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The next approximation would be to represent it as a linear interpolation between
the end point values,
x
−
x
i
x
i
f
i
+
1
f
i
.
s
(
x
)
=
f
i
+
−
(1.543)
x
i
+
1
−
The approximating function then appears as a series of trapeziums. While this
approximating function is now continuous, it has discontinuous first derivatives
at the nodes. This leads us to consider higher order approximation. As with all
higher order methods, a caveat is to compare the interpolation with the function
values, especially in poorly constrained intervals, as they can lead to divergences
from expected behaviours.
If we wish to match the function values and first derivatives at the end points
of the interval, we require four free parameters and
s
(
x
) becomes a cubic. If we
allow discontinuities in the second derivatives at the nodes, the cubic approximat-
ing function is called a
local
or
Hermite
spline. The term
spline
comes from the
name of a draughting tool used to draw curves that are smooth and have continuous
curvature. If the approximating function
s
(
x
) is continuous at the nodes, and has
there not only continuous first derivatives, but also continuous second derivatives,
it is called a
natural spline
. We will first consider interpolation and approximation
by natural splines.
1.6.1 Natural splines
By analogy to (1.543), the second derivatives at the end points of the interval
(
x
i
,
x
i
+
1
) will be correctly represented if
x
i
+
1
−
x
i
f
f
i
,
−
x
x
i
s
(
x
)
f
=
+
−
(1.544)
i
i
+
1
where each superscript prime indicates di
ff
erentiation. This leads us to postulate
the form
x
i
x
i
+
1
−
x
−
x
i
f
i
+
1
f
i
s
(
x
)
=
f
i
+
−
K
f
i
x
i
f
(
x
−
x
i
)(
x
−
x
i
+
1
)
6
x
−
x
i
x
i
+
1
−
f
+
+
+
−
,
(1.545)
i
i
+
1
for an approximating function that matches both function values and second deriv-
atives at the end points of the interval (
x
i
,
x
i
+
1
). Note that, while the nodes
x
j
and
function values
f
j
are known, the constant
K
and second derivatives
f
j
are as yet
unknown. Then,
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