## NLEIGS

 swMATH ID: 22547 Software Authors: Güttel S., Van Beeumen R., Meerbergen K., Michiels W. Description: 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. Homepage: http://twr.cs.kuleuven.be/research/software/nleps/nleigs.php Related Software: NLEVP; RKToolbox; Matlab; SLEPc; JDQZ; JDQR; FEAST; NEP-PACK; Eigtool; PETSc; ARPACK; DDE-BIFTOOL; NEP; deal.ii; DGGHRD; DGGHR3; Chebfun; MUMPS; UMFPACK; TDS-CONTROL Cited in: 35 Publications
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### Cited by 62 Authors

 7 Jarlebring, Elias 6 Güttel, Stefan 5 Dopico, Froilán M. 5 Michiels, Wim 5 Van Beeumen, Roel 4 Marcaida, Silvia 4 Meerbergen, Karl 3 Bai, Zhaojun 3 Engström, Christian 3 Mele, Giampaolo 3 Quintana, María C. 3 Xiao, Jinyou 2 Amparan, A. 2 Araújo C., Juan C. 2 Berljafa, Mario 2 Campos, Carmen 2 Mackey, D. Steven 2 Román, Jose E. 2 Tisseur, Françoise 2 Zaballa, Ion 2 Zhang, Chuanzeng 1 Anguas, Luis Miguel 1 Araujo-Cabarcas, Juan Carlos 1 Bevilacqua, Roberto 1 Chen, Hongjia 1 Del Corso, Gianna M. 1 El-Guide, Mohamed 1 Elsworth, Steven 1 Gaaf, Sarah W. 1 Garrett, C. Kristopher 1 Ge, Lixin 1 Gemignani, Luca 1 Hollister, Richard 1 Huang, Tsungming 1 Huang, Xin 1 Karlsson, Johan 1 Kononenko, Oleksiy 1 Li, Rencang 1 Li, Zenghai 1 Liang, Tengfei 1 Lietaert, Pieter 1 Lu, Ding 1 Marques, Osni A. 1 Meng, Shuangshuang 1 Mengi, Emre 1 Międlar, Agnieszka 1 Ng, Cho-Kuen 1 Ng, Esmond G. 1 Pérez, Javier J. 1 Perović, Vasilije 1 Ringh, Emil 1 Runborg, Olof 1 Saad, Yousef 1 Sakurai, Tetsuya 1 Su, Yangfeng 1 Van Dooren, Paul M. 1 Wang, Junpeng 1 Wen, Lihua 1 Xiao, Liling 1 Xu, Kuan 1 Yang, Chao 1 Zheng, Changjun
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### Cited in 15 Serials

 7 Linear Algebra and its Applications 5 SIAM Journal on Scientific Computing 4 SIAM Journal on Matrix Analysis and Applications 3 Journal of Computational Physics 2 Computer Methods in Applied Mechanics and Engineering 2 ACM Transactions on Mathematical Software 2 International Journal for Numerical Methods in Engineering 2 Journal of Scientific Computing 1 IMA Journal of Numerical Analysis 1 BIT 1 Journal of Computational and Applied Mathematics 1 Numerical Linear Algebra with Applications 1 ETNA. Electronic Transactions on Numerical Analysis 1 Engineering Analysis with Boundary Elements 1 Acta Numerica
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### Cited in 11 Fields

 33 Numerical analysis (65-XX) 13 Linear and multilinear algebra; matrix theory (15-XX) 7 Systems theory; control (93-XX) 6 Partial differential equations (35-XX) 5 Optics, electromagnetic theory (78-XX) 3 Approximations and expansions (41-XX) 3 Operator theory (47-XX) 1 Functions of a complex variable (30-XX) 1 Mechanics of particles and systems (70-XX) 1 Fluid mechanics (76-XX) 1 Statistical mechanics, structure of matter (82-XX)