计算近似微分多项式的GCRD
Computing GCRDs of Approximate Differential Polynomials
AI总结：
本文提出了一种计算近似微分多项式近似最大公右因子（GCRD）的算法，该算法基于线性化微分 Sylvester 矩阵和块 SVD，已用 Maple 实现并验证了其鲁棒性。
摘要：
引入了具有近似多项式系数的微分（Ore）型多项式，这些多项式提供了具有强代数结构的近似微分算子的有用表示，已在精确符号环境中成功应用。本文提出了一种计算两个近似微分多项式近似最大公右因子（GCRD）的算法，该算法直观上是求解两个输入算子共同解的微分算子。更正式地说，给定近似微分多项式 f 和 g，本文展示了如何找到“附近”的多项式 f~ 和 g~，它们具有非平凡的 GCRD。这里“附近”是在合适定义的范数下。该算法是 Corless 等人（1995）为常规多项式近似 GCD 提出的基于 SVD 方法的推广。我们对适当“线性化”的微分 Sylvester 矩阵应用块 SVD。该算法已在 Maple 中实现，并展示了其鲁棒性。
英文摘要：
Differential (Ore) type polynomials with approximate polynomial coefficients are introduced. These provide a useful representation of approximate differential operators with a strong algebraic structure, which has been used successfully in the exact, symbolic, setting. We then present an algorithm for the approximate Greatest Common Right Divisor (GCRD) of two approximate differential polynomials, which intuitively is the differential operator whose solutions are those common to the two inputs operators. More formally, given approximate differential polynomials $f$ and $g$, we show how to find "nearby" polynomials $\widetilde f$ and $\widetilde g$ which have a non-trivial GCRD. Here "nearby" is under a suitably defined norm. The algorithm is a generalization of the SVD-based method of Corless et al. (1995) for the approximate GCD of regular polynomials. We work on an appropriately "linearized" differential Sylvester matrix, to which we apply a block SVD. The algorithm has been implemented in Maple and a demonstration of its robustness is presented.
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