Interpolation vs. Extrapolation

Difference Between Interpolation And Extrapolation

Interpolation and Extrapolation, techniques by which new information can be obtained from certain given information.

Interpolation is a technique for determining new values that lie between certain given values. In most cases, the problem involves tables of approximations for logarithmic or trigonometric functions where it is desired to find a reasonable value between values listed in the table. Since the values found in the table are approximations, the interpolated value will be an approximation. For example, in a 4-place common logarithm table we find the entries: log 6.5 = 0.8129 log 6.6 = 0.8195. How do we find the value of log 6.53? The simplest method is to note the proportional change in the number entries and the corresponding logarithm values. Thus 6.53 is 3/10 of the distance from 6.50 to 6.60, and therefore log 6.53 is approximately equal to 3/10 of the distance from 0.8129 to 0.8195. Thus log 6.53 = 0.8129 + 0.3 (0.8195 − 0.8129) = 0.8149. This method is called linear interpolation because the assumption of proportionality is geometrically equivalent to using a straight line as an approximation to the true graph in a small interval.

This is illustrated in the accompanying figure. Suppose the curved line is the graph of the function y = f(x), and we know the two values y1 = f(x1) and y2 = f(x2). We want to find the value for y + f(x) for some x0 between x1 and x2. (In the previous example, the function is the logarithmic function, y = log x, where x1 = 6.5, x2 = 6.6, and x0 = 6.53.) By drawing a straight line between P and Q, we see that for a given x-value, x0, the error will be small, namely the distance A to B. If P and Q are sufficiently close together, the error will be small enough to be disregarded and the y-value as A may be taken as the value of f(x0). From analytic geometry, we know that any point (x,y) on the straight line through the points P and Q must satisfy the proportion

In extrapolation, the basic idea is the same as in interpolation, except here we are interested in getting an approximation to the function that will allow us to go beyond the given values. For example, log 2 and log 3 could be used to find an approximate value of log 3.1 by extrapolation: log 3.1 ≅ log 3 + 1/10 (log 3 − log 2).

 

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