Convolutional number-theoretic method to optimise integer matrix multiplication
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There have been several algorithms designed to optimise matrix multiplication. From schoolbook method with complexity O(n^3) to advanced tensor-based tools with O(n^2.373) [2], much work has been to reduce the number of steps used in the recursion [1,2], and some group-theoretic interpretations as well [3], that conjecture that there exists a quadratic algorithm to perform matrix multiplication. But in the latter, there seems to be no substantiation as a full-fledged algorithm with matrix multiplication exponent ω = 2. In this we introduce a technique that represents vectors of integers appropriately and combines them to make dot product a constant-time operation for powerful processors. For an n × n matrix, we present a method where we iteratively repeat this dot product for each of the element in the resultant matrix. Preprocessing and computation makes it a quadratic algorithm with a considerable constant of proportionality. Later, extensions to this have been discussed on other matters of integers.
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