Narrow Results

Stencil codes such as the Jacobi, Gauß-Seidel, and red-black Gauß-Seidel kernels are among the most time-consuming routines in many scientific and engineering applications. The performance of these codes critically depends on an efficient usage of caches, and can be improved by tiling. Several tiling schemes have been suggested in the literature; this paper gives an overview and comparison. Then, in the main part, we prove a lower bound on the number of cold and capacity misses. Finally, we analyze a particular tiling scheme, and show that it is off the lower bound by a factor of at most ten. Our results show up limitations to the speedup that can be gained by future research.

The Siromoney matrix model is a simple and elegant model for describing two-dimensional digital picture languages. The notion of attaching indices to nonterminals in a generative grammar, introduced and investigated by Aho, is considered in the vertical phase of a Siromoney matrix grammar (SMG). The advantage of this study is that the new model retains the simplicity and elegance of SMG but increases the generative power and enables us to describe pictures not generable by SMG. Besides certain closure properties and hierarchy results, applications of these two-dimensional grammars to describe tilings, polyominoes, distorted patterns and parquet deformations are studied.