Multiprocessing linear algebra algorithms on the CRAY X-MP-2: Experiences with small granularity

Abstract

This paper gives a brief overview of the CRAY X-MP-2 general-purpose multiprocessor system and discusses how it can be used effectively to solve problems that have small granularity. An implementation is described for linear algebra algorithms that solve systems of linear equations when the matrix is general and when the matrix is symmetric and positive definite. OVERVIEW OF THE SYSTEM"Multiprocessor" is a term that has been used for years. Our definition follows those of [8], [9], and [lo].The CRAY X-MP is a follow-up to the CRAY-1S system offered by CRAY Research, Inc. The CRAY X-MP family is a general-purpose multiprocessor systeln. It inherits the basic vector functions of CRAY-lS, with major architectural improvements for each individual processor. The interprocessor communication mechanism and the provision of Solid-State Disk device(SSD) are new designs that create tremendous potential in the realm of high-speed computing.The CRAY X-MP-2 system is the first product of the CRAY X-MP family.

Multitasking a Navier-Stokes algorithm on the CRAY-2

“…The exceptions are some experiments on the CRAY X-MP and CRAY-2. Some results of macrotasking (previously called multitasking) on the CRAY X-MP family were reported by Larson [1984], Chen et al [1984], Seager [1986], and Mandell [1987]. Larson reported speedups of 1.86 to 1.89 on the CRAY X-MP/2 for a particle-in-cell code, a weather forecasting code, and a three-dimensional seismic migration code.…”

“…Larson reported speedups of 1.86 to 1.89 on the CRAY X-MP/2 for a particle-in-cell code, a weather forecasting code, and a three-dimensional seismic migration code. Chen et al [1984] used two sets of multitasking routines on the CRAY X-MP/2 to study the solution of symmetric linear systems using LU decomposition and Cholesky factorization algorithms. They reported speedups of 0.76 to 1.80 by using the standard multitasking routines and speedups of 1.39 to 1.92 by using a set of assembly language (CAL) routines.…”

MIMD algorithms and their implementation

“…The fine granularity of SIMD means that the parallelization acts on single data elements as opposed to matrices of even larger structures in MIMD. The concept of granularity is rather well understood for problems of linear algebra [2], but is more vague for other problems.…”

Microtasking general purpose partial differential equation software on the CRAY X-MP

“…ASC-8519354. 1986; Bank and Rose 1982;Chan and Keller 1982] for resulting nonlinear algebraic equations; and multigrid linear algebra techniques [Bank and Douglas 1985;Bank and Dupont 1981] incorporating a version of the Yale Sparse Matrix Package [Eisenstat et al 1982]. Use of such a code as a testbed to study multiprocessing performance differs from approaches focussing on application-specific software and from those directing attention to single important components of such software (e.g., [Chen et al 1984;Seager 1986]), both in its objectives and what it can achieve.…”