http://numericalmethods.eng.usf.edu 1 lagrangian interpolation computer engineering majors authors:...
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http://numericalmethods.eng.usf.edu 1
Lagrangian Interpolation
Computer Engineering Majors
Authors: Autar Kaw, Jai Paul
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM Undergraduates
Lagrange Method of Interpolation
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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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Example A robot arm with a rapid laser scanner is doing a quick quality
check on holes drilled in a rectangular plate. The hole centers in the plate that describe the path the arm needs to take are given below.
If the laser is traversing from x = 2 to x = 4.25 in a linear path, find the value of y at x = 4 using the Lagrange method for linear interpolation.
x (m) y (m)
2 7.24.25 7.15.25 6.07.81 5.09.2 3.510.6 5.0
Figure 2 Location of holes on the rectangular plate.
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Linear Interpolation
1
0
)()()(i
ii xyxLxy
)()()()( 1100 xyxLxyxL
5 0 5 107.08
7.1
7.12
7.14
7.16
7.18
7.27.2
7.1
y s
f range( )
f x desired
x s1
10x s0
10 x s range x desired
2.7,00.2 00 xyx
1.7,25.4 11 xyx
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Linear Interpolation (contd)
1
00 0
0 )(
jj j
j
xx
xxxL
10
1
xx
xx
1
10 1
1 )(
jj j
j
xx
xxxL
01
0
xx
xx
25.400.2,1.700.225.4
00.22.7
25.400.2
25.4
)( 101
00
10
1
xxx
xyxx
xxxy
xx
xxxy
in.1111.7
)1.7(88889.0)2.7(11111.0
)1.7(00.225.4
00.200.4)2.7(
25.400.2
25.400.4)00.4(
y
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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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Example A robot arm with a rapid laser scanner is doing a quick quality
check on holes drilled in a rectangular plate. The hole centers in the plate that describe the path the arm needs to take are given below.
If the laser is traversing from x = 2 to x = 4.25 in a linear path, find the value of y at x = 4 using the Lagrange method for quadratic interpolation.
x (m) y (m)
2 7.24.25 7.15.25 6.07.81 5.09.2 3.510.6 5.0
Figure 2 Location of holes on the rectangular plate.
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Quadratic Interpolation
2
00 0
0 )(
jj j
j
xx
xxxL
20
2
10
1
xx
xx
xx
xx
2
10 1
1 )(
jj j
j
xx
xxxL
21
2
01
0
xx
xx
xx
xx
2
20 2
2 )(
jj j
j
xx
xxxL
12
1
02
0
xx
xx
xx
xx
2 2.5 3 3.5 4 4.5 5 5.56
6.5
7
7.5
87.56258
6
y s
f range( )
f x desired
5.252 x s range x desired
2.7,00.2 oo xyx
1.7,25.4 11 xyx
0.6,25.5 22 xyx
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Quadratic Interpolation (contd)
)()()()( 212
1
02
01
21
2
01
00
20
2
10
1 xyxx
xx
xx
xxxy
xx
xx
xx
xxxy
xx
xx
xx
xxxy
0.6
25.425.500.225.5
25.400.400.200.41.7
25.525.400.225.4
25.500.400.200.42.7
25.500.225.400.2
25.500.425.400.400.4
y
0.615385.01.71111.12.7042735.0
in.2735.7
The absolute relative approximate error a obtained between the results from the first and second order
polynomial is
1002735.7
1111.72735.7
a
%2327.2
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Comparison Table
Order of Polynomial
1 2
Location (in.) 7.1111 7.2735
Absolute Relative Approximate Error
---------- 2.2327%
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Example A robot arm with a rapid laser scanner is doing a quick quality
check on holes drilled in a rectangular plate. The hole centers in the plate that describe the path the arm needs to take are given below.
