Shmuel Winograd

Matrix multiplication via arithmetic progressions

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 is 2.376.

The Organization of Computations for Uniform Recurrence Equations

A set equations in the quantities <italic>a<subscrpt>i</subscrpt></italic>(<italic>p</italic>), where <italic>i</italic> = 1, 2, · · ·, <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 following property holds: If <italic>p</italic> and <italic>q</italic> are in <italic>R</italic> and <italic>w</italic> is an integer <italic>n</italic>-vector, then <italic>a<subscrpt>i</subscrpt></italic>(<italic>p</italic>) depends directly on <italic>a<subscrpt>j</subscrpt></italic>(<italic>p</italic> - <italic>w</italic>) if and only if <italic>a<subscrpt>i</subscrpt></italic>(<italic>q</italic>) depends directly on <italic>a<subscrpt>j</subscrpt></italic>(<italic>q</italic> - <italic>w</italic>). Finite-difference approximations to systems of partial differential equations typically lead to such recurrence equations. The structure of such a system is specified by a <italic>dependence graph G</italic> having <italic>m</italic> vertices, in which the directed edges are labeled with integer <italic>n</italic>-vectors. For certain choices of the set <italic>R</italic>, necessary and sufficient conditions on <italic>G</italic> are given for the existence of a schedule to compute all the quantities <italic>a<subscrpt>i</subscrpt></italic>(<italic>p</italic>) explicitly from their defining equations. Properties of such schedules, such as the degree to which computation can proceed “in parallel,” are characterized. These characterizations depend on a certain iterative decomposition of a dependence graph into subgraphs. Analogous results concerning implicit schedules are also given.

On computing the discrete Fourier transform

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.

Fast algorithms for the discrete cosine transform

Several fast algorithms for computing discrete cosine transforms (DCTs) and their inverses on multidimensional inputs of sizes which are powers of 2 are introduced. Because the 1-D 8-point DCT and the 2-D 8*8-point DCT are so widely used, they are discussed in detail. Algorithms for computing scaled DCTs and their inverses are also presented. These have applications in compression of continuous tone image data, where the DCT is generally followed by scaling and quantization. >

Arithmetic complexity of computations

Three examples General background Product of polynomials FIR filters Product of polynomials modulo a polynomial Cyclic convolution and discrete Fourier transform.

On the Asymptotic Complexity of Matrix Multiplication

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

Some bilinear forms whose multiplicative complexity depends on the field of constants

\omega

On multiplication of 2 × 2 matrices

, the exponent for matrix multiplication, is a limit point, that is, it cannot be realized by any single algorithm. We also use these results to construct a new algorithm which shows that

On the Time Required to Perform Addition

\omega < 2.495548

A New Algorithm for Inner Product

.