Approximating the p th root by composite rational functions
Approximating the p th root by composite rational functions
复制标题
用复合有理函数逼近 p 次方根
DOI：
10.1016/j.jat.2021.105577
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发表时间：
2021
影响因子：
0.9
通讯作者：
Nakatsukasa, Yuji
中科院分区：
文献类型：
--
作者：
Gawlik, Evan S.;Nakatsukasa, Yuji
有理逼近理论的一个里程碑式的结果表明，[0，1]上的x1 scinp可以用一个具有根指数精度的（n，n）型有理函数来逼近。受Zolotarev函数（对于平方根和符号函数）的递归最优性性质的启发，我们研究了用r k（x，r k-1（x，r k-2（（x，r 1（x，1）形式的复合有理函数逼近x 1 p。虽然这类有理函数不再包含极小极大（最佳）逼近p≥ 3，我们证明了它达到约p次根指数收敛的程度。此外，至关重要的是，相对于自由度的数量，收敛是双指数的，这表明复合有理函数能够近似x1 scinp和相关函数（例如，|X|和部门职能），具有卓越的效率。
A landmark result from rational approximation theory states that x 1∕ p on [0, 1] can be approximated by a type-(n, n) rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev functions (for the square root and sign functions), we investigate approximating x 1∕ p by composite rational functions of the form r k (x, r k− 1 (x, r k− 2 (⋯(x, r 1 (x, 1))))). While this class of rational functions ceases to contain the minimax (best) approximant for p≥ 3, we show that it achieves approximately p th-root exponential convergence with respect to the degree. Moreover, crucially, the convergence is doubly exponential with respect to the number of degrees of freedom, suggesting that composite rational functions are able to approximate x 1∕ p and related functions (such as| x| and the sector function) with exceptional efficiency.