数值分析中的等价定理：积分、微分与插值
Equivalence Theorems in Numerical Analysis : Integration, Differentiation and Interpolation
AI总结：
本文提出并证明了数值方法中一致性、收敛性和稳定性之间的Lax-Richtmyer型等价定理，并通过多项式插值、数值微分、数值积分和蒙特卡洛积分等实例展示了其应用。
摘要：
我们证明，如果一个数值方法被表示为作用于数据且依赖于参数（通常是离散化大小的度量）的算子序列，那么一致性、收敛性和稳定性可以通过一个Lax-Richtmyer型等价定理联系起来——一致的方法收敛当且仅当它是稳定的。我们将一致性定义为在稠密子空间上的收敛性，将稳定性定义为离散适定性。在某些应用中，收敛性比一致性或稳定性更难证明，因为收敛性需要解的知识。等价定理在这种情况下很有用。我们给出了多项式插值、数值微分、使用求积公式的数值积分和蒙特卡洛积分的等价定理的具体实例。
英文摘要：
We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discretization, then consistency, convergence and stability can be related by a Lax-Richtmyer type equivalence theorem -- a consistent method is convergent if and only if it is stable. We define consistency as convergence on a dense subspace and stability as discrete well-posedness. In some applications convergence is harder to prove than consistency or stability since convergence requires knowledge of the solution. An equivalence theorem can be useful in such settings. We give concrete instances of equivalence theorems for polynomial interpolation, numerical differentiation, numerical integration using quadrature rules and Monte Carlo integration.
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。
