Polynomial two-parameter eigenvalue problems and matrix pencil methods   for stability of delay-differential equations
  Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called matrix pencil methods, the general ideas used as well as the proofs differ considerably. Moreover, the available theory hardly reveals the relations between the different methods.   In this work, a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem: the polynomial two-parameter eigenvalue problem, of which the quadratic two-parameter eigenvalue problem is a special case. This framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods.   We also recognize a few new matrix pencil variants to determine DDE stability.   Finally, the recognition of the new types of eigenvalue problem opens a door to efficient computation of DDE stability. 
Elias Jarlebring, Michiel E. Hochstenbach