INTERPOLATION

in mathematics, the determination or estimation of the value of a function f(x) from certain known values of the function. If x0 xn, f(x) is said to be an extrapolation. If x0, . . . , xn are given, along with corresponding values y0, . . . , yn, interpolation may be regarded as the determination of a function y = f(x) whose graph passes through the n + 1 points (xi, yi) (i = 0, 1, . . . , n). There are infinitely many such functions, but the simplest is a polynomial y = p(x) = b0 + b1x + . . . + bnxn. If p(xi) = yi for i = 0, . . . , n, p(x) is called the polynomial interpolation through the points (xi, yi). There is exactly one such interpolating polynomial of degree n or less. Polynomial approximation is useful even if the actual function f(x) satisfying f(xi) = yi is not a polynomial, for it often happens that the value of the polynomial p(x) is a good estimate for the value of f(x). If two points (x0, y0) and (x1, y1) are given, the polynomial interpolation is called the linear interpolation, for here p(x) = a + bx. The methods of analytic geometry give b = (y1 - y0)/(x1 - x0) that is, the slope of the line y = p(x) and the equation y = p(x) = y0 + b(x - x0). To illustrate, suppose the population of a city is known to have been 304,000 in 1970 and 335,000 in 1980. The population in 1974 and in 1987 can be estimated by linear interpolation. (The latter estimate is an extrapolation.) We have x0 = 1970, x1 = 1980, y0 = 304, and y1 = 335, where the y's are measured in thousands. Thus b = (y1 - y0)/(x1 - x0) = 31/10 = 3.1, and the interpolating line is y = 304 + 3.1(x - 1970). Setting x = 1974, we find y = 316.4 thousand as the population estimate in 1974, and setting x = 1987 we estimate 356.7 thousand as the population in that year. This latter extrapolation is also called the straight-line projection. It would be foolhardy to take the above numbers as anything more than reasonable estimates. Few people would consider a straight-line projection to the year 2100 as meaningful, or in the other direction an extrapolation back to the year 1492. (The latter gives an estimated population of about minus 1,000,000.) For n + 1 points, the polynomial interpolation is especially easy to compute if the xi's are equally spaced. Let xi + 1 - xi = h and calculate the differences Dyi = yi + 1 - yi (i = 0, . . . , n - 1), then the second differences D2yi = Dyi + 1 - Dyi (i = 0, . . . , n - 2), and so on. Then take a0 = y0, a1 = Dy0, a2 = D2y2, etc. Then the polynomial interpolation through the points (xi, yi) is This is called Newton's formula after its discoverer, Sir Isaac Newton. To illustrate this formula, suppose that a table of values gives f(3) = 1.3106, f(3.1) = 1.3200, and f(3.2) = 1.3310, and estimate f(3.15) by quadratic interpolation. Here, h = 0.1 and x0 = 3. The y's are successively 1.3106, 1.3200, 1.3310. The first differences are .0094, .0110, and the second difference is .0016. Newton's formula gives (The first two terms give the linear approximation 1.3247.) Similar formulas can be found if the xi's are not equally spaced, but the computation is more cumbersome.