Chap_18.ppt on interpolation basics undersatanding

by Lale Yurttas, Texas A
&M University
Chapter 18 1
Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Interpolation
Chapter 18
• Estimation of intermediate values between precise
data points. The most common method is:
• Although there is one and only one nth-order
polynomial that fits n+1 points, there are a variety of
mathematical formats in which this polynomial can be
expressed:
– The Newton polynomial
– The Lagrange polynomial
n
n x
a
x
a
x
a
a
x
f 



 
2
2
1
0
)
(

&M University
Chapter 18 2
Figure 18.1

&M University
Chapter 18 3
Newton’s Divided-Difference
Interpolating Polynomials
Linear Interpolation/
• Is the simplest form of interpolation, connecting two data
points with a straight line.
• f1(x) designates that this is a first-order interpolating
polynomial.
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
Linear-interpolation
formula
Slope and a
finite divided
difference
approximation to
1st
derivative

&M University
Chapter 18 4
Figure
18.2

&M University
Chapter 18 5
Quadratic Interpolation/
• If three data points are available, the estimate is
improved by introducing some curvature into the line
connecting the points.
• A simple procedure can be used to determine the
values of the coefficients.
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&M University
Chapter 18 6
General Form of Newton’s Interpolating Polynomials/
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Bracketed function
evaluations are finite
divided differences

&M University
Chapter 18 7
Errors of Newton’s Interpolating Polynomials/
• Structure of interpolating polynomials is similar to the Taylor
series expansion in the sense that finite divided differences are
added sequentially to capture the higher order derivatives.
• For an nth
-order interpolating polynomial, an analogous
relationship for the error is:
• For non differentiable functions, if an additional point f(xn+1) is
available, an alternative formula can be used that does not
require prior knowledge of the function:
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1
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1
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n
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x
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R 
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 
 

Is somewhere
containing the unknown
and he data

&M University
Chapter 18 8
Lagrange Interpolating Polynomials
• The Lagrange interpolating polynomial is simply a
reformulation of the Newton’s polynomial that
avoids the computation of divided differences:









n
i
j
j j
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i
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x
f
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x
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0
0
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(
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(

&M University
Chapter 18 9
  
  
  
  
  
  
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1
2
1
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1
0
0
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0
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1
x
f
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











•As with Newton’s method, the Lagrange version has an
estimated error of:


 

n
i
i
n
n
n x
x
x
x
x
x
f
R
0
0
1 )
(
]
,
,
,
,
[ 

&M University
Chapter 18 10
Figure 18.10

&M University
Chapter 18 11
Coefficients of an Interpolating
Polynomial
• Although both the Newton and Lagrange
polynomials are well suited for determining
intermediate values between points, they do not
provide a polynomial in conventional form:
• Since n+1 data points are required to determine n+1
coefficients, simultaneous linear systems of
equations can be used to calculate “a”s.
n
x x
a
x
a
x
a
a
x
f 



 
2
2
1
0
)
(

&M University
Chapter 18 12
n
n
n
n
n
n
n
n
n
n
x
a
x
a
x
a
a
x
f
x
a
x
a
x
a
a
x
f
x
a
x
a
x
a
a
x
f
















2
2
1
0
1
2
1
2
1
1
0
1
0
2
0
2
0
1
0
0
)
(
)
(
)
(
Where “x”s are the knowns and “a”s are the unknowns.

&M University
Chapter 18 13
Figure 18.13

&M University
Chapter 18 14
Spline Interpolation
• There are cases where polynomials can lead to
erroneous results because of round off error
and overshoot.
• Alternative approach is to apply lower-order
polynomials to subsets of data points. Such
connecting polynomials are called spline
functions.

&M University
Chapter 18 15
Figure 18.14

&M University
Chapter 18 16
Figure 18.15

&M University
Chapter 18 17
Figure 18.16

&M University
Chapter 18 18
Figure 18.17

Chap_18.ppt on interpolation basics undersatanding

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Chap_18.ppt on interpolation basics undersatanding