幂级数求逆的准确性与稳定性
Accuracy and stability of inversion of power series
AI总结：
本文研究了幂级数求逆的数值稳定性，通过三角回代法证明其稳定性并给出误差界，探讨了根 deflate 对多项式根求解和有限差分权重计算的影响。
摘要：
本文考虑了幂级数 $p(x)=1+b_{1}x+b_{2}x^{2}+\cdots$ 的数值求逆，以计算满足 $p(x)q(x)=1$ 的逆级数 $q(x)$。数值求逆是三角回代法的特例，自Wilkinson (1961) 的经典工作以来，其数值稳定性已广受认可。本文证明了幂级数求逆的数值稳定性，并获得了误差界。一系列示例表明这些界非常有效。当 $p(x)$ 为多项式且 $x=a$ 为根且 $p(a)=0$ 时，通过简单除法 $p(x)/(x-a)$ 进行根 deflate 可引发与多项式根求解和有限差分权重计算相关的不稳定性。当 $p(x)$ 为多项式时，计算出的逆 $q(x)$ 的精度与 $p(x)$ 的伪零点有关。
英文摘要：
This article considers the numerical inversion of the power series $p(x)=1+b_{1}x+b_{2}x^{2}+\cdots$ to compute the inverse series $q(x)$ satisfying $p(x)q(x)=1$. Numerical inversion is a special case of triangular back-substitution, which has been known for its beguiling numerical stability since the classic work of Wilkinson (1961). We prove the numerical stability of inversion of power series and obtain bounds on numerical error. A range of examples show these bounds to be quite good. When $p(x)$ is a polynomial and $x=a$ is a root with $p(a)=0$, we show that root deflation via the simple division $p(x)/(x-a)$ can trigger instabilities relevant to polynomial root finding and computation of finite-difference weights. When $p(x)$ is a polynomial, the accuracy of the computed inverse $q(x)$ is connected to the pseudozeros of $p(x)$.
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。
