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The purpose of this paper is to highlight the performance issues of the matrix transposition algorithms for large matrices, relating to the Translation Lookaside Buffer (TLB) cache. The existing optimisation techniques such as coalesced access and the use of shared memory, regardless of their necessity and benefits, are not sufficient enough to neutralise the problem. As the data problem size increases, these optimisations do not exploit data locality effectively enough to counteract the detrimental effects of TLB cache misses. We propose a new optimisation technique that counteracts the performance degradation of these algorithms and seamlessly complements current optimisations. Our optimisation is based on detailed analysis of enumeration schemes that can be applied to either individual matrix entries or blocks (sub-matrices). The key advantage of these enumeration schemes is that they do not incur matrix storage format conversion because they operate on canonical matrix layouts. In addition, several cache-efficient matrix transposition algorithms based on enumeration schemes are offered—an improved version of the in-place algorithm for square matrices, out-of-place algorithm for rectangular matrices and two 3D involutions. We demonstrate that the choice of the enumeration schemes and their parametrisation can have a direct and significant impact on the algorithm’s memory access pattern. Our in-place version of the algorithm delivers up to 100% performance improvement over the existing optimisation techniques. Meanwhile, for the out-of-place version we observe up to 300% performance gain over the NVidia’s algorithm. We also offer improved versions of two involution transpositions for the 3D matrices that can achieve performance increase up 300%. To the best of our knowledge, this is the first effective attempt to control the logical-to-physical block association through the design of enumeration schemes in the context of matrix transposition. [ABSTRACT FROM AUTHOR]

We give a novel proof that primitive recursive functions are closed under nested simple recursion. This new presentation is supplied with a detailed proof which can be easily formalized in small fragments of Peano Arithmetic. [ABSTRACT FROM AUTHOR]