# How to solve this kind of eigenvalue problem?

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 $q$?

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

How to solve this problem? Thank you.

Jo

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Mark44
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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.

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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