Cubic Splines
James Keesling
1
Definition of Cubic Spline
Given a function
f
(
x
) defined on an interval [
a, b
] we want to fit a curve through the points
{
(
x
0
, f
(
x
0
))
,
(
x
1
, f
(
x
1
))
, . . . ,
(
x
n
, f
(
x
n
))
}
as an approximation of the function
f
(
x
).
We
assume that the points are given in order
a
=
x
0
< x
1
< x
2
<
· · ·
< x
n
=
b
and let
h
i
=
x
i
+1

x
i
. The method of approximation we describe is called
cubic spline interpolation
.
The cubic spline is a function
S
(
x
) on [
a, b
] with the following properties.
S
(
x
)
[
x
i
,x
i
+1
]
=
S
i
(
x
) is a cubic polynomial for
i
= 0
,
1
,
2
, . . . , n

1
S
i
(
x
i
) =
f
(
x
i
) for
i
= 0
,
1
,
2
, . . . , n

1
S
i
(
x
i
+1
) =
f
(
x
i
+1
) for
i
= 0
,
1
,
2
, . . . , n

1
S
0
i
(
x
i
+1
) =
S
0
i
+1
(
x
i
+1
) for
i
= 0
,
1
,
2
, . . . , n

2
S
00
i
(
x
i
+1
) =
S
00
i
+1
(
x
i
+1
) for
i
= 0
,
1
,
2
, . . . , n

2
S
00
0
(
x
0
) =
S
00
n

1
(
x
n
) [
free boundary condition
] or
S
0
0
(
x
0
) =
f
0
(
x
0
) and
S
0
n

1
(
x
n
) =
f
0
(
x
n
) [
clamped boundary condition
]
2
Determining the Coefficients of the Cubic Polynomials
Since each
S
i
(
x
) =
a
i
+
b
i
·
(
x

x
i
) +
c
i
·
(
x

x
i
)
2
+
d
i
·
(
x

x
i
)
3
has four constants to be
determined, we have 4
n
unknowns and the above conditions give us 4
n
equations. For the
free boundary case we can simplify the solutions of the equations to the following.
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 Summer '08
 Kyung
 Statistics, Polynomial interpolation, Spline interpolation, Cubic Hermite spline, cubic spline

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