Geodesic
The solution of spectral problems for polynomial matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVI, Tome 296 (2003), pp. 122-138

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For polynomial matrices of full rank, including matrices of the form $A-\lambda I$ and $A-\lambda B$, numerical methods for solving the following problems are suggested: find the divisors of a polynomial matrix whose spectra coincide with the roots of known divisors of its characteristic polynomial; compute the greatest common divisor of a sequence of polynomial matrices; solve the inverse eigenvalue problem for a polynomial matrix. The methods proposed are based on the $\Delta W$ and $\Delta V$ factorizations of polynomial matrices. Applications of these methods to the solution of certain algebraic problems are considered. 
@article{ZNSL_2003_296_a7,
     author = {V. N. Kublanovskaya},
     title = {The solution of spectral problems for polynomial matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {122--138},
     publisher = {mathdoc},
     volume = {296},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_296_a7/}
}
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V. N. Kublanovskaya. The solution of spectral problems for polynomial matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVI, Tome 296 (2003), pp. 122-138. http://geodesic.mathdoc.fr/item/ZNSL_2003_296_a7/