does not make clear exactly what you are supposed to do.
The routine for finding the dominant eigenvalue (using the power method) I indicated earlier. Just how far you go with the sequence of vectors depends on the required accuracy of the eigenvalue. Have you been given a desired accuracy ?

To find the smallest (in magnitude) eigenvalue (again using the power method), requires you to use the inverse of the matrix. In practice the LU decomposition of A should be used, but that isn't made clear.

How are you supposed to find the third eigenvalue ? Again it isn't made clear. You could continue with the power method theme, shifting to the midpoint of the two eigenvalues found already and then again using the inverse of A ? Alternatively you might calculate the characteristic polynomial and deduce the third eigenvalue from that ?

You are now asked to find the eigenvalues nearest to and furthest from 4. Can you not just read those from the three that have just been calculated ? Maybe you are supposed to use a shift of 4 and to repeat the earlier calculations ? If it is to be an exercise in shifting, what's the point, haven't you already (possibly) demonstrated that to find the middle eigenvalue ? I don't know what is required.

The issue of convergence is not relevant, you should have instructions to calculate the eigenvalues to some degree of accuracy and that is what you should do.