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

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Discussion Overview

The discussion revolves around solving modified eigenvalue problems, specifically the equations Lq = λq + a and Lq = λq + bλ², where 'a' and 'b' are terms with the same dimension as the eigenfunction 'q'. The scope includes theoretical approaches and potential methods for tackling these problems.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Jo mentions that standard eigenvalue problems can be solved using the eig command in Matlab but seeks methods for modified problems where additional terms are present.
  • Jo refers to an article by Mattheij and Soderlind, suggesting it provides a solution for the first problem but expresses uncertainty about the second problem involving a squared eigenvalue.
  • Some participants question the effectiveness of the article, noting it seems to complicate the problem without offering clear guidance.
  • Jo expresses a desire for elegant solutions or "tricks" to solve these modified eigenvalue problems, indicating a preference for iterative methods or clever approaches.
  • Jo acknowledges the complexity of the second problem due to the squared term and suggests that further reading of the referenced paper may be necessary.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to solve the modified eigenvalue problems. There are differing opinions on the utility of the referenced article, and the discussion remains unresolved regarding effective methods for the second problem.

Contextual Notes

There are limitations in the discussion, including incomplete understanding of the referenced article and the potential need for additional assumptions or methods not yet explored by participants.

jollage
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I know that eigenvalue problem like [itex]Lq=\lambda q[/itex] could be easily solved by eig command in Matlab.

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

Thanks a lot in advance.

Jo
 
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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 [itex]Lq=\lambda q + b\lambda^2[/itex], where b has the same dimension with q. Note that now [itex]\lambda[/itex] is squared and unknown.

How to solve this problem? Thank you.

Jo
 
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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.
 
Mark44 said:
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
 
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