On the exact and approximate bilinear complexities of multiplication of 4×2 and 2×2 matrices

Abstract

In 1960, Karatsuba found (paper [1]) an algorithm for multiplying two n-bit numbers in O(nlog2 3) operations. One of the authors (V. B. Alekseev) learned about this result in the summer of 1963, when, after completing the ninth grade at school, he arrived at a summer mathematical school in the environs of Moscow organized by Academician A. N. Kolmogorov. One of the invited teachers at this school was young A. A. Karatsuba. His result had made a huge impression on everybody and determined, to a large extent, the scientific interests of the first-named author. Karatsuba showed that, for simple common arithmetic and algebraic operations, there exist nonstandard asymptotically faster algorithms. The search for such algorithms has become an important problem of mathematics. For the multiplication of numbers, this problem has been solved almost completely, while the situation with matrix multiplication has turned out to be much more complicated. This paper in largely devoted to the problem of multiplying a 4× 2matrix by a 2× 2matrix. We show that the bilinear complexity of this problem equals 14 over any field. On the other hand, we also show that, for this problem, there exist approximate bilinear algorithms with bilinear complexity 13. For the problem of multiplying a 5× 2matrix by a 2× 2matrix, we construct an approximate bilinear algorithm with bilinear complexity 16 over any field of characteristic different from 2.