7. _Derivation of Formulae._--The advancing-difference formula (1) may
be written, in the symbolical notation of finite differences,
u_[theta] = (1 + [Delta])^([theta])u0 = E^([theta])u0 (13);
and it is an extension of the theorem that if n is a positive integer
n(n - 1)
u_n = u0 + n[Delta]u0 + --------[Delta]^2u0 + ... (14),
2!
the series being continued until the terms vanish. The formula (14) is
identically true: the formula (13) or (1) is only formally true, but
its applicability to concrete cases is due to the fact that the series
in (1), when taken for a definite number of terms, differs from the
true value of u_([theta]) by a "remainder" which in most cases is very
small when this definite number of terms is properly chosen.
Everett's formula (9), and the central-difference formula obtained by
substituting from (4) in (2), are modifications of a standard formula
[theta]([theta] - 1)
u_[theta] = u0 + [theta][delta]u_(1/2) + --------------------[delta]^2u0 +
2!
([theta] + 1)[theta]([theta] - 1)
----------------------------------[delta]^3u_(1/2) +
3!
([theta] + 1)[theta]([theta] - 1)([theta] - 2)
----------------------------------------------[delta]^4u0 + ... (15)
4!
which may similarly be regarded as an extension of the theorem that,
if n is a positive integer,
n(n - 1) (n + 1)n(n - 1)
u_n = u0 + n[delta]u_(1/2) + --------[delta]^2u0 + ---------------[delta]^3u_(1/2) + ... (16).
2! 3!
There are other central-difference formulae besides those mentioned
above; the general symbolical expression is
u_[theta] = (cosh [theta]hD + sinh [theta]hD)u0 (17),
where
cosh (1/2)hD = [mu], sinh (1/2)hD = (1/2)[delta] (18).
8. _Comparative Accuracy._--Central-difference formulae are usually
more accurate than advancing-difference formulae, whether we consider
the inaccuracy due to omission of the "remainder" mentioned in the
last paragraph or the error due to the approximative character of the
tabulated values. The latter is the more important. If each tabulated
value of u is within [+-](1/2)[rho] of the corresponding true value,
and if the differences used in the formulae are the tabular
differences, i.e. the actual successive differences of the tabulated
values of u, then the ratio of the limit of error of u_([theta]), as
calculated from the first r terms of the series in (1), to (1/2)[rho]
is the sum of the first r terms of the series
1 + o + [theta](1 - [theta]) + [theta](1 - [theta])(2 - [theta]) +
(7/12)[theta](1 - [theta])(2 - [theta])(3 - [theta]) +
(1/4)[theta](1 - [theta])(2 - [theta])(3 - [theta])(4 - [theta]) +
(31/360)[theta](1 - [theta])...(5 - [theta])+ ...,
while the corresponding ratio for the use of differences up to
[delta]^(2p)u0 inclusive in (4) or up to [delta]^(2p)u1 and o^(2p)u0
in (9) (i.e. in effect, up to [delta]^(2p + 1)u(1/2)) is the sum of
the first p + 1 terms of the series
[theta](1 - [theta]) (1 + [theta])[theta](1 - [theta])(2 -[theta])
1 + -------------------- + --------------------------------------------- +
1.1 (2!)^2
(2 + [theta])(1 + [theta])[theta](1 - [theta])(2 - [theta])(3 -[theta])
----------------------------------------------------------------------- + ...,
(3!)^2
it being supposed in each case that [theta] lies between 0 and 1. The
following table gives a comparison of the respective limits of error;
the lines I. and II. give the errors due to the advancing-difference
and the central-difference formulae, and the coefficient [rho] is
omitted throughout.