Approximation of eigenvalue with inverse iteration method

In summary, the matrix has the following elements: 2 and 1/2, 1/2 and 3/2, and 1 and 1/2. The approximate value of $\lambda_2$ is 1.2.
  • #1
mathmari
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Hey! :giggle:

We have the matrix $\begin{pmatrix}2 & 1/2 & 1 \\ 1/2 & 3/2 & 1/2 \\ 1 & 1/2 & 2\end{pmatrix}$.
We take as initial approximation of $\lambda_2$ the $1.2$. We want to calculate this value approximately using the inverse iteration (2 steps) using as starting vector $x^{(0)}=\begin{pmatrix}1 \\ 1\\ 1\end{pmatrix}$.

At the inverse iteration method do we have to use at each step the Rayleigh-Quotient or only at the beginning and then just the power iteration ?
I think to get a better approximation that we have to use the Rayleigh-Quotient at each step. Is that correct?

:unsure:
 
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  • #2
Hey mathmari!

It seems to me that we cannot use the power iteration method, since it can only find the eigenvalue with the greatest magnitude.
Instead we can use the inverse iteration method, which finds an eigenvector given an approximation of the eigenvalue, which does not improve the eigenvalue.
Or we can use the Rayleigh quotient iteration method, which is an extension of the inverse iteration method. It improves the eigenvalue as well as the eigenvector.

Since the initial eigenvalue approximation is still a bit off, it seems to me that it's best to use the Rayleigh quotient iteration for every iteration.
Or else we should try some variants and see what works best. :unsure:
 
  • #3
Klaas van Aarsen said:
It seems to me that we cannot use the power iteration method, since it can only find the eigenvalue with the greatest magnitude.
Instead we can use the inverse iteration method, which finds an eigenvector given an approximation of the eigenvalue, which does not improve the eigenvalue.
Or we can use the Rayleigh quotient iteration method, which is an extension of the inverse iteration method. It improves the eigenvalue as well as the eigenvector.

Since the initial eigenvalue approximation is still a bit off, it seems to me that it's best to use the Rayleigh quotient iteration for every iteration.
Or else we should try some variants and see what works best. :unsure:

Ok! Thank you for your answer! 🤩
 

Related to Approximation of eigenvalue with inverse iteration method

What is the inverse iteration method for approximating eigenvalues?

The inverse iteration method is an iterative algorithm used to approximate the eigenvalues of a square matrix. It is based on the power iteration method and uses the inverse of the matrix instead of the original matrix.

How does the inverse iteration method work?

The inverse iteration method starts with an initial guess for the eigenvalue and corresponding eigenvector. It then iteratively updates the eigenvalue and eigenvector using the inverse of the matrix and the current estimate. The process continues until the eigenvalue converges to a desired accuracy.

What are the advantages of using the inverse iteration method for eigenvalue approximation?

The inverse iteration method is generally faster than other methods for approximating eigenvalues, such as the power iteration method. It also allows for the approximation of specific eigenvalues, rather than just the dominant eigenvalue. Additionally, it can handle matrices with repeated eigenvalues.

What are the limitations of the inverse iteration method?

The inverse iteration method may not converge if the initial guess for the eigenvalue is too far from the actual value. It also requires the matrix to be invertible, which may not always be the case. Additionally, the algorithm may be sensitive to rounding errors and may not always produce accurate results.

How can the accuracy of the inverse iteration method be improved?

The accuracy of the inverse iteration method can be improved by using a better initial guess for the eigenvalue, such as an estimate based on the Gershgorin circle theorem. It can also be improved by using a more precise method for inverting the matrix, such as LU decomposition.

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