A collection of parallel linear equations routines for the Denelcor HEP

This paper describes the implementation and performance results for a few standard linear algebra routines on the Denelcor HEP computer. The algorithms used here are based on high-level modules that facilitate portability and perform efficiently in a xvide range of environments:The modules are chosen to be of a large enough computational granularity so that reasonably optimum performance may be insured. The design of algorithms with such fundamental modules in mind will also facilitate their replacement by others more suited to gain the desired performance on a particular computer architecture. We have been using the Denelcor HEP (Heterogenous Element Processor) to implement a modest set of parallel routines to handle some common problems that arise when dealing with dense matrices in linear algebra: matrix multiplication, Cholesky decomposition of a positive definite matrix, LU factorization with partial pivoting, and QR factorization of a general matrix. Jordan [3] describes the architecture and programming environment of the Denelcor HEP, and Stewart [5] provides a complete description of the algorithms discussed here. Part of the experiment was to examine the ease of taking a collection of algorithms, expressed in terms of high-level modules, and implementing then on a computer with parallel constructions, such as the Denelcor HEP. Our hope was to gain near-optimum performance from these routines by implementing only the underlying modules using parallel constructs. We look on our experience as an experiment in producing portable algorithms that have a high level of granularity in their structure and high performance on a wide variety of computer architectures. The basic algorithms used here are the same as those reported in a paper by Dongarra and Eisenstat [1] (with the exception of QR factorization). These algorithms are based on standard procedures in linear algebra. They have been written to retain much of the original .mathemati-cal formulation and are based on matrix-vector operations. Designing the algorithms in terms of such operations is the hard part of an implementation. By understanding the algorithm in terms