线性常微分算子的算法
Algorithms for Linear Ordinary Differential Operators
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中文总结 AI 辅助
该研究在Scratchpad II中实现微分环上的线性常微分算子，给出域系数下相关算法，讨论多项式系数环的伪除法与Ore局部化，将算子算术用于常微分方程因式分解并给出多类系数算子示例。
中文摘要 AI 辅助
我们在Scratchpad II中实现了微分环上的线性常微分算子，这些算子作用于配备相容导子的模。该系统的抽象数据类型功能允许将此类算子表示并作为一等对象操作，同时保留算子应用的常用记号。对于域上的系数，我们给出了左、右除法、最大公因子、最小公倍子及扩展欧几里得算法的构造性算法；右部构造可通过对应左部构造在反环中获得。我们还讨论了多项式系数环上的伪除法及Ore局部化，其产生右分式域。最后，我们将该算子算术应用于常微分方程的因式分解，利用相关的Riccati方程和牛顿多边形分析因子的可能奇异部分。示例包括具有常数、初等函数、有理函数及矩阵系数的算子。
英文摘要
We describe an implementation in Scratchpad II of linear ordinary differential operators over a differential ring, acting on a module equipped with a compatible derivation. The abstract data type facilities of the system allow such operators to be represented and manipulated as first-class objects while retaining the usual notation for operator application. For coefficients in a field, we give constructive algorithms for left and right division, greatest common divisors, least common multiples, and an extended Euclidean algorithm; the right-hand constructions may be obtained from the corresponding left-hand constructions in the opposite ring. We also discuss pseudo-division over polynomial coefficient rings and an Ore localization yielding a right field of fractions. Finally, we apply this operator arithmetic to factorization of ordinary differential equations, using the associated Riccati equation and Newton polygons to analyze possible singular parts of factors. Examples include operators with constant, elementary-function, rational-function, and matrix coefficients.
