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Re: Polynomial eigenvalue problem
*To*: mathgroup at smc.vnet.net
*Subject*: [mg58725] Re: [mg58699] Polynomial eigenvalue problem
*From*: "Carl K. Woll" <carl at woll2woll.com>
*Date*: Fri, 15 Jul 2005 14:14:07 -0400 (EDT)
*References*: <200507150702.DAA23070@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
----- Original Message -----
From: "Alan" <info at optioncity.REMOVETHIS.net>
To: mathgroup at smc.vnet.net
Subject: [mg58725] [mg58699] Polynomial eigenvalue problem
> Does anyone know a Mathematica solution
> for the polynomial eigenvalue problem (PEP)?
>
> The PEP problem of order p . N is:
> Find the eigenvalues lambda, and eigenvectors v (each an N-vector) that
> solve
>
> (A_0 + lambda A_1 + lambda^2 A_2 + ... + lambda^p A_p) x = 0,
>
> where:
> p is a given integer,
> each coefficients A_i is a given N x N (real or complex) square matrix.
>
> There are p . N solution pairs (lambda, x) if the problem is not
> ill-posed.
> If both A_0 and A_p are singular the problem is potentially ill-posed.
>
> Thanks for any help,
> alan
>
In version 5, the linear problem can be directly solved using Eigensystem if
A_0 is nonsingular:
Eigensystem[{A_0, -A_1}]
For higher order polynomials, one idea is to proceed the same way as one
does with the usual eigenvalue problem. First some test data:
randmatrix[n_] := Table[Random[], {n}, {n}]
SeedRandom[1]
a[0] = randmatrix[3];
a[1] = randmatrix[3];
a[2] = randmatrix[3];
mat = a[0] + lambda a[1] + lambda^2 a[2];
Get the eigenvalues by setting determinant to zero:
eigs = Sort[NSolve[Det[mat]]];
Get corresponding eigenvectors by using NullSpace:
eigvecs = Join @@ NullSpace /@ (mat /. Union[eigs]);
The Sort and Union commands are there in case there are repeated
eigenvalues.
Let's check that our eigenvalues and eigenvectors satisfy your equation:
In[21]:=
Table[(mat/.eigs[[i]]) . eigvecs[[i]], {i,Length[eigs]}]//Chop
Out[21]=
{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}
Let's compare our results for the linear case:
mat = a[0] + lambda a[1];
In[23]:=
NSolve[Det[mat]]
Out[23]=
{{lambda -> -1.29964}, {lambda -> -0.97271}, {lambda -> 1.38013}}
In[24]:=
Eigenvalues[{a[0],-a[1]}]
Out[24]=
{1.38013, -1.29964, -0.97271}
Carl Woll
Wolfram Research
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