Overview

Given a set of n + 1 numbers y _i defined at n + 1 nodes x _i, where x ₀ < x ₁ < < x _n, it is required to find a value y which may plausibly be said to approximate the exact value, if it were known, corresponding to the point x. The given values may be from a table, or may result from an expensive piece of computation, or they may be the result of experimental observations. Indeed, we may be interested in the latter case to discover a plausible functional relationship between x and y which would enable us, over some range of the variables concerned, to write y = f( x), to some approximation. If the value of x lies in the range x ₀ < x < x _n of known values, we are seeking an interpolated value of y; if it lies outside this range, we seek instead an extrapolated value. For reasons that will become apparent, it is very much more difficult to extrapolate reliably than it is to interpolate. For the moment it is perhaps sufficient to observe that, even for interpolation, there must be an assumption that any functional relationship between y and x is not only continuous but sufficiently slowly varying, otherwise there is no reason to suppose, given x, that