ABLE: An Adaptive Block Lanczos Method for Non-Hermitian Eigenvalue Problems. This work presents an adaptive block Lanczos method for large-scale non-Hermitian Eigenvalue problems (henceforth the ABLE method). The ABLE method is a block version of the non-Hermitian Lanczos algorithm. There are three innovations. First, an adaptive blocksize scheme cures (near) breakdown and adapts the blocksize to the order of multiple or clustered eigenvalues. Second, stopping criteria are developed that exploit the semiquadratic convergence property of the method. Third, a well-known technique from the Hermitian Lanczos algorithm is generalized to monitor the loss of biorthogonality and maintain semibiorthogonality among the computed Lanczos vectors. Each innovation is theoretically justified. Academic model problems and real application problems are solved to demonstrate the numerical behaviors of the method.

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  1. De la Cruz Cabrera, Omar; Matar, Mona; Reichel, Lothar: Centrality measures for node-weighted networks via line graphs and the matrix exponential (2021)
  2. Pozza, Stefano; Pranić, Miroslav: The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization (2021)
  3. Alqahtani, Hessah; Reichel, Lothar: Generalized block anti-Gauss quadrature rules (2019)
  4. Alqahtani, Hessah; Reichel, Lothar: Simplified anti-Gauss quadrature rules with applications in linear algebra (2018)
  5. Alqahtani, Hessah; Reichel, Lothar: Multiple orthogonal polynomials applied to matrix function evaluation (2018)
  6. Barkouki, H.; Bentbib, A. H.; Heyouni, Mohammed; Jbilou, K.: An extended nonsymmetric block Lanczos method for model reduction in large scale dynamical systems (2018)
  7. Lietaert, Pieter; Meerbergen, Karl; Tisseur, Françoise: Compact two-sided Krylov methods for nonlinear eigenvalue problems (2018)
  8. Fenu, Caterina; Higham, Desmond J.: Block matrix formulations for evolving networks (2017)
  9. Arrigo, Francesca; Benzi, Michele; Fenu, Caterina: Computation of generalized matrix functions (2016)
  10. Barkouki, Houda; Bentbib, A. H.; Jbilou, K.: An adaptive rational block Lanczos-type algorithm for model reduction of large scale dynamical systems (2016)
  11. Campos, Carmen; Roman, Jose E.: Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems (2016)
  12. Cong, Yuhao; Li, Dongping: Block Krylov subspace methods for approximating the linear combination of (\varphi)-functions arising in exponential integrators (2016)
  13. Meng, Jing; Li, Hou-Biao; Jing, Yan-Fei: A new deflated block GCROT((m,k)) method for the solution of linear systems with multiple right-hand sides (2016)
  14. Reichel, Lothar; Rodriguez, Giuseppe; Tang, Tunan: New block quadrature rules for the approximation of matrix functions (2016)
  15. Li, Ren-Cang; Ye, Qiang: Simultaneous similarity reductions for a pair of matrices to condensed forms (2014)
  16. Paige, Christopher C.; Panayotov, Ivo; Zemke, Jens-Peter M.: An augmented analysis of the perturbed two-sided Lanczos tridiagonalization process (2014)
  17. Calandra, Henri; Gratton, Serge; Lago, Rafael; Vasseur, Xavier; Carvalho, Luiz Mariano: A modified block flexible GMRES method with deflation at each iteration for the solution of non-Hermitian linear systems with multiple right-hand sides (2013)
  18. Mori, Daisuke; Yamamoto, Yusaku: Backward error analysis of the AllReduce algorithm for Householder QR decomposition (2012)
  19. Niu, Qiang; Lu, Linzhang: Deflated block Krylov subspace methods for large scale eigenvalue problems (2010)
  20. Quillen, Patrick; Ye, Qiang: A block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems (2010)

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