A fast algorithm for solving linearly recurrent sequences

TL;DR

This paper introduces a faster algorithm for computing the D-th term of linearly recurrent sequences, improving efficiency over previous methods by refining the complexity bounds using polynomial multiplication techniques.

Contribution

It presents a refined algorithm that computes sequence terms more efficiently, reducing the complexity from previous optimal results by incorporating the degree of the squarefree part of the annihilating polynomial.

Findings

Achieves computation of the D-th term in O( M( d̄ ) log D + M( d ) log d ) operations

Refines the previous optimal complexity of O( M( d ) log D )

Utilizes polynomial multiplication to improve efficiency

Abstract

We present an algorithm which computes the $D^{th}$ term of a sequence satisfying a linear recurrence relation of order $d$ over a field $K$ in $O( \mathsf{M}(\bar d)\log(D) + \mathsf{M}(d)\log(d))$ operations in $K$, where $\bar d \leq d$ is the degree of the squarefree part of the annihilating polynomial of the recurrence and $\mathsf{M}$ is the cost of polynomial multiplication in $K$. This is a refinement of the previously optimal result of $O( \mathsf{M}(d)\log(D) )$ operations, due to Fiduccia.