Interpolation
Chapter 18
• Estimation of intermediate values between precise
data points. The most common method is:
f ( x)  a0  a1x  a2 x2   an xn
• 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
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Figure 18.1
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Newton’s Divided-Difference
Interpolating Polynomials
Linear Interpolation/
• Is the simplest form of interpolation, connecting two data
points with a straight line.
f1 ( x)  f ( x0 ) f ( x1 )  f ( x0 )

x  x0
x  x0
Slope and a
finite divided
difference
approximation to
1st derivative
f ( x1 )  f ( x0 )
f1 ( x)  f ( x0 ) 
( x  x0 ) Linear-interpolation
x  x0
formula
• f1(x) designates that this is a first-order interpolating
polynomial.
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Figure
18.2
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Quadratic Interpolation/
• If three data points are available, the estimate is
improved by introducing some curvature into the line
connecting the points.
f 2 ( x)  b0  b1 ( x  x0 )  b2 ( x  x0 )(x  x1 )
• A simple procedure can be used to determine the
values of the coefficients.
x  x0
b0  f ( x0 )
x  x1
f ( x1 )  f ( x0 )
b1 
x  x0
x  x2
f ( x2 )  f ( x1 ) f ( x1 )  f ( x0 )

x2  x1
x1  x0
b2 
x2  x0
5
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General Form of Newton’s Interpolating Polynomials/
f n ( x)  f ( x0 )  ( x  x0 ) f [ x1 , x0 ]  ( x  x0 )(x  x1 ) f [ x2 , x1 , x0 ]
   ( x  x0 )(x  x1 )  ( x  xn 1 ) f [ xn , xn 1 ,  , x0 ]
b0  f ( x0 )
b1  f [ x1 , x0 ]
b2  f [ x2 , x1 , x0 ]

bn  f [ xn , xn 1 ,  , x1 , x0 ]
f [ xi , x j ] 
f ( xi )  f ( x j )
f [ xi , x j , xk ] 
xi  x j
f [ xi , x j ]  f [ x j , xk ]
Bracketed function
evaluations are finite
divided differences
xi  xk

f [ xn , xn 1 ,  , x1 , x0 ] 
f [ xn , xn 1 ,  , x1 ]  f [ xn 1 , xn  2 ,  , x0 ]
Chapter
xn 18x0
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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:
f ( n1) ( )
Rn 
( x  x0 )(x  x1 )( x  xn )
(n  1)!
 Is somewhere
containing the unknown
and he data
• 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:
Rn  f [ xn1, xn , xn1 ,, x0 ](x  x0 )(x  x1 )( x  xn )
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Lagrange Interpolating Polynomials
• The Lagrange interpolating polynomial is simply a
reformulation of the Newton’s polynomial that
avoids the computation of divided differences:
n
f n ( x)   Li ( x) f ( xi )
i 0
n
Li ( x)  
j 0
j i
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x  xj
xi  x j
Chapter 18
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x  x0
x  x1
f1 ( x) 
f ( x0 ) 
f ( x1 )
x0  x1
x1  x0


x  x0 x  x2 
x  x1 x  x2 
f 2 ( x) 
f ( x0 ) 
f ( x1 )
x0  x1 x0  x 2 
x1  x0 x1  x 2 

x  x0 x  x1 

f ( x2 )
x2  x0 x2  x1 
•As with Newton’s method, the Lagrange version has an
estimated error of:
n
Rn  f [ x, xn , xn1 ,, x0 ] ( x  xi )
i 0
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Figure 18.10
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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:
f ( x)  a0  a1x  a2 x   ax x
2
n
• Since n+1 data points are required to determine n+1
coefficients, simultaneous linear systems of equations
can be used to calculate “a”s.
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f ( x0 )  a0  a1 x0  a x   a x
2
2 0
n
n 0
f ( x1 )  a0  a1 x1  a x   a x
2
2 1
n
n 1

f ( xn )  a0  a1 xn  a x   a x
2
2 n
n
n n
Where “x”s are the knowns and “a”s are the unknowns.
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Figure 18.13
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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.
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Figure 18.14
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Figure 18.15
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Figure 18.16
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Figure 18.17
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