线性递归序列的快速（多）求值：改进及应用
Fast (Multi-)Evaluation of Linearly Recurrent Sequences: Improvements and Applications
摘要：
对于线性递归向量序列P[n+1] = A（n）* P[n]，考虑计算第n项P[n]或L ≤ n个任意项P[n_1]，. P[n_L]，对于常系数A（n）=A的情况和对于元素在n中为多项式的矩阵A（n）。我们改进和推广了已知的算法，并给出了新的应用，特别是证明了：例如，任何经典正交多项式族（p_n）在给定x处的求值在O（n^{1/2} log n）个运算内，与所考虑的族（p_n）无关。* 对于任意L个索引n_1，.，n_L <= n，值p_{n_i}（x）可以使用O（n^{1/2} log n + L log（n/L））算术运算同时计算;同样，这个运行时间界限也是一致的。* 每一个超几何（或更一般地说，完整）函数承认近似评估的绝对误差e>0内O（（log（1/e）^{1/2} loglog（1/e））-而不是O（log（1/e））-算术步骤。* 给定m和特征为零的域上的d次多项式p，p^m到项X^n的系数可以在O（d^2 M（n^{1/2}））步内计算，其中M（n）表示乘以两个n次多项式的成本。* 对于p^m的任意L<=n^{1/2}个期望系数到项X^{n_i}，n_1，.，n_L <= n。
英文摘要：
For a linearly recurrent vector sequence P[n+1] = A(n) * P[n], consider the problem of calculating either the n-th term P[n] or L<=n arbitrary terms P[n_1],...,P[n_L], both for the case of constant coefficients A(n)=A and for a matrix A(n) with entries polynomial in n. We improve and extend known algorithms for this problem and present new applications for it. Specifically it turns out that for instance * any family (p_n) of classical orthogonal polynomials admits evaluation at given x within O(n^{1/2} log n) operations INDEPENDENT of the family (p_n) under consideration. * For any L indices n_1,...,n_L <= n, the values p_{n_i}(x) can be calculated simultaneously using O(n^{1/2} log n + L log(n/L)) arithmetic operations; again this running time bound holds uniformly. * Every hypergeometric (or, more generally, holonomic) function admits approximate evaluation up to absolute error e>0 within O((log(1/e)^{1/2} loglog(1/e)) -- as opposed to O(log(1/e)) -- arithmetic steps. * Given m and a polynomial p of degree d over a field of characteristic zero, the coefficient of p^m to term X^n can be computed within O(d^2 M(n^{1/2})) steps where M(n) denotes the cost of multiplying two degree-n polynomials. * The same time bound holds for the joint calculation of any L<=n^{1/2} desired coefficients of p^m to terms X^{n_i}, n_1,...,n_L <= n.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
