Fast Algorithms for Manipulating Formal Power Series

Relaxed algorithms for p-adic numbers

“…Using zealous techniques, the resolution of recursive equations can often be done using a Newton iteration, which doubles the precision at every step [BK78]. Although this leads to good asymptotic time complexities in n, such Newton iterations require the computation and inversion of the Jacobian of Φ, leading to a non trivial dependence of the asymptotic complexity on the size of Φ as an expression.…”

Code Generation for Polynomial Multiplication

“…The short product is a truncated product: to A, B in R[x] n , it associates C = AB mod x n ∈ R[x] n ; it was introduced and described in [14,10], and finds a natural role in many algorithms involving power series operations, such as those relying on Newton iteration [4]. The situation is similar to that of the transposed product: the previous references describe Karatsuba's version in detail, but hardly mention other algorithms in the divide-and-conquer family.…”

A probabilistic algorithm to test local algebraic observability in polynomial time

“…In [15,5], the authors show that such an operator is quadratic. We sketch its construction and neglect the technical details for the sake of simplicity.…”

“…Thus, the product, the exponential and, if A 0 is invertible, the inverse of matrices with coefficients in a series ring can be computed at precision j with the classical Newton operator (see 4.7 in [23] and § 5.2 in [5] for more details). Furthermore, it is a basic fact from the theory of linear ordinary system that if AẆ + A ′ W = 0 and A is invertible then W = exp A −1 A ′ is a matricial solution of this system.…”

“…Hence, at each step, the number of arithmetic operations necessary to evaluate this slp on power series truncated at order j, is bounded by O M (j)(N (n + ℓ) + (n + m)L) . Furthermore, the determination of the first j terms of the solution series of a system of linear ODE (5) requires O M (j)(N (n) + N (n + ℓ)) arithmetic operations by the well-known method of integrating factors (see § 5.2 in [5] for more details). So, as M (j) + M ⌊j/2⌋ + · · · = O M (j) and as our Newton operator is quadratic, the arithmetic complexity of the computations of the jacobian matrix ∂(…”