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T. J. Ashby1, A. D. Kennedy2, and S. M. Watt3 1 Institute for Computer Systems Ar hite ture, University of Edinburgh, S otland, UK T.Ashby ed.a .uk 2 Parti le Physi s Theory, University of Edinburgh, S otland, UK adk ph.ed.a .uk 3 Computer S ien e Department, The University of Western Ontario, London, Canada Stephen.Watt uwo. a Abstra t. An important step in latti e QCD, and various super omputing appli ations in general, onsists of updating the values asso iated with sites on a regular latti e based on a neighbourhood of ea h site des ribed by a sten il entred on it. The path from a neighbour to the entre of the sten il may also play a part in the update by the a tion of link transformations on values that traverse a given link. This paper des ribes a method to automati ally generate ode and arry out an important lass of optimisations for su h problems on large parallel omputers, appli able to sten ils expressed as paths on any given regular latti e type and dimension. Our te hnique is based on the theory of aÆne Coxeter groups and revolves around re overing and manipulating the generators of the translation subgroup orresponding to the desired latti e. The method has been prototyped in the Aldor language. 1 Introdu tion Physi al systems are often homogenous or isotropi . Su h symmetries may be dis rete, as for a rystal latti e, or ontinuous as in the spa e-time va uum of quantum eld theories. In the latter ase numeri al omputations make use of a dis rete approximation to render the problem nite. It is ne essary that the physi s of these systems, whi h is often en oded in an a tion, respe t these symmetries. While the simplest form of the a tion may be easy to write by hand, it is often desirable to to use higher-order dis retization s hemes (\improved a tions") to redu e the dis retization errors, and these an be ome extremely ompli ated. The numeri al simulation of su h physi al systems is one of the prin ipal onsumers of time on large-s ale parallel omputers, and the intrinsi parallelism of the underlying physi s often leads to the most natural and e e tive way of parallelizing the omputations. This leads to a signi ant programming problem, as the mapping of the symmetries of the physi al system to those of a parallel omputer often depends on the spe i ma hine topology used, and on the detailed parameters of the