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http://numericalmethods.eng.usf.edu 1
Newton’s Divided Difference Polynomial
Method of Interpolation
Civil Engineering Majors
Authors: Autar Kaw, Jai Paul
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM
Undergraduates
Newton’s Divided Difference Method of
Interpolation
http://numericalmethods.eng.usf.edu
http://numericalmethods.eng.usf.edu3
What is Interpolation ?
Given (x0,y0), (x1,y1), …… (xn,yn), find the value of ‘y’ at a value of ‘x’ that is not given.
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Interpolants
Polynomials are the most common choice of interpolants because they are easy to:
EvaluateDifferentiate, and Integrate.
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Newton’s Divided Difference Method
Linear interpolation: Given pass a linear interpolant through the data
where
),,( 00 yx ),,( 11 yx
)()( 0101 xxbbxf
)( 00 xfb
01
011
)()(
xx
xfxfb
http://numericalmethods.eng.usf.edu6
ExampleTo maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using Newton’s Divided Difference method for linear interpolation.
Temperature vs. depth of a lake
Temperature Depth
T (oC) z (m)19.1 019.1 -119 -2
18.8 -318.7 -418.3 -518.2 -617.6 -711.7 -89.9 -99.1 -10
http://numericalmethods.eng.usf.edu7
Linear Interpolation
15 10 5 011
12
13
14
15
16
17
1817.6
11.7
y s
f range( )
f x desired
x s1
10x s0
10 x s range x desired
)()( 010 zzbbzT
,80 z 7.11)( 0 zT
,71 z 6.17)( 1 zT
)( 00 zTb
7.11
01
011
)()(
zz
zTzTb
87
7.116.17
9.5
http://numericalmethods.eng.usf.edu8
Linear Interpolation (contd)
15 10 5 011
12
13
14
15
16
17
1817.6
11.7
y s
f range( )
f x desired
x s1
10x s0
10 x s range x desired
)()( 010 zzbbzT
),8(9.57.11 z 78 z
At 5.7z
)85.7(9.57.11)5.7( T
C 65.14
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Quadratic InterpolationGiven ),,( 00 yx ),,( 11 yx and ),,( 22 yx fit a quadratic interpolant through the data.
))(()()( 1020102 xxxxbxxbbxf
)( 00 xfb
01
011
)()(
xx
xfxfb
02
01
01
12
12
2
)()()()(
xx
xx
xfxf
xx
xfxf
b
http://numericalmethods.eng.usf.edu10
ExampleTo maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using Newton’s Divided Difference method for quadratic interpolation.
Temperature vs. depth of a lake
Temperature Depth
T (oC) z (m)19.1 019.1 -119 -2
18.8 -318.7 -418.3 -518.2 -617.6 -711.7 -89.9 -99.1 -10
http://numericalmethods.eng.usf.edu11
Quadratic Interpolation (contd)
9 8.5 8 7.5 78
10
12
14
16
1817.6
9.89238
y s
f range( )
f x desired
79 x s range x desired
))(()()( 102010 zzzzbzzbbzT
,90 z 9.9)( 0 zT
,81 z 7.11)( 1 zT
,72 z 6.17)( 2 zT
http://numericalmethods.eng.usf.edu12
Quadratic Interpolation (contd)
9.9)( 00 zTb
8.178
9.97.11)()(
01
011
zz
zTzTb
02
01
01
12
12
2
)()()()(
zz
zz
zTzT
zz
zTzT
b
97
98
9.97.11
87
7.116.17
2
8.19.5
05.2
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Quadratic Interpolation (contd)
))(()()( 102010 zzzzbzzbbzT
),8)(9(05.2)9(8.19.9 zzz 79 z
