Interpolation
Chapter 18
• Estimation of intermediate values between precise
data points. The most common method is:
f ( x )  a 0  a1 x  a 2 x    a n x
2
n
• 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
Chapter 18
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Figure 18.1
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Figure
18.2
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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 ( x 0 )
x  x0

f1 ( x )  f ( x 0 ) 
f ( x1 )  f ( x 0 )
x1  x 0
f ( x1 )  f ( x 0 )
x1  x 0
Slope and a
finite divided
difference
approximation to
1st derivative
( x  x 0 ) Linear-interpolation
formula
• f1(x) designates that this is a first-order interpolating
polynomial.
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Chapter 18
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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  x 0 )  b 2 ( x  x 0 )( x  x1 )
• A simple procedure can be used to determine the
values of the coefficients.
x  x0
b0  f ( x 0 )
x  x1
b1 
f ( x1 )  f ( x 0 )
x1  x 0
f ( x 2 )  f ( x1 )
x  x2
b2 
x 2  x1

f ( x1 )  f ( x 0 )
x1  x 0
x2  x0
7
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Chapter 18
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by Lale Yurttas, Texas
A&M University
Chapter 18
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General Form of Newton’s Interpolating Polynomials/
f n ( x )  f ( x 0 )  ( x  x 0 ) f [ x1 , x 0 ]  ( x  x 0 )( x  x1 ) f [ x 2 , x1 , x 0 ]
   ( x  x 0 )( x  x1 )  ( x  x n 1 ) f [ x n , x n 1 ,  , x 0 ]
b0  f ( x 0 )
b1  f [ x1 , x 0 ]
b 2  f [ x 2 , x1 , x 0 ]

b n  f [ x n , x n 1 ,  , x 1 , x 0 ]
f [ xi , x j ] 
f ( xi )  f ( x j )
f [ xi , x j , xk ] 
Bracketed function
evaluations are finite
divided differences
xi  x j
f [ xi , x j ]  f [ x j , xk ]
xi  xk

f [ x n , x n  1 ,  , x1 , x 0 ] 
f [ x n , x n  1 ,  , x1 ]  f [ x n  1 , x n  2 ,  , x 0 ]
Chapter
x 18x
n
0
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Chapter 18
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Chapter 18
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Chapter 18
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Chapter 18
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Chapter 18
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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:
Rn 
f
( n 1)
( )
( n  1)!
( x  x 0 )( x  x1 )  ( x  x n )
 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:
R n  f [ x n 1 , x n , x n 1 ,  , x 0 ]( x  x 0 )( x  x1 )  ( x  x n )
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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
fn ( x) 
 L ( x) f ( x )
i
i
i0
n
Li ( x ) 

j0
ji
x  xj
xi  x j
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f1 ( x ) 
x  x1
x 0  x1
f ( x0 ) 
x  x0
x1  x 0
f ( x1 )
 x  x 0  x  x 2 
 x  x1  x  x 2 
f2 (x) 
f ( x0 ) 
 x 0  x1  x 0  x 2 
 x1  x 0  x1  x 2 
 x  x 0  x  x1 

f ( x2 )
 x 2  x 0  x 2  x 1 
f ( x1 )
•As with Newton’s method, the Lagrange version has an
estimated error of:
n
R n  f [ x , x n , x n 1 ,  , x 0 ] ( x  x i )
i0
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Figure 18.10
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f1 ( x )  f ( x 0 )
x  x0

f1 ( x )  f ( x 0 ) 
f ( x1 )  f ( x 0 )
x1  x 0
f ( x1 )  f ( x 0 )
x1  x 0
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( x  x0 )
21
Chapter 18
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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 )  a 0  a1 x  a 2 x    a x 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 ( x 0 )  a 0  a1 x 0  a 2 x   a n x
2
0
f ( x1 )  a 0  a 1 x1  a 2 x   a n x
2
1
n
0
n
1

f ( x n )  a 0  a1 x n  a 2 x   a n x
2
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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Quadratic Splines
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Chapter 18
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Chapter 18
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Cubic Splines
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Chapter 18
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Chapter 18
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Chapter 18
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Chapter 18
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Chapter 18
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Chapter 18
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