Quadratic Eigen value problems?

[Søren Hauberg]

Hi,
   I just stumbled upon a cool algorithm that I'd like to try out. It
requires that I solve a Quadratic Eigen Value problem. The article
mentions that I can do this in Matlab using the 'polyeig' function.
Unfortunately it seems that Octave doesn't have this function :-(
   Does anybody know how to solve such problems with Octave?

Søren
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[Fumihiro CHIBA]

Hi,  S!)ren

consider the following problem:
lambda^2 A u + lambda B  u + C u == 0,
where 0 is zero vector; lambda is unknown scalar;
u is unknwon vector; A,B and C are given matrices.

introduce auxiliary vector v as follows.
v= lambda u.
the problem is transformed to:
lambda A v + lambda B u == -C u,
lambda u == v.

matrix representation of this problem is
lambda [[B A];[I 0]] * [u;v] == [-C; I][u:v],
where I is unit matrix.
this is a generalized eigenvalue problem.
octave function qz can be applied to this problem.

Reference:
SIAM Review Vol. 43 No. 2 pp. 235--286,
F. Tisseur and K. Meerbergen, "The quadratic eigenvalue problem"

2007-07-12, 15:52 JST,  "S!)ren Hauberg" <[hidden email]> wrote:
>Hi,
>   I just stumbled upon a cool algorithm that I'd like to try out. It
>requires that I solve a Quadratic Eigen Value problem. The article
>mentions that I can do this in Matlab using the 'polyeig' function.
>Unfortunately it seems that Octave doesn't have this function :-(
>   Does anybody know how to solve such problems with Octave?
>
>S!)ren

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[Søren Hauberg]

Hi,
   Thanks a lot for the answer! I should be able to implement the
algorithm now.

Thanks,
   Søren

Fumihiro CHIBA skrev:

> Hi,  S!)ren
>
> consider the following problem:
> lambda^2 A u + lambda B  u + C u == 0,
> where 0 is zero vector; lambda is unknown scalar;
> u is unknwon vector; A,B and C are given matrices.
>
> introduce auxiliary vector v as follows.
> v= lambda u.
> the problem is transformed to:
> lambda A v + lambda B u == -C u,
> lambda u == v.
>
> matrix representation of this problem is
> lambda [[B A];[I 0]] * [u;v] == [-C; I][u:v],
> where I is unit matrix.
> this is a generalized eigenvalue problem.
> octave function qz can be applied to this problem.
>
> Reference:
> SIAM Review Vol. 43 No. 2 pp. 235--286,
> F. Tisseur and K. Meerbergen, "The quadratic eigenvalue problem"
>
> 2007-07-12, 15:52 JST,  "S!)ren Hauberg" <[hidden email]> wrote:
>> Hi,
>>   I just stumbled upon a cool algorithm that I'd like to try out. It
>> requires that I solve a Quadratic Eigen Value problem. The article
>> mentions that I can do this in Matlab using the 'polyeig' function.
>> Unfortunately it seems that Octave doesn't have this function :-(
>>   Does anybody know how to solve such problems with Octave?
>>
>> S!)ren
>

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[Fumihiro CHIBA]

Sorry, I mistook.

#Correction:
On Thursday, July 12, 2007, at 05:18PM, "Fumihiro CHIBA" <[hidden email]> wrote:
>lambda [[B A];[I 0]] * [u;v] == [-C; I][u:v],
>where I is unit matrix.
lambda [[B A]; [I 0]] * [u; v] == [-C 0; 0 I]*[u; v],
where I is unit matrix.
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[David Bateman-3]

Søren Hauberg wrote:
> Hi,
>    Thanks a lot for the answer! I should be able to implement the
> algorithm now.
>
> Thanks,
>    Søren
>  
Hey this is the basis on the polyeig function itself. If you are going
to implement something maybe writing polyeig itself might be a good idea :-)

D.

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