  Quartic eigenvalue problem $(\lambda^4 A + \lambda^3 B + \lambda^2C + \lambda
D + E)x = \mathbf{0}$ naturally arises e.g. when solving the Orr-Sommerfeld
equation in the analysis of the stability of the {Poiseuille} flow, in
theoretical analysis and experimental design of locally resonant phononic
plates, modeling a robot with electric motors in the joints, calibration of
catadioptric vision system, or e.g. computation of the guided and leaky modes
of a planar waveguide. This paper proposes a new numerical method for the full
solution (all eigenvalues and all left and right eigenvectors) that is based on
quadratification, i.e. reduction of the quartic problem to a spectraly
equivalent quadratic eigenvalue problem, and on a careful preprocessing to
identify and deflate zero and infinite eigenvalues before the linearized
quadratification is forwarded to the QZ algorithm. Numerical examples and
backward error analysis confirm that the proposed algorithm is superior to the
available methods.
