The speed-up of the multiplication of matrices of a fixed size

Abstract

The algorithms for NXN MM in O(N^2.496) arithmetics presented in Part 1 become superior over the straightforward algorithm only for enormously large N, compare Remark 6.4 and the proof of Theorem 7.2. Those asymptotically fastest algorithms become extremely inefficient if they are applied to the multiplication of matrices of moderate sizes. Thus we need to devise some special algorithms for the latter problem. Their design and analysis with some natural extensions to the study of other related computational problems will be the subject of this part of the monograph.

In particular we will recall the bilinear algorithms and λ-algorithms of the currently least ranks and λ-ranks for nXn MM for all specific moderate n (see Sections 31 and 33), will consider the extensions to the classes of commutative quadratic algorithms and λ-algorithms (in Sections 32 and 33) and of the general arithmetical algorithms and λ-algorithms (in Section 34), and will recall some correlation between these algorithms and bilinear ones. We will show that the commutative quadratic λ-algorithms can be surprinsingly efficient already for multiplication of matrices of small sizes (and also for the evaluation of polynomials,see Example 34.2 in Section 34) although such λ-algorithms (and even more general arithmetical algorithms and λ-algorithms, see Section 34) help little for the asymptotic acceleration of MM. We will recall some lower bounds in Sections 35–38 for the sake of completeness and in order to demonstrate some typical algebraic techniques for that problem (basic active substitution argument, (linear) independence argument, reduction modulo the ideals defined by cubic terms, matrix-vector representation and tensor representation of quadratic algorithms, matrix representation of linear algorithms, duality and some other techniques of reduction of problems and algorithms to each other). Finally we will introduce a quantity for measuring asynchronicity of linear and bilinear algorithms in Section 39, and will demonstrate a new promising extension of the class of arithmetical λ-algorithms in Section 40.