INVERSE INTERPOLATION 1o. Inverse Interpolation: Simple Cases.—In all cases of inverse interpolation we have to pay special attention to the degree of accuracy of the result. This depends largely, in ordinary cases, on the initial figure of the first difference.
If the first difference is practically constant, inverse interpola tion is very simple. The direct formula where, if e lies between o and 1, may be either or We can calculate the expression in } by taking an ap proximate value of 0; and then 0 is found more accurately by dividing by this expression.
II. General Formula.—If third or higher differences have to be taken into account there are two methods.
(i.) We can use the general formula (6), viz.: and proceed by successive approximations. By transposition and division, we get We begin with an approximate value of 0. Substituting this on
the right-hand side, we get a closer approximation; and so on.
(ii.) The above method is only applicable when the intervals in x are equal. An alternative method, suitable for cases of un equal intervals, but not always applicable, is to invert the table i.e., to interchange the x column and the u column and find divided differences of the x's—and proceed by direct interpolation.
BIBLIOGRAPHY.—Edmund T. Whittaker and G. Robinson, The Calculus of Observations (1924), the chapters dealing with interpola tion are published separately, as A Short Course in Interpolation. J. F. Steffensen, Interpolation (1927). See also Duncan C. Fraser, Newton's Interpolation Formulas (1927) and Isaac Newton, Memorial volume (1927). See also CALCULUS OF DIFFERENCES. (W. F. S )