This preview shows pages 1–3. Sign up to view the full content.
05.06.1
Chapter 05.05
Spline Method of Interpolation
After reading this chapter, you should be able to:
1.
interpolate data using spline interpolation, and
2.
understand why spline interpolation is important.
What is interpolation?
Many times, data is given only at discrete points such as
,
,
0
0
y
x
1
1
,
y
x
, .
.....
,
1
1
,
n
n
y
x
,
n
n
y
x
,
.
So, how then does one find the value of
y
at any other value of
x
?
Well, a
continuous function
x
f
may be used to represent the
1
n
data values with
x
f
passing
through the
1
n
points (Figure 1).
Then one can find the value of
y
at any other value of
x
.
This is called
interpolation
.
Of course, if
x
falls outside the range of
x
for which the data is given, it is no longer
interpolation but instead is called
extrapolation
.
So what kind of function
x
f
should one choose?
A polynomial is a common
choice for an interpolating function because polynomials are easy to
(A)
evaluate,
(B)
differentiate, and
(C)
integrate
relative to other choices such as a trigonometric and exponential series.
Polynomial interpolation involves finding a polynomial of order
n
that passes
through the
1
n
points.
Several methods to obtain such a polynomial include the direct
method, Newton’s divided difference polynomial method and the Lagrangian interpolation
method.
So is the spline method yet another method of obtaining this
th
n
order polynomial.
…… NO!
Actually, when
n
becomes large, in many cases, one may get oscillatory behavior
in the resulting polynomial.
This was shown by Runge when he interpolated data based on a
simple function of
2
25
1
1
x
y
on an interval of [–1, 1].
For example, take six equidistantly spaced points in [–1, 1] and find
y
at these points as given in Table 1.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document05.04.2
Chapter 05.04
Figure 1
Interpolation of discrete data.
Table 1
Six equidistantly spaced points in [–1, 1].
Now through these six points, one can pass a fifth order polynomial
,
10
6731
.
5
10
0004
.
1
7308
.
1
10
3651
.
3
2019
.
1
10
1378
.
3
)
(
1
11
2
3
11
4
5
11
5
x
x
x
x
x
x
f
1
1
x
through the six data points.
On plotting the fifth order polynomial (Figure 2) and the original
function, one can see that the two do not match well.
One may consider choosing more
points in the interval [–1, 1] to get a better match, but it diverges even more (see Figure 3),
where 20 equidistant points were chosen in the interval [–1, 1] to draw a 19th order
polynomial.
In fact, Runge found that as the order of the polynomial becomes infinite, the
polynomial diverges in the interval of
726
.
0
1
x
and
1
726
.
0
x
.
So what is the answer to using information from more data points, but at the same
time keeping the function true to the data behavior?
The answer is in spline interpolation.
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 Kaw,A

Click to edit the document details