On Modified Divided Differences I

Although divided differences of a function are of basic importance in the theory of numerical analysis, they are not nearly as useful as ordinary differences in application. In the first place, they are much harder to generate ; not only because they involve unequal intervals of the argument, but because divisions are involved, often by small numbers. Furthermore, they are harder to interpret than ordinary differences, and the problem of making judgments at each step about the number of meaningful significant figures in the successive differences is one of considerable difficulty for high-speed computing machines, or for I.B.M. machines. We shall show that part of this difficulty at least can be overcome by modifying the definition of divided differences. These modified divided differences take on many of the characteristics of ordinary differences, based on equal intervals between successive arguments, and the familiar theory relating to ordinary differences becomes a special case of the more general theory. Moreover, the decimal point can remain fixed, and all differences can be carried to the same number of decimals as the entries themselves. Such modified differences can therefore serve as useful tools in analyzing functions which are available only at unequal intervals of the argument. Definitions. Let a function u(x) be given for a sequence of arguments #o, *i, • • •, xn, all included in a closed region X. No assumptions need be made about the ordering in the sequence. For brevity, let w* = u(xk). The accepted definition1 for the first divided difference of uk and u¡ is