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Quartic eigenvalue problem $(\lambda^4 A + \lambda^3 B + \lambda^2C + \lambdaD + E)x = \mathbf{0}$ naturally arises e.g. when solving the Orr-Sommerfeldequation in the analysis of the stability of the {Poiseuille} flow, intheoretical analysis and experimental design of locally resonant phononicplates, modeling a robot with electric motors in the joints, calibration ofcatadioptric vision system, or e.g. computation of the guided and leaky modesof a planar waveguide. This paper proposes a new numerical method for the fullsolution (all eigenvalues and all left and right eigenvectors) that is based onquadratification, i.e. reduction of the quartic problem to a spectralyequivalent quadratic eigenvalue problem, and on a careful preprocessing toidentify and deflate zero and infinite eigenvalues before the linearizedquadratification is forwarded to the QZ algorithm. Numerical examples andbackward error analysis confirm that the proposed algorithm is superior to theavailable methods.

An algorithm for the complete solution of the quartic eigenvalue problem