Numerische Mathematik

, Volume 76, Issue 3, pp 279-308

First online:

An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

  • Zhaojun BaiAffiliated withDepartment of Mathematics, University of Kentucky, Lexington, KY 40506, USA
  • , James DemmelAffiliated withComputer Science Division and Mathematics Department, University of California, Berkeley, CA 94720, USA
  • , Ming GuAffiliated withDepartment of Mathematics and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA

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We discuss an inverse-free, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix \(A\), or a pair of left and right deflating subspaces of a regular matrix pencil \(A - \lambda B\). This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm.

Mathematics Subject Classification (1991):65F15