Specifications

University of Pretoria etd – Combrinck, M (2006)
enough to reconstruct with confidence and an exact comparison of this method with the
other methods is not included in this study. Tartaras et. al. (2000) also applied smoothing
of data “… prior to and following differentiation”. This is a common technique applied to
reduce noise but ultimately alters data.
Numerical differentiation is an unstable method, because although the accuracy of the
formulas increases with smaller values of “h” (distance between successive nodes), this also
causes the round-off error to increase. Divisions by small numbers tend to exaggerate
round-off errors and should be avoided if possible. A better way of increasing the accuracy
is to use formulas derived from higher order polynomials with the same inter-node
distance “
h”, e.g. the three-point and five-point Lagrange polynomial-based formulas
(Burden and Faires, 1993).
Three different strategies for differentiation are compared in this study.
4.4.2.2.1 Lagrange three-point formula
The Lagrange three-point numerical differentiation formula is described in most text books
on numerical analysis and a detailed derivation can be found in Burden and Faires (1993).
In short, derivative values of discretely sampled data points are approximated by the
analytical derivative of a second order Lagrange polynomial fitted through three
consecutive data points, i.e.
[]
)for 10.4(.};;{,)()()()()()(
)(
))((
))((
)(
))((
))((
)(
))((
))((
)(
111111
1
111
1
11
11
1
111
1
+−++−−
+
+−+
−
+−
+−
−
+−−
+
∈++=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−−
−−
+
−−
−−
+
−−
−−
≈
′
iiiiiiiiix
i
iiii
ii
i
iiii
ii
i
iiii
ii
x
xxxxxfxLxfxLxfxLD
xf
xxxx
xxxx
xf
xxxx
xxxx
xf
xxxx
xxxx
Dxf
Effectively the method reduces to a sum of weighted function values and for the special
case of calculating f’(x
i
) (the derivative at the centre point) for equally spaced data points it
reduces to
).
1
(
)
1
(
2
1
)(0)
1
(
2
11
)(
−
−=
⎟
⎠
⎞
⎜
⎝
⎛
+
⋅+⋅+
−
⋅−≈
′
i
x
i
xh
i
xf
i
xf
i
xf
h
i
xf
where
(4.11)
End points of equally spaced values have weights of (-1.5; 2; -0.5) and (0.5; -2; 1.5) for
first and last points respectively while the weights for unequally spaced data points have to
be calculated for every point using equation 4.10. Errors (and noise) in the calculated
derivative are dependent on sampling interval, errors in data values (small errors in data are
39