Able: An adaptive block Lanczos method for non-Hermitian Eigenvalue problems

Zhaojun Bai, David Day, Qiang Ye

Research output: Contribution to journalArticlepeer-review

64 Scopus citations


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.

Original languageEnglish
Pages (from-to)1060-1082
Number of pages23
JournalSIAM Journal on Matrix Analysis and Applications
Issue number4
StatePublished - Jul 1999


  • Eigenvalue problem
  • Lanczos method
  • Non-Hermitian matrices
  • Spectral transformation

ASJC Scopus subject areas

  • Analysis


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