Arthur C. Norman

Implementing LISP in a high-level language

The problems of producing an efficient but robust implementation of a high‐level language by building on existing compilers are discussed with reference to a LISP system consisting of an interpreter, a compiler and a large collection of support routines.

Implementing a polynomial factorization and GCD package

This paper describes the construction of a rational function package where the GCD and factorization routines are well integrated and consistent with each other and both use state of the art algorithms. The work represents an exercise in producing a service rather than an experimental piece of code, where portability, reliability and clear readable code are important aims in addition to the obvious desire for speed. Measurements on the initial version of our package showed that even though it was based on the best of previously published methods its performance was uneven. The causes of the more notable bottle necks and the steps we took to avoid them are explained here and illustrate how apparently very fine details of coding can sometimes have gross effects on a systems overall behaviour.

A note on compacting garbage collection

A variation of the Haddon and Waite compacting garbage collector is presented that needs only bounded workspace, but which has typical runtime proportional to the size of the heap rather than n log n for a heap of size n. The algorithm has been measured in the context of a LISP system where it has been seen to behave close to its optimum. The relationship of this algorithm to one due to Lang and Weigbreit is also explained.

Not seeing the roots for the branches: multivalued functions in computer algebra

We discuss the multiple definitions of multivalued functions and their suitability for computer algebra systems. We focus the discussion by taking one specific problem and considering how it is solved using different definitions. Our example problem is the classical one of calculating the roots of a cubic polynomial from the Cardano formulae, which contain fractional powers. We show that some definitions of these functions result in formulae that are correct only in the sense that they give candidates for solutions; these candidates must then be tested. Formulae that are based on single-valued functions, in contrast, are efficient and direct.

CABAL: polynomial and power series algebra on a parallel computer

The work on designing or adapting algebra systems to exploit parallel hmdware that we have seen to date has concentrated on trying the speed-up that can come from concurrent exploration of branches in search trees or overlapped computation in demanding algorithms. We in this paper, on the other hand, are almost exclusively concerned with the fact that large-scale parallel computers are frequently configured with very large amounts of primary memory. Our system distributes very large formulae across the multiple processors in such a parallel computer in such a way that calculations that would otherwise fail for lack of memory can be completed. Our initial system described here is designed to perform simple polynomial and power series calculations, but even in such apparently limited domains the easing of memory restrict ions can sometimes prove useful. Results from our system, called CABAL (Cambridge And Bath ALgebra-system), are included both from rune on sm~l networks of conventional workstations and using a Hitachi SR2201 massively parallel super-computer. The main result is a demonstration of the feasibility of the design.

Interfacing REDUCE to Java

For some time it has been clear that algebra systems ought not to exist as isolated software packages, but should be viewed more as components in a more general scientific problem-solving environment. A so-called “software bus” would then link perhaps several algebra engines, each with special areas of strength, to separate tools to support numerical calculation, visualisation, domain-specific scripting or other capabilities. The term “bus” as used here derives from the same word used to describe a mode of transport, and as such is an abbreviation for omnibus — stressing the fact that to be useful it must be universally available. The language and system Java[5] presents itself as a candidate for such general acceptance by many different classes of application, so this paper is a preliminary investigation of both how Java code can be linked to REDUCE and what might be done by exploiting such a link.

Alkahest III: automatic analysis of periodic weakly nonlinear ODEs

In previous papers by the current authors various attempts that have been made in the automatic derivation of periodic solutions to weakly nonlinear differential equations have been reported. The equations in question are all perturbations of y′+y = 0, related to the equations that occur in the study of celestial mechanics. In the current paper the analysis of these equations has been taken further, with the automatic system, named Alkahest III, being able to determine the solution to equations even when the users initial conditions are invalid. The system can produce an approximate solution itself, and there are facilities to write an algebra program for REDUCE or CAMAL to generate higher order solutions.

A Parallel Symbolic Computation Environment: Structures and Mechanics

We describe a set of representations for polynomials and sparse matrices suited for use with fine-grain parallelism on a distributed memory multiprocessor system. Our aim is to support use of supercomputers with this style of architecture to perform computations that would exceed the main memory capacity of more traditional computers: although such systems have very high performance communication networks it is still essential to avoid letting any one part of the network become a bottleneck. We use randomised data placement both to avoid hot-spots in the communication patterns and to balance (in a probabilistic sense) the memory load placed upon each processing element. The expected application areas for such a system will be those where intermediate expression swell means that the huge primary memory available on MPP systems will be needed if the smaller final result is to be successfully computed.

A comparison of the Vaxima and Reduce factorization packages

During 1979 the programmed version of the multivariate factorization algorithm described in [6] stabilized and replaced the old factoring algorithm in Macsyma as the standard factorizer. It has thus graduated from being a program written as part of a research project into being a tool that anyone with a copy of Macsyma may rely on. In 1981, a factoring package [4] was being prepared for distribution as a module in a forthcoming release of Reduce. During September 1982 the two authors were able to make some direct comparisons between their codes by running both factorizers on the same set of examples on the same computer at the same time. This note reports on the comparative timings so obtained.

Memory tracing of algebraic calculations

We present a software tool which allows us to visualise details of the use of memory during the execution of an algebra system. We apply this to gain a better understanding of the behaviour of REDUCE, and hence to make proposals for ways in which the execution can be improved. The same tool will soon be used in the performance engineering of a version of Axiom.