Moreover, some key aspects of the components are presented for the first time, for example, Section 4.2, describing the interface between GAP and SymGridPar2.

Applications are typically large and potentially highly parallel, making them good candidates for parallel execution at a variety of scales from multi-core to high-performance computing systems.

However, much existing work on parallel computing is based around numeric rather than symbolic computations.

In particular, symbolic computing presents particular problems in terms of varying granularity and irregular task sizes that do not match conventional approaches to parallelisation.

It also presents problems in terms of the structure of the algorithms and data.

This paper describes a new implementation of the free open-source GAP computational algebra system that places parallelism at the heart of the design, dealing with the key scalability and cross-platform portability problems.

We provide three system layers that deal with the three most important classes of hardware: individual shared memory multi-core nodes, mid-scale distributed clusters of (multi-core) nodes and full-blown high-performance computing systems, comprising large-scale tightly connected networks of multi-core nodes.

This requires us to develop new cross-layer programming abstractions in the form of new domain-specific skeletons that allow us to seamlessly target different hardware levels.

Our results show that, using our approach, we can achieve good scalability and speedups for two realistic exemplars, on high-performance systems comprising up to 32 000 cores, as well as on ubiquitous multi-core systems and distributed clusters.

The work reported here paves the way towards full-scale exploitation of symbolic computation by high-performance computing systems, and we demonstrate the potential with two major case studies.

Symbolic computation has underpinned several key advances in Mathematics and Computer Science, for example, in number theory, cryptography and coding theory.

Computational algebra is an important class of symbolic computation, where applications are typically characterised by complex and expensive computations.

This makes them good candidates for execution on both large and small-scale parallel systems, including high-performance computing (HPC) systems.

With the widespread.

The size and complexity of the problems that are being tackled means, however, that there is also a significant demand for symbolic computation on clusters of distributed-memory systems and potentially for full-blown HPC systems.

The HPC-GAP systems described in this paper aim to meet those diverse needs in a way that is idiomatic for the GAP programmer and easy for the domain specialist to exploit.

Firstly, symbolic algorithms tend to employ complex data and control structures that are not generally studied by the parallel programming community.

These properties present significant challenges to conventional parallelisation techniques.

Finally, the developers of symbolic computing applications are not parallelism experts: they are typically mathematicians or other domain experts, who usually do not possess a background in computer science and who have limited time and inclination to learn complex system interfaces.

What is needed is a simple, but highly scalable, way to deal with massive parallelism in symbolic computing systems, which is consistent with existing symbolic programming approaches.

Our work establishes symbolic computation as a new and exciting application domain for HPC.

It also provides a vade mecum for the process of producing effective high-performance versions of large legacy systems.

The primary research contributions of this paper are as follows.

These are a thread-safe multi-core implementation, GAP5 (Section 2); an MPI binding to exploit clusters (Section 3) and the SymGridPar2 framework that provides symbolic computation at HPC scale (Section 4).

Collectively, these allow us to address scalability at multiple levels of abstraction up to large-scale HPC systems (Section 5).

Specifically, we consider Orbit calculations and a Hecke Algebra representation theory calculation on the HECToR HPC system.

The datasets for these, and other experiments in the paper, are available at 2 (Section 6).

Many of the individual components of HPC-GAP and associated case studies have been previously reported piecemeal: with descriptions of GAP5 3, SymGridPar2 4, the MPI-GAP case study 5 and the SymGridPar2 case study 6.

A key part of the systematic presentation of the HPC-GAP components is to provide a common SumEuler benchmark to facilitate comparison of the programming models and performance at different scales.

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