NLEIGS: A class of fully rational Krylov methods for nonlinear eigenvalue problems. A new rational Krylov method for the efficient solution of nonlinear eigenvalue problems, $A(lambda)x = 0$, is proposed. This iterative method, called fully rational Krylov method for nonlinear eigenvalue problems (abbreviated as NLEIGS), is based on linear rational interpolation and generalizes the Newton rational Krylov method proposed in [R. Van Beeumen, K. Meerbergen, and W. Michiels, SIAM J. Sci. Comput., 35 (2013), pp. A327--A350]. NLEIGS utilizes a dynamically constructed rational interpolant of the nonlinear function $A(lambda)$ and a new companion-type linearization for obtaining a generalized eigenvalue problem with special structure. This structure is particularly suited for the rational Krylov method. A new approach for the computation of rational divided differences using matrix functions is presented. It is shown that NLEIGS has a computational cost comparable to the Newton rational Krylov method but converges more reliably, in particular, if the nonlinear function $A(lambda)$ has singularities nearby the target set. Moreover, NLEIGS implements an automatic scaling procedure which makes it work robustly independently of the location and shape of the target set, and it also features low-rank approximation techniques for increased computational efficiency. Small- and large-scale numerical examples are included. From the numerical experiments we can recommend two variants of the algorithm for solving the nonlinear eigenvalue problem.

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  1. Bevilacqua, R.; Del Corso, G. M.; Gemignani, L.: Orthogonal iterations on companion-like pencils (2022)
  2. Amparan, A.; Dopico, F. M.; Marcaida, S.; Zaballa, I.: On minimal bases and indices of rational matrices and their linearizations (2021)
  3. Araújo C., Juan C.; Engström, Christian: On spurious solutions encountered in Helmholtz scattering resonance computations in (\mathbbR^d) with applications to nano-photonics and acoustics (2021)
  4. Campos, Carmen; Roman, Jose E.: NEP. A module for the parallel solution of nonlinear eigenvalue problems in SLEPc (2021)
  5. Chen, Hongjia; Xu, Kuan: On the backward error incurred by the compact rational Krylov linearization (2021)
  6. Araujo C., Juan C.; Campos, Carmen; Engström, Christian; Roman, Jose E.: Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structures (2020)
  7. Dopico, Froilán M.; Marcaida, Silvia; Quintana, María C.; Van Dooren, Paul: Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems (2020)
  8. El-Guide, Mohamed; Miȩdlar, Agnieszka; Saad, Yousef: A rational approximation method for solving acoustic nonlinear eigenvalue problems (2020)
  9. Anguas, Luis M.; Dopico, FroiláN M.; Hollister, Richard; Mackey, D. Steven: Van Dooren’s index sum theorem and rational matrices with prescribed structural data (2019)
  10. Dopico, Froilán M.; Marcaida, Silvia; Quintana, María C.: Strong linearizations of rational matrices with polynomial part expressed in an orthogonal basis (2019)
  11. Elsworth, Steven; Güttel, Stefan: Conversions between barycentric, RKFUN, and Newton representations of rational interpolants (2019)
  12. Liang, Tengfei; Wang, Junpeng; Xiao, Jinyou; Wen, Lihua: Coupled BE-FE based vibroacoustic modal analysis and frequency sweep using a generalized resolvent sampling method (2019)
  13. Amparan, A.; Dopico, F. M.; Marcaida, S.; Zaballa, I.: Strong linearizations of rational matrices (2018)
  14. Araujo-Cabarcas, Juan Carlos; Engström, Christian; Jarlebring, Elias: Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map (2018)
  15. Elias Jarlebring, Max Bennedich, Giampaolo Mele, Emil Ringh, Parikshit Upadhyaya: NEP-PACK: A Julia package for nonlinear eigenproblems - v0.2 (2018) arXiv
  16. Lietaert, Pieter; Meerbergen, Karl; Tisseur, Françoise: Compact two-sided Krylov methods for nonlinear eigenvalue problems (2018)
  17. Mele, Giampaolo; Jarlebring, Elias: On restarting the tensor infinite Arnoldi method (2018)
  18. Perović, Vasilije; Mackey, D. Steven: Linearizations of matrix polynomials in Newton bases (2018)
  19. Ringh, Emil; Mele, Giampaolo; Karlsson, Johan; Jarlebring, Elias: Sylvester-based preconditioning for the waveguide eigenvalue problem (2018)
  20. Van Beeumen, Roel; Marques, Osni; Ng, Esmond G.; Yang, Chao; Bai, Zhaojun; Ge, Lixin; Kononenko, Oleksiy; Li, Zenghai; Ng, Cho-Kuen; Xiao, Liling: Computing resonant modes of accelerator cavities by solving nonlinear eigenvalue problems via rational approximation (2018)

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