THE SPECTRAL DECOMPOSITION OF NONSYMMETRIC MATRICES ON DISTRIBUTED-MEMORY PARALLEL COMPUTERS

The implementation and performance of a class of divide-and-conquer al gorithms for computing the spectral decomposition of nonsymmetric matr ices on distributed memory parallel computers are studied in this pape r. After presenting a general framework, we focus on a spectral divide -and-conquer (SDC) algorithm with Newton iteration. Although the algor ithm requires several times as many floating point operations as the b est serial QR algorithm, it can be simply constructed from a-small set of highly parallelizable matrix building blocks within Level 3 basic linear algebra subroutines (BLAS). Efficient implementations of these building blocks are available on a wide range of machines. In some ill -conditioned cases, the algorithm may lose numerical stability, but th is can easily be detected and compensated for. The algorithm reached 3 1% efficiency with respect to the underlying PUMMA matrix multiplicati on and 82% efficiency with respect to the underlying ScaLAPACK matrix inversion on a 256 processor Intel Touchstone Delta system, and 41% ef ficiency with respect to the matrix multiplication in CMSSL on a 32 no de Thinking Machines CM-5 with vector units. Our performance model pre dicts the performance reasonably accurately. To take advantage of the geometric nature of SDC algorithms, we have designed a graphical user interface to let the user choose the spectral decomposition according to specified regions in the complex plane.