Numerical Differentiation
Numerical Differentiation
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DOI：
10.1201/9781420007602.ch11
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发表时间：
2019-05
期刊：
影响因子：
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通讯作者：
Carl Christian Kjelgaard Mikkelsen
中科院分区：
文献类型：
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作者：
Carl Christian Kjelgaard Mikkelsen
这些在科学应用中经常出现，通常是由于测量过程中存在固有误差。如果试图使用标准数值技术，如中心差分公式，来计算基础函数的近似导数值，很明显，这些导数的误差将大大超过原始函数数据中存在的测量误差。事实上，人们早就知道，在不限制均匀扰动类型的情况下，人们可以构造出原始和扰动导数值之差任意大的例子。因此，从实际的角度来看，真正的问题是：假设存在某种均匀扰动，人们试图描绘一类合适的扰动，目的是能够在该类中计算一个近似导数，其误差大小与原始均匀扰动的误差大小近似。这里的“合适的类”通常取决于应用程序，但在特定情况下通常是明确的。
such as these arise frequently in scientific applications, generally as a result of the inherent errors present in measurement processes. If one were to attempt to use standard numerical techniques, such as the central difference formula, to compute approximated derivative values for the underlying function, it is evident that the errors in these derivatives would greatly exceed the measurement error present in the original function data. In fact, it has long been known that with no restrictions on the type of uniform perturbation that one allows, one can construct examples in which the difference between the original and perturbed derivative values is arbitrarily large. The real problem from a practical standpoint therefore is this: assuming some kind of uniform perturbation, one seeks to delineate a class of suitable perturbations, with the intention of being able to compute, within the class, an approximate derivative whose magnitude of error is approximately that of the original uniform perturbation. The “suitable class ” here would typically depend on the application, but is generally clear in specific cases.