Polynomial eigenvalue problem

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

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

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

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

solves
the polynomial eigenvalue problem of degree `p`

$$\left({A}_{0}+\lambda {A}_{1}+\dots +{\lambda}^{P}{A}_{p}\right)x=0$$

where polynomial degree `p`

is a non-negative
integer, and `A0,A1,...Ap`

are input matrices of
order `n`

. The output consists of a matrix `X`

of
size `n`

-by-`n*p`

whose columns
are the eigenvectors, and a vector `e`

of length `n*p`

containing
the eigenvalues.

If `lambda`

is the `j`

th eigenvalue
in `e`

, and `x`

is the `j`

th
column of eigenvectors in `X`

, then `(A0 + lambda*A1 + ... + lambda^p*Ap)*x`

is approximately `0`

.

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

is
a vector of length `n*p`

whose elements are the eigenvalues
of the polynomial eigenvalue problem.

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

also
returns a vector `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 multiple eigenvalues.

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

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

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

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