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
where polynomial degree
p is a non-negative
A0,A1,...Ap are input matrices of
n. The output consists of a matrix
n*p whose columns
are the eigenvectors, and a vector
e of length
lambda is the
x is the
column of eigenvectors in
(A0 + lambda*A1 + ... + lambda^p*Ap)*x is approximately
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
condition numbers for the eigenvalues. At least one of
be nonsingular. Large condition numbers imply that the problem is
close to a problem with multiple eigenvalues.
Based on the values of
several special cases:
p = 0, or
the standard eigenvalue problem:
p = 1,
polyeig(A,B) is the generalized eigenvalue problem:
n = 1, or
a0,a1,...,ap is the standard polynomial
roots([ap ... a1 a0]).
singular the problem is potentially ill-posed. Theoretically, the
solutions might not exist or might not be unique. Computationally,
the computed solutions might be inaccurate. If one, but not both,
Ap is singular, the
problem is well posed, but some of the eigenvalues might be zero or
Note that scaling
A0,A1,..,Ap to have
equal 1 may increase the accuracy of
general, however, this cannot be achieved. (See Tisseur  for more detail.)
 Dedieu, Jean-Pierre Dedieu and Francoise Tisseur, "Perturbation theory for homogeneous polynomial eigenvalue problems," Linear Algebra Appl., Vol. 358, pp. 71-94, 2003.
 Tisseur, Francoise and Karl Meerbergen, "The quadratic eigenvalue problem," SIAM Rev., Vol. 43, Number 2, pp. 235-286, 2001.
 Francoise Tisseur, "Backward error and condition of polynomial eigenvalue problems" Linear Algebra Appl., Vol. 309, pp. 339-361, 2000.