Higher Order Polynomial Eigenvalue Problems

where is a matrix polynomial defined as

in which the are square by matrices. A thorough study of the mathematical properties of matrix polynomials can be found in [194].

In order to make the eigenvalue
problem well defined, these matrices have to satisfy certain properties;
in particular should be nonsingular.
Similar to the quadratic problem, these problems can also be linearized
to

where

The relation between and is given by . is a block companion matrix of the PEP. The generalized eigenproblem can be solved with one of the methods discussed in Chapter 8. A disadvantage of this approach is that one has to work with larger matrices of order , and these matrices also have eigenpairs, of course. This implies that one has to check which of the computed eigenpairs satisfies the original polynomial equation. Ruhe [372] (see also Davis [102]) discussed methods that directly handle the problem (9.24), for instance, with Newton's method. For larger values of one may expect all sorts of problems with the convergence of these techniques. In §9.2.5, we have discussed a method that can be used to attack problems with large . In that approach, one first projects the given problem (9.24) onto a low-dimensional subspace and obtains a similar problem of low dimension. This low-dimensional polynomial eigenproblem can then be solved with one of the approaches mentioned above. In [221] a fourth-order polynomial problem has been solved successfully, using this reduction technique.