A simple and fast algorithm for computing exponentials of power series

International audienceAs was initially shown by Brent, exponentials of truncated power series can be computed using a constant number of polynomial multiplications. This note gives a relatively simple algorithm with a low constant factor

A superfast solver for Sylvester's resultant linear systems generated by a stable and an anti-stable polynomial

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A Simple and Fast Algorithm for Computing the NN-th Term of a Linearly Recurrent Sequence

We present a simple and fast algorithm for computing the $N$-th term of a given linearly recurrent sequence. Our new algorithm uses $O(\mathsf{M}(d) \log N)$ arithmetic operations, where $d$ is the order of the recurrence, and $\mathsf{M}(d)$ denotes the number of arithmetic operations for computing the product of two polynomials of degree $d$. The state-of-the-art algorithm, due to Charles Fiduccia (1985), has the same arithmetic complexity up to a constant factor. Our algorithm is simpler, faster and obtained by a totally different method. We also discuss several algorithmic applications, notably to polynomial modular exponentiation, powering of matrices and high-order lifting.Comment: 34 page

High-performance code generation for polynomials and power series

Newton iteration is a versatile tool. In this thesis, we investigate its applications to the computation of power series solutions of first-order non-linear differential equations. To speed-up such computations, we first focus on improving polynomial multi­ plication and its variants: plain multiplication, transposed multiplication and short multiplication, for Karatsuba’s algorithm and its generalizations. Instead of rewriting code for different multiplication algorithms, a general approach is designed to output computer-generated code based on multiplication graph representations. Next, we investigate the existing Newton iteration algorithms for differential equa­ tion solving problems. To improve their efficiency, we recall how one can reduce the amount of useless computations by using transposed multiplication and short mul­ tiplication. We provide an optimized code generator that applies these techniques automatically to a given differential equation