最短双向线性递归
Shortest Two-way Linear Recurrences
摘要：
设$s$是长度为$n$的域上的有限序列。众所周知，如果$s$满足具有非零常数项的$d$阶的线性递归，则$s$的逆也满足$d$阶的递归（具有逆序的系数）。A. Salagean提出了一种算法来找到这样一个最短的“双向”递归--它可能比最短长度的$s$的线性递归$\LC_n$更长。本文给出了一个新的、更简单的计算最短双向线性递推式的算法。首先，我们证明了我们用来构造一个最小多项式迭代的多项式对总是相对素的，我们还给出了扩展的乘数。然后，我们结合度下界与一个简单的重写发表的算法，由于作者，以获得我们更简单的算法。最短长度的增加是$\max\{n+1-2\LC_n,0\}$。
英文摘要：
Let $s$ be a finite sequence over a field of length $n$. It is well-known that if $s$ satisfies a linear recurrence of order $d$ with non-zero constant term, then the reverse of $s$ also satisfies a recurrence of order $d$ (with coefficients in reverse order). A recent article of A. Salagean proposed an algorithm to find such a shortest 'two-way' recurrence -- which may be longer than a linear recurrence for $s$ of shortest length $\LC_n$. We give a new and simpler algorithm to compute a shortest two-way linear recurrence. First we show that the pairs of polynomials we use to construct a minimal polynomial iteratively are always relatively prime; we also give the extended multipliers. Then we combine degree lower bounds with a straightforward rewrite of a published algorithm due to the author to obtain our simpler algorithm. The increase in shortest length is $\max\{n+1-2\LC_n,0\}$.
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