A Jacobi-Davidson method for two-real-parameter nonlinear eigenvalue problems arising from delay-differential equations

The critical delays of a delay-differential equation can be computed by solving a nonlinear two-parameter eigenvalue problem. The solution of this two-parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR-type method for solving such quadratic eigenvalue problem that only computes real valued critical delays, i.e. complex critical delays, which have no physical meaning, are discarded. For large scale problems, we propose new correction equations for a Newton type or Jacobi-Davidson style method, that also forces real valued critical delays. We present three different equations: one real valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi-Davidson style correction equation which is suitable for an iterative linear system solver. We also present an alternative to the recurrence relation in the case the Jacobi-Davidson method cannot be continued with a real unconverged critical delay. We show numerical examples for large scale problems arising from PDEs