Graphics Programs Reference
In-Depth Information
Interpolation and Curve Fitting
3
Given the
n
datapoints (
x
i
,
y
i
)
,
i
=
1
,
2
,...,
n
, estimate
y
(
x
).
3.1
Introduction
Discrete data sets, or tables of the form
x
1
x
2
x
3
···
x
n
y
1
y
2
y
3
···
y
n
arecommonly involvedintechnicalcalculations. The source of the data may beex-
perimentalobservationsor numericalcomputations. There is adistinctionbetween
interpolation and curve fitting. Ininterpolationweconstruct a curvethrough the data
points. Indoing so, we make the implicit assumption that the datapoints are accurate
and distinct. Curve fitting is applied to data thatcontain scatter (noise), usuallydueto
measurementerrors. Here we wanttofind a smooth curvethat approximates the data
in somesense. Thus the curve does not havetohit the datapoints. This difference
betweeninterpolation and curve fitting is illustrated in Fig. 3.1.
3.2
Polynomial Interpolation
Lagrange's Method
The simplest formof an interpolant is apolynomial. It is always possible to construct a
unique polynomial
P
n
−
1
(
x
)ofdegree
n
−
1 that passes through
n
distinct datapoints.
103
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