newton’s divided difference polynomial method of interpolation
DESCRIPTION
Newton’s Divided Difference Polynomial Method of Interpolation. Civil Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. - PowerPoint PPT PresentationTRANSCRIPT
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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
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Newton’s Divided Difference Method of
Interpolation
http://numericalmethods.eng.usf.edu
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
![Page 18: Newton’s Divided Difference Polynomial Method of Interpolation](https://reader035.vdocument.in/reader035/viewer/2022062314/56813c5d550346895da5de3f/html5/thumbnails/18.jpg)
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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
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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
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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
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Comparison Table
Order of Polynomial
1 2 3
Temperature (oC) 65.14 138.14 725.14
Absolute Relative Approximate Error
---------- %6251.3 %9898.3
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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
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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