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. Arrigo, Francesca; Benzi, Michele; Fenu, Caterina: Computation of generalized matrix functions (2016)
  2. Barkouki, Houda; Bentbib, A.H.; Jbilou, K.: An adaptive rational block Lanczos-type algorithm for model reduction of large scale dynamical systems (2016)
  3. Campos, Carmen; Roman, Jose E.: Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems (2016)
  4. 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)
  5. Reichel, Lothar; Rodriguez, Giuseppe; Tang, Tunan: New block quadrature rules for the approximation of matrix functions (2016)
  6. Li, Ren-Cang; Ye, Qiang: Simultaneous similarity reductions for a pair of matrices to condensed forms (2014)
  7. Paige, Christopher C.; Panayotov, Ivo; Zemke, Jens-Peter M.: An augmented analysis of the perturbed two-sided Lanczos tridiagonalization process (2014)
  8. 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)
  9. Mori, Daisuke; Yamamoto, Yusaku: Backward error analysis of the AllReduce algorithm for Householder QR decomposition (2012)
  10. Niu, Qiang; Lu, Linzhang: Deflated block Krylov subspace methods for large scale eigenvalue problems (2010)
  11. Quillen, Patrick; Ye, Qiang: A block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems (2010)
  12. Xu, Wei-Wei; Li, Wen; Ching, Wai-Ki; Chen, Yan-Mei: Backward errors for eigenproblem of two kinds of structured matrices (2010)
  13. Gutknecht, Martin H.; Schmelzer, Thomas: The block grade of a block Krylov space (2009)
  14. Baglama, James: Augmented block Householder Arnoldi method (2008)
  15. Gugercin, Serkan: An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems (2008)
  16. Gutknecht, Martin H.; Schmelzer, Thomas: Updating the QR decomposition of block tridiagonal and block Hessenberg matrices (2008)
  17. Zhou, Yunkai; Saad, Yousef: Block Krylov-Schur method for large symmetric eigenvalue problems (2008)
  18. Hoffnung, Leonard; Li, Ren-Cang; Ye, Qiang: Krylov type subspace methods for matrix polynomials (2006)
  19. Lopez, L.; Simoncini, V.: Preserving geometric properties of the exponential matrix by block Krylov subspace methods (2006)
  20. Morgan, Ronald B.: Restarted block-GMRES with deflation of eigenvalues (2005)

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