Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method.

*(English)*Zbl 1237.65065Summary: Spectral discretization methods are well established methods for the computation of characteristic roots of time-delay systems. In this paper a method is presented for computing all characteristic roots in a given right half plane. In particular, a procedure for the automatic selection of the number of discretization points is described. This procedure is grounded in the connection between a spectral discretization and a rational approximation of exponential functions. First, a region that contains all desired characteristic roots is estimated. Second, the number of discretization points is selected in such a way that in this region the rational approximation of the exponential functions is accurate. Finally, the characteristic roots approximations, obtained from solving the discretized eigenvalue problem, are corrected up to the desired precision by a local method. The effectiveness and robustness of the procedure are illustrated with several examples and compared with DDE-BIFTOOL.

##### MSC:

65L03 | Numerical methods for functional-differential equations |

65L15 | Numerical solution of eigenvalue problems involving ordinary differential equations |

34K28 | Numerical approximation of solutions of functional-differential equations (MSC2010) |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

##### Keywords:

delay differential equations; characteristic root; spectral discretization; numerical examples; rational approximation of exponential functions; eigenvalue problem
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\textit{Z. Wu} and \textit{W. Michiels}, J. Comput. Appl. Math. 236, No. 9, 2499--2514 (2012; Zbl 1237.65065)

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