Orders of approximation
Scale analysis · Big O notation
Curve fitting · False precision
Approximation · Generalization error
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In science, engineering, and other quantitative disciplines, orders of approximation refer 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 term in the series expansion used in the approximation. 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. If a quantity is constant within the whole interval, approximating it with a second-order Taylor series will not increase the accuracy. Thus the numbers zeroth, first, second etc. used formally in the above meaning do not directly give information about percent error or significant figures.
This formal usage of order of approximation corresponds to the order of the power series representing the error, which is the first first nonzero higher derivative of the error. The expressions: a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth are used as fixed phrases.
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.
Formally, an nth-order approximation is one where the order of magnitude of the error is at most , or  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.
Zeroth-order approximation is the term scientists use for a first rough answer. Many simplifying assumptions are made, and when a number is needed, an order-of-magnitude answer (or zero significant figures) is often given. For example, you might say "the town has a few thousand residents", when it has 3,914 people in actuality. This is also sometimes referred to as an order-of-magnitude approximation. The zero of "zeroth-order" represents the fact that even the only number given, "a few", is itself loosely defined.
A zeroth-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be constant, or a flat line with no slope: a polynomial of degree 0. For example,
is an approximate fit to the data, obtained by simply averaging the x-values and the y-values then deriving a multiplicative function for that average. Other methods for selecting a constant approximation can be used, such as taking the average of x-values and y-values and then working out the average difference between them:
First-order approximation is the term scientists use for a slightly better answer. Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has 4×103 or four thousand residents"). In the case of a first-order approximation, at least one number given is exact. In the zeroth order example above, the quantity "a few" was given but in the first order example, the number "4" is given.
A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a linear approximation, straight line with a slope: a polynomial of degree 1. For example,
is an approximate fit to the data. In this example there is a zeroth order approximation that is the same as the first order but the method of getting there is different; i.e. a wild stab in the dark at a relationship happened to be as good as an 'educated guess'.
Second-order approximation is the term scientists use for a decent-quality answer. Few simplifying assumptions are made, and when a number is needed, an answer with two or more significant figures ("the town has 3.9×103 or thirty-nine hundred residents") is generally given. In mathematical finance, second-order approximations are known as convexity corrections. As in the examples above, the term "2nd order" refers to the number of exact numerals given for the imprecise quantity. In this case, "3" and "9" are given as the two successive levels of precision, instead of simply the "4" from the first order, or "a few" from the zeroth-order found in the examples above.
A second-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a quadratic polynomial, geometrically, a parabola: a polynomial of degree 2. For example,
is an approximate fit to the data. In this case, with only three data points, a parabola is an exact fit.
While higher-order approximations exist and are crucial to a better understanding and description of reality, they are not typically referred to by number.
Continuing the above, a third-order approximation would be required to perfectly fit four data points, and so on. See polynomial interpolation.
These terms are also used colloquially by scientists and engineers to describe phenomena that can be neglected as not significant (e.g. "Of course the rotation of the Earth affects our experiment, but it's such a high-order effect that we wouldn't be able to measure it" or "At these velocities, relativity is a fourth-order effect that we only worry about at the annual calibration.") In this usage, the ordinality of the approximation is not exact, but is used to emphasize its insignificance; the higher the number used, the less important the effect. The terminology, in this context, represents a high level of precision required to account for an effect which is inferred to be very small when compared to the overall subject matter. The higher the order, the more precision is required to measure the effect, and therefore the smallness of the effect in comparison to the overall measurement.