II. FINITE DIFFERENCE METHOD 1 Difference formulae

By proceeding in analogous way, higher order derivatives can be approximated.
For example, approximating the function f by a second degree polynomial
through the points (x ¡ h, f(x ¡ h)), (x, f(x)) and (x + h, f(x + h)),
it turns out that D+(D−f) is a central difference approximation of f ′′ (x) :
= 1
h
[ 1
h
D+(D−f) = ( D−f(x + h) ¡ D−f(x)
h
1
(f(x + h) ¡ f(x)) ¡ (f(x) ¡ f(x ¡ h))] =
h
= 1
h 2 [f(x + h) ¡ 2f(x) + f(x ¡ h)] » f ′′ (x).
The study of the truncation error can be introduced by an example: if the
first derivative of the function e x is approximated by the forward (central)
difference operator in the point x = 1, the truncation errors D+e x ¡ e x ,
(Dce x ¡ e x ) for increments h, h/2, h/4,... are written in the following table.
The ratio between an approximation and the subsequent one is represented
by r:
h D+e x ¡ e x r Dce x ¡ e x r
0.4 0.624013 2.14 0.0730696 4.02
0.2 0.290893 2.06 0.0181581 4.00
0.1 0.140560 2.03 0.0045327 4.00
0.05 0.069103 2.01 0.0011327 4.00
0.025 0.034263 0.0002831
If the forward difference operator is applied, the error is approximately
halved when the increment is halved. If the case of central difference, the
error is approximately reduced by a factor 4 when h is halved. Such results
suggest that the errors behave linearly as h ! 0 in the first case, and
quadratically, O(h 2 ), in the other case.
Thm. II.1.1 (On truncation error)Let f be sufficiently regular and let ET
be its truncation error. Then ET = O(h) in case of a forward difference,
ET = O(h 2 ) in case of a central difference, as h ! 0. Besides, for the central
difference D+(D−) approximating f ′′ (x), the truncation error is O(h 2 ) as
h ! 0.
PROOF.
2