lagrangian interpolation
DESCRIPTION
Lagrangian Interpolation. Civil Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Lagrange Method of Interpolation http://numericalmethods.eng.usf.edu. What is Interpolation ?. - PowerPoint PPT PresentationTRANSCRIPT
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Lagrangian Interpolation
Civil Engineering Majors
Authors: Autar Kaw, Jai Paul
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM Undergraduates
Lagrange 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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Lagrangian Interpolation
Lagrangian interpolating polynomial is given by
n
iiin xfxLxf
0
)()()(
where ‘ n ’ in )(xf n stands for the thn order polynomial that approximates the function )(xfy
given at )1( n data points as nnnn yxyxyxyx ,,,,......,,,, 111100 , and
n
ijj ji
ji xx
xxxL
0
)(
)(xLi is a weighting function that includes a product of )1( n terms with terms of ij
omitted.
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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 the Lagragian 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
1
0
)()()(i
ii zTzLzT
)()()()( 1100 zTzLzTzL
7.11,8 00 zTz
6.17,7 11 zTz
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Linear Interpolation (contd)
10
11
00 0
0 )(zz
zz
zz
zzzL
jj j
j
01
01
10 1
1 )(zz
zz
zz
zzzL
jj j
j
78),6.17(87
8)7.11(
78
7)()()( 1
01
00
10
1
zzz
zTzz
zzzT
zz
zzzT
C65.14
)6.17(5.0)7.11(5.0)6.17(87
85.7)7.11(
78
75.7)5.7(
T
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Quadratic InterpolationFor the second order polynomial interpolation (also called quadratic interpolation), we
choose the velocity given by
2
0
)()()(i
ii tvtLtv
)()()()()()( 221100 tvtLtvtLtvtL
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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 the Lagragian 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
9.9,9 oo zTz
7.11,8 11 zTz
6.17,7 22 zTz
2
00 0
0 )(
jj j
j
zz
zzzL
20
2
10
1
zz
zz
zz
zz
2
10 1
1 )(
jj j
j
zz
zzzL
21
2
01
0
zz
zz
zz
zz
2
20 2
2 )(
jj j
j
zz
zzzL
12
1
02
0
zz
zz
zz
zz
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Quadratic Interpolation (contd)
212
1
02
01
21
2
01
00
20
2
10
1)( zTzz
zz
zz
zzzT
zz
zz
zz
zzzT
zz
zz
zz
zzzT
C138.14
6.17375.07.1175.09.9125.0
6.178797
85.795.77.11
7898
75.795.79.9
7989
75.785.75.7
T
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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Cubic InterpolationFor the third order polynomial (also called cubic interpolation), we choose the
temperature given by
3
0
)()()(i
ii zTzLzT
)()()()()()()()( 33221100 zTzLzTzLzTzLzTzL
9 8.5 8 7.5 7 6.5 68
10
12
14
16
18
2019.19774
9.44745
y s
f range( )
f x desired
69 x s range x desired
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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 the Lagragian 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.edu15
Cubic Interpolation (contd)
9 8.5 8 7.5 7 6.5 68
10
12
14
16
18
2019.19774
9.44745
y s
f range( )
f x desired
69 x s range x desired
9.9,9 oo zTz 7.11,8 11 zTz
6.17,7 22 zTz 2.18,6 33 zTz
3
00 0
0 )(
jj j
j
zz
zzzL
30
3
20
2
10
1
zz
zz
zz
zz
zz
zz
3
10 1
1 )(
jj j
j
zz
zzzL
31
3
21
2
01
0
zz
zz
zz
zz
zz
zz
3
20 2
2 )(
jj j
j
zz
zzzL
32
3
12
1
02
0
zz
zz
zz
zz
zz
zz
3
30 3
3 )(
jj j
j
zz
zzzL
23
2
13
1
03
0
zz
zz
zz
zz
zz
zz
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Cubic Interpolation (contd)
323
2
13
1
03
02
30
3
10
1
02
0
131
3
21
2
01
00
30
3
20
2
10
1
zTzz
zz
zz
zz
zz
zzzT
zz
zz
zz
zz
zz
zz
zTzz
zz
zz
zz
zz
zzzT
zz
zz
zz
zz
zz
zzzT
30 zzz
2.18768696
75.785.795.76.17
678797
65.785.795.7
7.11687898
65.775.795.79.9
697989
65.775.785.75.7
T
2.180625.06.175625.07.115625.09.90625.0
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 °C 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
2.18768696
7896.17
678797
689
7.11687898
6799.9
697989
678
32
zzzz
zzzzzz
zzzzzzzT
69,4.91.71
69,7.41.7158.262
2
2
2
zzdz
Td
zzzdz
dT
Simply setting this expression equal to zero, we get
m5638.7
69,4.91.710
z
zz
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/lagrange_method.html
