DIRECT INTERPOLATION (UNEQUAL INTERVALS) 9. When the differences of consecutive x's are not all equal, we use formulae analogous to ordinary interpolation-formulae, but with the differences prepared in a special way. The best known formulae are Newton's (original) formula and Lagrange's formula; the former uses "divided" differences.
Let the tabulated values of x be , a, b, c, d, , the cor responding values of u being denoted by , f(a), f(b), f(c), f(d) Then our table is of the form: The quantities f(a,b) in the first line are called divided differences of the first order; those in the second line are divided differences of the second order; and so on. If u is a polynomial of degree k in the divided differences of order k+I are all o. It should be Observed that the value of any divided difference is independent of the order in which the u's involved in it are ar ranged; thus f (a , b) = f (b , a) , f (a , b , c) = f (a , c , b) , etc.
Newton's formula for unequal intervals is the series continuing until the differences are zero or negligible. This formula covers all those with which we have so far dealt; thus the Newton-Gauss formula (§ 7) is obtained by taking the is in the order o, I, I, 2, -2, . . . .
The method was subsequently modified by Newton, in a manuscript written in 1676 but not known to the world until 1926. Another method, similar to this modified method, but not identical with it, is that of "adjusted differences." Lagrange's formula is This gives exactly the same result as Newton's formula. It is not so convenient for calculation as the latter, and it does not show how many u's have to be taken into account; but it is important in dealing with the theory of the subject.