At ,5.7z
)85.7)(95.7(05.2)95.7(8.19.9)5.7( T
C 138.14 The absolute relative approximate error a obtained between the results
from the first and second order polynomial is
100138.14
65.14138.14
a
%6251.3
http://numericalmethods.eng.usf.edu14
General Form
))(()()( 1020102 xxxxbxxbbxf
where
Rewriting))(](,,[)](,[][)( 1001200102 xxxxxxxfxxxxfxfxf
)(][ 000 xfxfb
01
01011
)()(],[
xx
xfxfxxfb
02
01
01
12
12
02
01120122
)()()()(
],[],[],,[
xx
xx
xfxf
xx
xfxf
xx
xxfxxfxxxfb
http://numericalmethods.eng.usf.edu15
General FormGiven )1( n data points, nnnn yxyxyxyx ,,,,......,,,, 111100 as
))...()((....)()( 110010 nnn xxxxxxbxxbbxf
where
][ 00 xfb
],[ 011 xxfb
],,[ 0122 xxxfb
],....,,[ 0211 xxxfb nnn
],....,,[ 01 xxxfb nnn
http://numericalmethods.eng.usf.edu16
General formThe third order polynomial, given ),,( 00 yx ),,( 11 yx ),,( 22 yx and ),,( 33 yx is
))()(](,,,[
))(](,,[)](,[][)(
2100123
1001200103
xxxxxxxxxxf
xxxxxxxfxxxxfxfxf
0b
0x )( 0xf 1b
],[ 01 xxf 2b
1x )( 1xf ],,[ 012 xxxf 3b
],[ 12 xxf ],,,[ 0123 xxxxf
2x )( 2xf ],,[ 123 xxxf
],[ 23 xxf
3x )( 3xf
http://numericalmethods.eng.usf.edu17
ExampleTo maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using Newton’s Divided Difference method for cubic interpolation.
Temperature vs. depth of a lake
Temperature Depth
T (oC) z (m)19.1 019.1 -119 -2
18.8 -318.7 -418.3 -518.2 -617.6 -711.7 -89.9 -99.1 -10
http://numericalmethods.eng.usf.edu18
Example
The temperature profile is chosen as
))()(())(()()( 2103102010 zzzzzzbzzzzbzzbbzT
We need to choose four data points that are closest to 5.7z
,90 z 9.9)( 0 zT
,81 z 7.11)( 1 zT
,72 z 6.17)( 2 zT
,63 z 2.18)( 3 zT
http://numericalmethods.eng.usf.edu19
Example
The values of the constants are obtained as
0b 9.9 1b 8.1 2b 05.2 3b 5667.1
0b
,90 z 9.9 1b
8.1 2b
,81 z 7.11 05.2 3b
9.5 5667.1
,72 z 6.17 65.2
6.0
,63 z 2.18
http://numericalmethods.eng.usf.edu20
Example
))()(())(()()( 2103102010 zzzzzzbzzzzbzzbbzT
)7)(8)(9(5667.1)8)(9(05.2)9(8.19.9 zzzzzz , −9 ≤ z ≤ −6
At ,5.7z
)75.7)(85.7)(95.7(5667.1
)85.7)(95.7(05.2)95.7(8.19.9)5.7(
T
C 725.14 The absolute relative approximate error a obtained between the
results from the second and third order polynomial is
100725.14
138.14725.14
a
%9898.3
http://numericalmethods.eng.usf.edu21
Comparison Table
Order of Polynomial
1 2 3
Temperature (oC) 65.14 138.14 725.14
Absolute Relative Approximate Error
---------- %6251.3 %9898.3
http://numericalmethods.eng.usf.edu22
ThermoclineWhat is the value of depth at which the thermocline exists?
The position where the thermocline exists is given where 02
2
dz
Td
.
69,5667.155.3558.2629.615
69,7895667.18905.298.19.932
zzzz
zzzzzzzzT
69,7.41.7158.262 2 zzzdz
dT
69,4.91.712
2
zzdz
Td
Simply setting this expression equal to zero, we get
69,4.910.710 zz
m5638.7z
Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit
http://numericalmethods.eng.usf.edu/topics/newton_divided_difference_method.html