Differential (Ore) type polynomials with “approximate” polynomial coefficients are introduced. These provide an effective notion of approximate differential operators, with a strong algebraic structure. We introduce the approximate greatest common right divisor problem (GCRD) of differential polynomials, as a non-commutative generalization of the well-studied approximate GCD problem. Given two differential polynomials, we present an algorithm to find nearby differential polynomials with a non-trivial GCRD, where nearby is defined with respect to a suitable coefficient norm. Intuitively, given two linear differential polynomials as input, the (approximate) GCRD problem corresponds to finding the (approximate) differential polynomial whose solution space is the intersection of the solution spaces of the two inputs. The approximate GCRD problem is proven to be locally well posed. A method based on the singular value decomposition of a differential Sylvester matrix is developed to produce an initial approximation of the GCRD. With a sufficiently good initial approximation, Newton iteration is shown to converge quadratically to an optimal solution. Finally, sufficient conditions for existence of a solution to the global problem are presented along with examples demonstrating that no solution exists when these conditions are not satisfied.

中文翻译：

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计算微分多项式的近似最大公有因数

介绍了具有“近似”多项式系数的微分（矿石）多项式。这些提供了具有强代数结构的近似微分算子的有效概念。我们介绍了微分多项式的近似最大公共权利除数问题（GCRD），作为对精心研究的近似GCD问题的非交换泛化。鉴于两个差分多项式，我们提出了一个算法来寻找附近有一个不平凡的GCRD，其中微分多项式附近相对于合适的系数范数定义。直观地，给定两个线性微分多项式作为输入，（近似）GCRD问题对应于找到其（解）微分多项式，其解空间是两个输入的解空间的交集。事实证明，近似的GCRD问题是局部合适的。开发了一种基于差分西尔维斯特矩阵奇异值分解的方法来产生GCRD的初始近似值。有了足够好的初始近似值，牛顿迭代就可以二次收敛到最优解。最后，给出了解决整体问题的充分条件，并举例说明了在不满足这些条件时不存在任何解决方案。