Inverse Free Parallel Spectral Divide and Conquer Algorithms for Nonsymmetric Eigenproblems

Abstract

We discuss two inverse free, highly parallel, spectral divide and conquer algorithms: one for computing an invariant subspace of a nonsymmetric matrix and another one for computing left and right deflating subspaces of a regular matrix pencil A 0 B. These two closely related algorithms are based on earlier ones of Bulgakov, Godunov and Malyshev, but improve on them in several ways. These algorithms only use easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition. The existing parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which is faster but can be less stable than the new algorithm. Appears also as ffl Research Report 94-01, Department of Mathematics, University of Kentucky. ffl University of California at Berkeley, Computer Science Division Report UCB//CSD-94-793. It is also available by anonymous ftp on tr-ftp.cs.berkeley.edu, in directory pub/tech-reports/cs/csd94 -793. ffl Lawrence Berkeley Laboratory..