An elementary, machine-independent, recursive algorithm for matrix multiplication C+=A*B provides implicit blocking at every level of the memory hierarchy and tests out faster than classically optimrd code, tracking hand-coded BLAS3 routines. Proof of concept is demonstrated by racing the in-place algorithm against manufacturer's hand-tuned BLAS3 routines; it can win.The recursive code bifurcates naturafly at the top level into independent block-oriented processes, that each writes to a disjoint and contiguous region of memory. Experience has shown that the indexing vastly improves the patterns of memory access at all levels of the memory hierarchy, independently of the sizes of caches or pages and without ud hoc programming. It afso exposed a weakness in SGI'S C compilers that merrily unroll loops for the superscafar R8000 processor, but do not analogously unfold the base cases of the most elementary recursions. Such deficiencies might deter future programmers from using this rich class of recursive algorithms.

Abstract. Definitions for the uniform representation of d-dimensional matrices serially in Morton-order (or Z-order) support both their use with cartesian indices, and their divide-and-conquer manipulation as quaternary trees. In the latter case, d-dimensional arrays are accessed as 2 d -ary trees. This data structure is important because, at once, it relaxes serious problems of locality and latency, and the tree helps schedule multiprocessing. It enables algorithms that avoid cache misses and page faults at all levels in hierarchical memory, independently of a specific runtime environment.This paper gathers the properties of Morton order and its mappings to other indexings, and outlines for compiler support of it. Statistics elsewhere show that the new ordering and block algorithms achieve high flop rates and, indirectly, parallelism without any low-level tuning.

Dilated integers form an ordered group of the cartesian indices into a d-dimensional array represented in Morton order. Efficient implementations of its operations can be found elsewhere; this paper offers efficient casting (type)conversions to and from ordinary integer representation. As Morton-order representation for two-and three-dimensional arrays attracts more users because of its excellent block locality, the efficiency of these conversions becomes important. They are essential for programmers who would use cartesian indexing there. Two algorithms for each casting conversion are presented here: including to-and-from dilated integers, for both d = 2 and d = 3. They fall into two families. One family uses newly compact table-lookup so cache capacity is better preserved. The other generalizes better to all d, using processor-local arithmetic that is newly presented as abstract d-ary and (d − 1)-ary recurrences. Test results for 2 and 3 dimensions generally favor the former.

Quadtree matrices using Morton-order storage provide natural blocking on every level of a memory hierarchy. Writing the natural recursive algorithms to take advantage of this blocking results in code that honors the memory hierarchy without the need for transforming the code. Furthermore, the divide-and-conquer algorithm breaks problems down into independent computations. These independent computations can be dispatched in parallel for straightforward parallel processing.Proof-of-concept is given by an algorithm for QR factorization based on Givens rotations for quadtree matrices in Morton-order storage. The algorithms deliver positive results, competing with and even beating the LAPACK equivalent.