SHORT METHODS IN MULTIPLICATION

review of radiographic methods in dental implantology

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Faster integer multiplication using short lattice vectors

We prove that $n$-bit integers may be multiplied in $O(n \log n \, 4^{\log^* n})$ bit operations. This complexity bound had been achieved previously by several authors, assuming various unproved number-theoretic hypotheses. Our proof is unconditional, and depends in an essential way on Minkowski's theorem concerning lattice vectors in symmetric convex sets.

Homotopy methods for multiplication modulo triangular sets

We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li, Moreno Maza and Schost, we propose an algorithm that relies on homotopy and fast evaluation-interpolation techniques. We obtain a quasi-linear time complexity for substantial families of examples, for which no such result was known before. Applications are given to notably addition of a...

Optimization Methods for Sparse Matrix-Vector Multiplication

Algebraic Methods for Fast Matrix Multiplication

An improvement upon the naive O(n3) algorithm for matrix multiplication was first presented by Strassen, obtaining the result in only O(n2.81) field operations [5]. This raises the question of what the best possible exponent k such that matrix multiplication can be carried out in at most O(nk) time is. Clearly k ≥ 2, since n2 is the size of the output. It is believed that the optimal k is exact...