Geology Reference
In-Depth Information
f
i
+
1
−
f
i
K
+
f
(
x
x
i
)
s
(
x
)
i
=
x
i
+
−
x
i
+
1
)
+
(
x
−
x
i
+
1
−
6
f
f
x
i
2 (
x
x
i
)
2
,
i
+
1
−
1
6
i
+
−
x
i
+
1
)(
x
−
x
i
)
+
(
x
−
(1.546)
x
i
+
1
−
and
f
f
x
i
x
x
i
)
.
f
K
+
−
1
3
i
i
+
1
i
s
(
x
)
=
+
−
x
i
+
1
+
2 (
x
−
(1.547)
3
x
i
+
1
−
f
f
Thus, to match second derivatives at the end points, the constant is
K
=
+
.
i
+
1
i
On the interval (
x
i
,
x
i
+
1
), the first derivative of the approximating function is
f
2
f
i
(
x
x
i
)
i
+
1
+
f
i
+
1
−
f
i
s
(
x
)
=
x
i
+
−
x
i
+
1
)
+
(
x
−
x
i
+
1
−
6
f
x
i
+
1
−
x
i
2 (
x
f
i
x
i
)
2
,
i
+
1
−
1
6
+
−
x
i
+
1
)(
x
−
x
i
)
+
(
x
−
(1.548)
while on the interval (
x
i
−
1
,
x
i
), it is
f
i
+
2
f
i
−
1
(
x
x
i
−
1
)
f
i
−
1
x
i
−
x
i
−
1
+
f
i
−
s
(
x
)
=
−
x
i
)
+
(
x
−
6
x
i
−
1
2 (
x
x
i
−
1
)
2
.
f
f
−
1
6
i
i
−
1
+
−
x
i
)(
x
−
x
i
−
1
)
+
(
x
−
(1.549)
x
i
−
To ensure continuity of first derivatives, expressions (1.548) and (1.549) are
equated at the common node,
x
=
x
i
,togive
x
i
−
1
)
f
x
i
−
1
)
f
x
i
)
f
(
x
i
−
+
2 (
x
i
+
1
−
+
(
x
i
+
1
−
i
−
1
i
i
+
1
6
6 (
x
i
+
1
−
x
i
−
1
)
6
x
i
+
1
=
x
i
−
1
f
i
−
1
−
x
i
−
1
)
f
i
+
f
i
+
1
.
(1.550)
x
i
−
(
x
i
+
1
−
x
i
)(
x
i
−
−
x
i
The condition (1.550) applies at all internal nodes (
i
=
2,...,
N
−
1), giving
N
−
2
equations in the
N
second derivatives.
Two further conditions are required to express the second derivatives at the nodes
entirely in terms of the function values there. In the conventional description of
natural splines
, the second derivatives at the end nodes are taken to vanish. This
corresponds to the spline draughting tool being clamped at the end points. Instead,
we choose to make the highest derivatives as smooth as possible, taking the third
derivatives to be the same in each of the first two intervals and in each of the last
two intervals. Di
erentiating expression (1.547), the third derivative on the interval
(
x
i
,
x
i
+
1
) is found to be
ff
f
f
−
i
+
1
i
s
(
x
)
=
x
i
.
(1.551)
x
i
+
1
−
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