Absolute and Relative Errors

2.1.2 Absolute and Relative Errors

Approximation is a central concept in almost all the uses of mathematics. One must often

be satisfied with approximate values of the quantities with which one works. Another type of approximation occurs when one ignores some quantities which are small compared to others. Such approximations are often necessary to ensure that the mathematical and numerical treatment of a problem does not become hopelessly complicated.

We make the following definition.

Definition 2.1.1.

Let ˜x be an approximate value whose exact value is x. Then the absolute error in ˜x is

4x = | ˜x − x|, 4x/x = |( ˜x − x)/x|.

In some books the error is defined with the opposite sign to what we use here. It makes almost no difference which convention one uses, as long as one is consistent. Note that x − ˜x is the correction which should be added to ˜x to get rid of the error. The correction and the absolute error then have the same magnitude but may have different signs.

In many situations one wants to compute a strict or approximate bound for the absolute or relative error. Since it is sometimes rather hard to obtain an error bound that is both strict and sharp, one sometimes prefers to use less strict but often realistic error estimates. These can be based on the first neglected term in some expansion, or on some other asymptotic considerations.

The notation x = ˜x ± ǫ means, in this book, | ˜x − x| ≤ ǫ. For example, if x = 0.5876 ± 0.0014, then 0.5862 ≤ x ≤ 0.5890, and | ˜x − x| ≤ 0.0014. In other texts, the same plus–minus notation is sometimes used for the “standard error” (see Sec. 2.3.3) or

bound and the relative error bound may be defined as bounds for

−p implies that components ˜x i with | ˜x i about p significant digits, but this is not true for components of smaller absolute value. An

alternative is to use componentwise relative errors,

(2.1.1) but this assumes that x i

max

| ˜x i −x i |/|x i |,

2.1. Basic Concepts in Error Estimation

91 We will distinguish between the terms accuracy and precision. By accuracy we mean

the absolute or relative error of an approximate quantity. The term precision will be reserved for the accuracy with which the basic arithmetic operations +, −, ∗, / are performed. For

floating-point operations this is given by the unit roundoff; see (2.2.8). Numerical results which are not followed by any error estimations should often, though not always, be considered as having an uncertainty of 1 2 of a unit in the last decimal place. In presenting numerical results, it is a good habit, if one does not want to go through the difficulty of presenting an error estimate with each result, to give explanatory remarks such as

• “All the digits given are thought to be significant.” • “The data have an uncertainty of at most three units in the last digit.”

• “For an ideal two-atom gas, c P /c V = 1.4 (exactly).”

We shall also introduce some notations, useful in practice, though their definitions are not exact in a mathematical sense:

a ≪ b (a ≫ b) is read “a is much smaller (much greater) than b.” What is meant by “much smaller”(or “much greater”) depends on the context—among other things, on the desired precision.

a ≈ b is read “a is approximately equal to b” and means the same as |a − b| ≪ c, where c is chosen appropriate to the context. We cannot generally say, for example, that 10 −6 ≈ 0.

a same as “a ≤ b or a ≈ b.”

Occasionally we shall have use for the following more precisely defined mathematical concepts:

f (x) = O(g(x)), x → a, means that |f (x)/g(x)| is bounded as x → a (a can be finite, +∞, or −∞).

f (x) = o(g(x)), x → a, means that lim x →a f (x)/g(x) = 0.

f (x) ∼ g(x), x → a, means that lim x →a f (x)/g(x) = 1.