论文标题
一种简单快速的算法，用于计算线性循环序列的$ n $ then项
A Simple and Fast Algorithm for Computing the $N$-th Term of a Linearly Recurrent Sequence
论文作者
论文摘要
我们提出了一种简单而快速的算法，用于计算给定线性复发序列的$ n $ th。我们的新算法使用$ o（\ Mathsf {M}（d）\ log n）$算术操作，其中$ d $是复发的顺序，而$ \ Mathsf {M}（M}（d）$表示计算两种级别级别$ d $ D $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $。由于Charles Fiduccia（1985），最先进的算法具有相同的算术复杂性，直至恒定因素。我们的算法更简单，更快，并且通过完全不同的方法获得。我们还讨论了几种算法应用，特别是针对多项式模块化启动，矩阵的动力和高阶提升。
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.
