Personnel at Rice U.
In addition to the standard drivers, we developed a special driver for computing a partial Singular Value Decomposition and also one for banded problems. We also added the capability to use complex (single or double precision) arithmetic. These additions greatly increased the applicability of ARPACK.
We implemented a matlab version called ``speig'' that will be adopted by Mathworks in the near future as the procedure for solving large sparse eigenproblems.
We have nearly completed a users guide for ARPACK.
Parallel ARPACK (PARPACK) is now provided as an enhancement to the serial ARPACK package. It's redesign has greatly simplified software maintenance and enforces a level of algorithmic consistency between the parallel and serial implementations.
The portability of and scalability of PARPACK has been demonstrated on multiple platforms including the SGI Power Challenge Cluster at Rice, the IBM SP2 at Maui, Sun stations, the Intel Paragon at Sandia, the CRAY T3D at the Ohio supercomputing center. This work is reported in the proceedings of the 1996 Copper Mt. Conference on Iterative Methods and will be a forthcomming CAAM Tech. Rept. (by K. Maschhoff and D. Sorensen).
PARPACK is available via anonymous ftp from Rice University and from Netlib under the SCALAPACK directory.
The deflation technique of "locking" reduces the serial bottleneck caused by operations on the projected matrix by providing a mechanism for keeping the active portion of this matrix small. Operations on the projected matrix due to the implicit restart mechanism are replicated on each processor and this redundancy presents a potential bottleneck which can prevent scalability when approximations to a large percentage of eigenvalues are required. The potential for such a bottleneck is hopefully diminished with this locking scheme in place.
The out-of-core solver is an extension of the symmetric ARPACK code with locking which accommodates situations where converged Ritz vectors must be stored on disk. Different strategies for maintaining orthogonalization to these out-of-core vectors are under investigation.
The out-of-core solver is currently being tested at the MPRCL at Sandia Nat. Labs on the Intel paragon.
Recent work by Lehoucq and Sorensen indicates that an unblocked Arnoldi/Lanczos method coupled with a deflation strategy can be used reliably to compute multiple and/or clustered eigenvalues provided an appropriate convergence criteria is selected. A blocked reduction may, however, prove to be more efficient in many situations.
Recently we have begun to evaluate the differences between the blocked and unblocked variants of Arnoldi/Lanczos methods in order to evaluate the necessity and feasibility of providing a blocked method in future versions of ARPACK. Basically we would like to assess whether the benefits of a blocked reduction outweigh the added computational complexity.
In our study we first consider the careful numerical implementation of block methods. Robust deflation, orthogonalization and restarting strategies are presented. In particular, we demonstrate how the Implicitly Restarted Arnoldi (IRA) method may extended to block formulations. Finally, we perform a series of careful numerical experiments to assess the differences between the blocked and unblocked variants. The goal of our study is to provide the numerical analyst and software developer a better understanding of the many issues involved.
A blocked version has been fully implemented in a Matlab test code and preliminary results of this study will be presented at the SIAM Conference on Sparse Matrices in October. This work is in a preliminary stage of development that we expect to complete near the end of the 96-97 academic year.
Software released during 95-96: