An Algorithm for the Complete Solution of the Quartic Eigenvalue Problem

The quartic eigenvalue problem (λ 4 A +λ 3 B 2 C D + E ) x = 0 naturally arises in a plethora of applications, such as when solving the Orr–Sommerfeld equation stability analysis Poiseuille flow, theoretical and experimental design locally resonant phononic plates, modeling robot with electric motors joints, calibration catadioptric vision system, or, for example, computation guided leaky modes planar waveguide. This article proposes new numerical method full solution (all eigenvalues all left right eigenvectors) that, starting suitable linearization, uses an initial, structure-preserving reduction designed to reveal deflate certain number zero infinite before final linearization is forwarded QZ algorithm. backward error phase bounded column wise each coefficient matrix, which advantageous if matrices are graded. Numerical examples show that proposed algorithm capable computing eigenpairs small residuals, it competitive available state-of-the-art methods.