Shmuel Winograd

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We present a new method for accelerating matrix multiplication asymptotically. This work builds on recent ideas of Volker Strassen, by using a basic trilinear form which is not a matrix product. We make novel use of the Salem-Spencer Theorem, which gives a fairly dense set of integers with no three-term arithmetic progression. Our resulting matrix exponent(More)
A set equations in the quantities <italic>a<subscrpt>i</subscrpt></italic>(<italic>p</italic>), where <italic>i</italic> = 1, 2, &#183; &#183; &#183;, <italic>m</italic> and <italic>p</italic> ranges over a set <italic>R</italic> of lattice points in <italic>n</italic>-space, is called a <italic>system of uniform recurrence equations</italic> if the(More)
New algorithms for computing the Discrete Fourier Transform of n points are described. For n in the range of a few tens to a few thousands these algorithms use substantially fewer multiplications than the best algorithm previously known, and about the same number of additions.
The main results of this paper have the following flavor: given one algorithm for multiplying matrices, there exists another, better, algorithm. A consequence of these results is that &#x03C9;, the exponent for matrix multiplication, is a limit point, that is, cannot be realized by any single algorithm. We also use these results to construct a new algorithm(More)
Construction of algorithms is a time honored mathematical activity. Euclid's algorithm for finding the greatest common divisor of two integers, as well as the many constructions by a ruler and compass are some of the fruits of the search for algorithms by the Greek mathematicians. In our days, we have the whole field of Numerical Analysis devoted to finding(More)
A problem related to the decentralized control of a multiple access channel is considered: Suppose <italic>k</italic> stations from an ensemble of <italic>n</italic> simultaneously transmit to a multiple access channel that provides the feedback 0, 1, or 2+, denoting <italic>k</italic> = 0, <italic>k</italic> = 1, or <italic>k</italic> &#8805; 2,(More)
In this paper we consider the system of bilinear forms which are defined by a product of two polynomials modulo a thirdP. We show that the number of multiplications depend on how the field of constants used in the algorithm splitsP. If $$P = \prod\nolimits_{i = 1}^k {P_i^{li} } $$ then 2 ·deg(P) − k multiplications are needed. (We assume thatP i is(More)