Order of approximation

In science, engineering, and other quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is. In formal expressions, the ordinal number used before the word order refers to the highest power in the series expansion used in the approximation. The expressions: a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth are used as fixed phrases. The expression a zero-order approximation is also common. Cardinal numerals are occasionally used in expressions like an order-zero approximation, an order-one approximation, etc. The omission of the word order leads to phrases that have less formal meaning. Phrases like first approximation or to a first approximation may refer to a roughly approximate value of a quantity. The phrase to a zeroth approximation indicates a wild guess. The expression order of approximation is sometimes informally used to mean the number of significant figures, in increasing order of accuracy, or to the order of magnitude. However, this may be confusing, as these formal expressions do not directly refer to the order of derivatives. The choice of series expansion depends on the scientific method used to investigate a phenomenon. The expression order of approximation is expected to indicate progressively more refined approximations of a function in a specified interval. The choice of order of approximation depends on the research purpose. One may wish to simplify a known analytic expression to devise a new application or, on the contrary, try to fit a curve to data points. Higher order of approximation is not always more useful than the lower one. For example, if a quantity is constant within the whole interval, approximating it with a second-order Taylor series will not increase the accuracy. In the case of a smooth function, the nth-order approximation is a polynomial of degree n, which is obtained by truncating the Taylor series to this degree.

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Order of approximation

Mathématiques

thumb|upright|Raisonnement mathématique sur un tableau. Les mathématiques (ou la mathématique) sont un ensemble de connaissances abstraites résultant de raisonnements logiques appliqués à des objets divers tels que les ensembles mathématiques, les nombres, les formes, les structures, les transformations ; ainsi qu'aux relations et opérations mathématiques qui existent entre ces objets. Elles sont aussi le domaine de recherche développant ces connaissances, ainsi que la discipline qui les enseigne.