Fast algorithms for matrix multiplication using pseudo-number-theoretic transforms

Abstract

We provide a novel approach to the design of fast algorithms for matrix multiplication. The operation of matrix multiplication is reformulated as a convolution, which is implemented using pseudo-number-theoretic transforms. Writing the convolution as multiplication of polynomials evaluated off the unit circle reduces the number of multiplications without producing any error, since the (integer) elements of the product matrix are known to be bounded. The new algorithms are somewhat analogous to the arbitrary precision approximation (APA) algorithms, but have the following advantages: (1) a simple design procedure is specified for them; (2) they do not suffer from roundoff error; and (3) reasons for their existence is clear. The new algorithms are also non-commutative, so that they may be applied recursively to block matrix multiplication. This work establishes a link between matrix multiplication and fast convolution algorithms and so opens another line of enquiry for the fast matrix multiplication problem. Some numerical examples illustrate the operation of the new proposed algorithms.