Matrix and vector operations on hypercube parallel processors

Abstract

We describe a library of parallel vector and matrix operations for hypercube multiprocessors that supports both full and sparse matrices. The library includes operations such as vector arithmetic, innerproducts, matrix transpose, matrix-vector and matrix-matrix multiplication and rank one updates. The library should be generally applicable to a wide range of architectures. Performance of the library routines depends on the ability to map various topological graphs onto the processor network. In the case of hypercubes we have used such mappings for binary trees, hierarchies of rings and rectangular grids. We describe algorithms for the solution of elliptic and hyperbolic equations on parallel computers, and present results of several implementations. The library is a fundamental tool in the development of the PDE solution algorithms and all machine dependencies of these algorithms are hidden in the linear algebra package. We show that these algorithms perform at high computational efficiency on both the Caltech and Intel hypercubes. Solution methods involved include preconditioned conjugate gradient, multigrid methods, and for hyperbolic problems, both explicit finite differences and the random choice method. These algorithms implement substantial parts of many fluid dynamics calculations.