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How to solve this kind of eigenvalue problem?

  1. Apr 6, 2013 #1
    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.

    Last edited by a moderator: Apr 6, 2013
  2. jcsd
  3. Apr 6, 2013 #2
    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.

    Last edited: Apr 6, 2013
  4. Apr 6, 2013 #3


    Staff: Mentor

    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.
  5. Apr 6, 2013 #4
    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.

    Last edited: Apr 6, 2013
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