Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

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.

Was this topic helpful?