Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 1

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

nxaxaxaaxf ++++= L2

210)(

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 2

Figure 18.1

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 3

Figure

18.2

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 4

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.

)()()(

)()(

)()()()(

0

01

0101

01

01

0

01

xxxx

xfxfxfxf

xx

xfxf

xx

xfxf

−−

−+=

−

−=

−

−

Linear-interpolation

formula

Slope and a

finite divided

difference

approximation to

1st derivative

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 5

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 6

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

7

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.

))(()()( 1020102 xxxxbxxbbxf −−+−+=

02

01

01

12

12

22

01

0111

000

)()()()(

)()(

)(

xx

xx

xfxf

xx

xfxf

bxx

xx

xfxfbxx

xfbxx

−

−

−−

−

−

==

−

−==

==

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 8

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

by Lale Yurttas, Texas

A&M University

Chapter 18 9

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 10

General Form of Newton’s Interpolating Polynomials/

0

02111011

011

0122

011

00

01110

012100100

],,,[],,,[],,,,[

],[],[],,[

)()(],[

],,,,[

],,[

],[

)(

],,,[)())((

],,[))((],[)()()(

xx

xxxfxxxfxxxxf

xx

xxfxxfxxxf

xx

xfxfxxf

xxxxfb

xxxfb

xxfb

xfb

xxxfxxxxxx

xxxfxxxxxxfxxxfxf

n

nnnnnn

ki

kjji

kji

ji

ji

ji

nnn

nnn

n

−

−=

−

−=

−

−=

=

=

=

=

−−−++

−−+−+=

−−−−

−

−−

KKK

M

L

M

LLL

Bracketed function

evaluations are finite

divided differences

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 11

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 12

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 13

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 14

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 15

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 16

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:

)())(()!1(

)(10

)1(

n

n

n xxxxxxn

fR −−−

+=

+

Lξ

)())(](,,,,[ 10011 nnnnn xxxxxxxxxxfR −−−≅ −+ LK

ξ Is somewhere

containing the unknown

and he data

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 17

Lagrange Interpolating Polynomials

• The Lagrange interpolating polynomial is simply a

reformulation of the Newton’s polynomial that

avoids the computation of divided differences:

∏

∑

≠=

=

−

−=

=

n

ijj ji

j

i

n

i

iin

xx

xxxL

xfxLxf

0

0

)(

)()()(

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 18

( )( )( )( )

( )( )( )( )

( )( )( )( )

)(

)()()(

)()()(

2

1202

10

1

2101

200

2010

212

1

01

00

10

11

xfxxxx

xxxx

xfxxxx

xxxxxf

xxxx

xxxxxf

xfxx

xxxf

xx

xxxf

−−

−−+

−−

−−+

−−

−−=

−

−+

−

−=

•As with Newton’s method, the Lagrange version has an

estimated error of:

∏=

− −=n

i

innn xxxxxxfR0

01 )(],,,,[ K

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 19

Figure 18.10

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 20

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 21

)()()(

)()(

)()()()(

0

01

0101

01

01

0

01

xxxx

xfxfxfxf

xx

xfxf

xx

xfxf

−−

−+=

−

−=

−

−

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 22

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 23

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

xxaxaxaaxf ++++= L2

210)(

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 24

n

nnnnn

n

n

n

n

xaxaxaaxf

xaxaxaaxf

xaxaxaaxf

+++=

+++=

+++=

L

M

L

L

2

210

1

2

121101

0

2

020100

)(

)(

)(

Where “x”s are the knowns and “a”s are the unknowns.

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 25

Figure 18.13

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 26

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.

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 27

Figure 18.14

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 28

Figure 18.15

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 29

Figure 18.16

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 30

Figure 18.17

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 31

Quadratic Splines

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 32

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 33

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 34

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 35

Cubic Splines

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 36

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 37

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 38

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 39

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 40

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 41

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 18 42