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We present a systematic, algebraically based, design methodology forefficient implementation of computer programs optimized over multiple levels ofthe processor/memory and network hierarchy. Using a common formalism todescribe the problem and the partitioning of data over processors and memorylevels allows one to mathematically prove the efficiency and correctness of agiven algorithm as measured in terms of a set of metrics (such asprocessor/network speeds, etc.). The approach allows the average programmer toachieve high-level optimizations similar to those used by compiler writers(e.g. the notion of "tiling").  The approach presented in this monograph makes use of A Mathematics of Arrays(MoA, Mullin 1988) and an indexing calculus (i.e. the psi-calculus) to enablethe programmer to develop algorithms using high-level compiler-likeoptimizations through the ability to algebraically compose and reduce sequencesof array operations. Extensive discussion and benchmark results are presentedfor the Fast Fourier Transform and other important algorithms.

Conformal Computing: Algebraically connecting the hardware/software  boundary using a uniform approach to high-performance computation for  software and hardware applications