Fast in-place accumulation

Abstract

International audienceThis paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some output variables are also input variables, linked by a linear dependency. Fundamental examples include the in-place accumulated multiplication of polynomials or matrices, C += AB (that is with only O(1) extra space). The difficulty is to combine in-place computations with fast algorithms: those usually come at the expense of (potentially large) extra temporary space, but with accumulation the output variables are not even available to store intermediate values. We first propose a novel automatic design of fast and in-place accumulating algorithms for any bilinear formulae (and thus for polynomial and matrix multiplication) and then extend it to any linear accumulation of a collection of functions. For this, we relax the in-place model to any algorithm allowed to modify its inputs, provided that those are restored to their initial state afterwards. This allows us to ultimately derive unprecedented in-place accumulating algorithms for fast polynomial multiplications and for Strassen-like matrix multiplications.We then consider the simultaneously fast and in-place computation of the Euclidean polynomial modular remainder R(X) ≡ A(X) mod B(X). Fast algorithms for this usually also come at the expense of a linear amount of extra temporary space. In particular, they require one to first compute and store the whole quotient Q(X) such that A = BQ+R. We here propose an *in-place* algorithm to compute the remainder only. If A and B have respective degree m+n and n, and M(k) denotes the complexity of a (not-in-place) algorithm to multiply two degree-k polynomials, our algorithm uses at most O((n/m) M(m) log(m)) arithmetic operations. In this particular case this is a factor log(n) more than the not-in-place algorithm. But if M(n) = Θ(n^{1+ε}) for some ε>0, then our algorithms do match the not-in-place complexity bound of O((n/m) M(m)). We also propose variants that compute – still in-place and with the same kind of complexity bounds – the over-place remainder A(X) ≡ A(X) mod B(X), the accumulated remainder R(X) += A(X) mod B(X) and the accumulated modular multiplication R(X) += A(X)C(X) mod B(X), that is multiplication in a polynomial extension of a finite field.To achieve this, we develop techniques for Toeplitz matrix operations, for generalized convolutions, short product and power series division and remainder whose output is also part of the input