If the laser is traversing from x = 2 to x = 4.25 in a linear path, find the value of y at x = 4 using a fifth order Lagrange polynomial.
x (m) y (m)
2 7.24.25 7.15.25 6.07.81 5.09.2 3.510.6 5.0
Figure 2 Location of holes on the rectangular plate.
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Fifth Order Interpolation
5
0
)()()(i
ii xyxLxy
)()()()()()()()()()()()( 554433221100 xyxLxyxLxyxLxyxLxyxLxyxL
2.7,00.2 oo xyx
1.7,25.4 11 xyx
0.6,25.5 22 xyx
0.5,81.7 33 xyx
5.3,20.9 44 xyx
0.5,60.10 55 xyx
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Fifth Order Interpolation (contd)
5
00 0
0 )(
jj j
j
xx
xxxL
50
5
40
4
30
3
20
2
10
1
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
5
10 1
1 )(
jj j
j
xx
xxxL
51
5
41
4
31
3
21
2
01
0
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
5
20 2
2 )(
jj j
j
xx
xxxL
52
5
42
4
32
3
12
1
02
0
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
5
30 3
3 )(
jj j
j
xx
xxxL
53
5
43
4
23
2
13
1
03
0
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
5
40 4
4 )(
jj j
j
xx
xxxL
54
5
34
3
24
2
14
1
04
0
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
5
50 5
5 )(
jj j
j
xx
xxxL
45
4
35
3
25
2
15
1
05
0
xx
xx
xx
xx
xx
xx
xx
xx
xx
xx
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Fifth Order Polynomial (contd)
)(
)(
)(
)(
)(
)()(
545
4
35
3
25
2
15
1
05
0
454
5
34
3
24
2
14
1
04
0
353
5
43
4
23
2
13
1
03
0
252
5
42
4
32
3
12
1
02
0
151
5
41
4
31
3
21
2
01
0
050
5
40
4
30
3
20
2
10
1
xyxx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xyxx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xyxx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xyxx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xyxx
xx
xx
xx
xx
xx
xx
xx
xx
xx
xyxx
xx
xx
xx
xx
xx
xx
xx
xx
xxxy
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Fifth Order Polynomial (contd)
)0.5()20.960.10)(81.760.10)(25.560.10)(25.460.10)(00.260.10(
)20.9)(81.7)(25.5)(25.4)(00.2(
)5.3()60.1020.9)(81.720.9)(25.520.9)(25.420.9)(00.220.9(
)60.10)(81.7)(25.5)(25.4)(00.2(
)0.5()60.1081.7)(20.981.7)(25.581.7)(25.481.7)(00.281.7(
)60.10)(20.9)(25.5)(25.4)(00.2(
)0.6()60.1025.5)(20.925.5)(81.725.5)(25.425.5)(00.225.5(
)60.10)(20.9)(81.7)(25.4)(00.2(
)1.7()60.1025.4)(20.925.4)(81.725.4)(25.525.4)(00.225.4(
)60.10)(20.9)(81.7)(25.5)(00.2(
)2.7()60.1000.2)(20.900.2)(81.700.2)(25.500.2)(25.400.2(
)60.10)(20.9)(81.7)(25.5)(25.4()(
xxxxx
xxxxx
xxxxx
xxxxx
xxxxx
xxxxxxy
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Fifth Order Polynomial (contd)
24.228
4.32065.37277.157378.30851.28
273.78
3.36946.42412.175781.33591.29
069.41
8.43514.49121.198453.3663.31
304.29
9.64735.69033.257222.43386.33
461.35
1.79975.81697.287983.46286.34
38.365
16994128622.377377.53611.37
2345
2345
2345
2345
2345
2345
xxxxx
xxxxx
xxxxx
xxxxx
xxxxx
xxxxxxy
6.102
,0072923.023091.07862.2855.15344.41898.30)( 5432
x
xxxxxxy
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Fifth Order Polynomial (contd)
6.102
,0072929.023093.07865.2858.15351.41900.30)( 5432
x
xxxxxxy
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
