Parallel Spectral Division via the Generalized Matrix Sign Function

In this paper we demonstrate the parallelism of the spectral division via the matrix sign function for the generalized nonsymmetric eigenproblem. We employ the so-called generalized Newton iterative scheme in order to compute the sign function of a matrix pair. A recent study has allowed considerable reduction (by 75%) in the computational cost of this iteration, making this approach competitive when compared to the traditional QZ algorithm. The matrix sign function is thus revealed as an eecient and reliable spectral division method for applications that only require partial information of the eigenspectrum. For applications which require complete information of the eigendis-tribution, the matrix sign function can be used as an initial divide-and-conquer method, combined with the QZ algorithm for the last stages. The experimental results on an IBM SP2 multicomputer demonstrate the parallel performance (eeciency around 60{80%) and scalability of this approach.