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Polynomial eigenvalue problem

`e = polyeig(A0,A1,...,Ap)`

```
[X,e] =
polyeig(A0,A1,...,Ap)
```

```
[X,e,s]
= polyeig(A0,A1,...,Ap)
```

returns
the eigenvalues for the polynomial eigenvalue problem of
degree `e`

= polyeig(`A0,A1,...,Ap`

)`p`

.

`[`

also returns
matrix `X`

,`e`

] =
polyeig(`A0,A1,...,Ap`

)`X`

, of size `n`

-by-`n*p`

,
whose columns are the eigenvectors.

`[`

additionally
returns vector `X`

,`e`

,`s`

]
= polyeig(`A0,A1,...,Ap`

)`s`

, of length `p*n`

,
containing condition numbers for the eigenvalues. At least one of `A0`

and `Ap`

must
be nonsingular. Large condition numbers imply that the problem is
close to a problem with repeated eigenvalues.

`polyeig`

handles the following simplified cases:`p = 0`

, or`polyeig(A)`

, is the standard eigenvalue problem,`eig(A)`

.`p = 1`

, or`polyeig(A,B)`

, is the generalized eigenvalue problem,`eig(A,-B)`

.`n = 0`

, or`polyeig(a0,a1,...,ap)`

, is the standard polynomial problem,`roots([ap ... a1 a0])`

, where`a0,a1,...,ap`

are scalars.

The `polyeig`

function uses the QZ factorization
to find intermediate results in the computation of generalized eigenvalues. `polyeig`

uses
the intermediate results to determine if the eigenvalues are well-determined.
See the descriptions of `eig`

and `qz`

for more information.

The computed solutions might not exist or be unique, and can
also be computationally inaccurate. If both `A0`

and `Ap`

are
singular matrices, then the problem might be ill-posed. If only one
of `A0`

and `Ap`

is singular, then
some of the eigenvalues might be `0`

or `Inf`

.

Scaling `A0,A1,...,Ap`

to have `norm(Ai)`

roughly
equal to `1`

might increase the accuracy of `polyeig`

.
In general, however, this improved accuracy is not achievable. (See
Tisseur [3] for
details).

[1] Dedieu, Jean-Pierre, and Francoise Tisseur. “Perturbation theory for homogeneous polynomial eigenvalue problems.”
*Linear Algebra Appl.* Vol. 358, 2003, pp. 71–94.

[2] Tisseur, Francoise, and Karl Meerbergen. “The quadratic eigenvalue problem.” *SIAM
Rev.* Vol. 43, Number 2, 2001, pp. 235–286.

[3] Francoise Tisseur. “Backward error and condition of polynomial eigenvalue problems.”
*Linear Algebra Appl.* Vol. 309, 2000, pp.
339–361.