How Do You Solve Modified Eigenvalue Problems Like Lq=\lambda q + a?

[jollage]

I know that eigenvalue problem like $Lq=\lambda q$ could be easily solved by eig command in Matlab.

But how to solve a problem like $Lq=\lambda q + a$, where $a$ has the same dimension with the eigenfunction $q$?

Thanks a lot in advance.

Jo

 

[jollage]

OK, by reading the article On Inhomogeneous Eigenvalue Problems by Mattheij and Soderlind, this kind of the eigenvalue problem could be solved.

Then I have a new problem, which is $Lq=\lambda q + b\lambda^2$, where b has the same dimension with q. Note that now $\lambda$ is squared and unknown.

How to solve this problem? Thank you.

Jo

 

[Mark44]

Doesn't the article you cited give any insight? It seems odd to me that the authors would trade one problem (solving for eigenvalues in Lq = λq + a) for what seems to be a harder problem, without providing some direction.

 

[jollage]

Doesn't the article you cited give any insight? It seems odd to me that the authors would trade one problem (solving for eigenvalues in Lq = λq + a) for what seems to be a harder problem, without providing some direction.

Dear Mark,

Thank you. My apologies. I only read half of that paper, till where I have found the way to attack the problem Lq = λq + a directly. The remaining part of the paper introduces a method called power iteration.

I thought this kind of problem could be easily and directly solved with some tricks that I still do not know. (like the quadratic eigenvalue problem to be solved with a new eigenfunction, which is elegant and smart). I have a feeling that problem like these two I posted here could always be solved using some iteration methods by guessing and adjusting then guessing... I want to know the elegant and smart way, the so-called tricks.

The second problem is more complex since now λ is squared in the third term while the first problem has only a constant additional term.

Maybe I should finish that paper. Otherwise if you have some experiences and tricks, please share. Thank you.

Jo

 

[CellCoree]

There are a few different approaches you could take to solve this type of eigenvalue problem. One method is to use the same eig command in Matlab, but add the vector a to the matrix L before solving for the eigenvalues and eigenvectors. This will give you the eigenvalues and eigenvectors for the modified problem Lq = \lambda q + a.

Another approach is to use a different numerical method, such as the power method or the QR algorithm, which can handle more complex eigenvalue problems like this one. These methods may require more coding and computation, but they can provide more accurate solutions for more complicated problems.

Additionally, if you are looking for a more analytical solution, you could try using techniques such as perturbation theory or variation methods to approximate the eigenvalues and eigenvectors. These methods can be more challenging, but they can provide valuable insights into the behavior of the system and can be useful for verifying numerical solutions.

Overall, the best approach will depend on the specific problem and the resources available. I recommend exploring different methods and consulting with colleagues or experts in the field to determine the most appropriate approach for your particular situation. Good luck with your eigenvalue problem!