http://numericalmethods.eng.usf.edu 1 lagrangian interpolation major: all engineering majors...
TRANSCRIPT
http://numericalmethods.eng.usf.edu 1
Lagrangian Interpolation
Major: All 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
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.
http://numericalmethods.eng.usf.edu4
Interpolants
Polynomials are the most common choice of interpolants because they are easy to:
EvaluateDifferentiate, and Integrate.
http://numericalmethods.eng.usf.edu5
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.
http://numericalmethods.eng.usf.edu6
Example The upward velocity of a rocket is given as a
function of time in Table 1. Find the velocity at t=16 seconds using the Lagrangian method for linear interpolation.
Table Velocity as a function of time
Figure. Velocity vs. time data for the rocket example
(s) (m/s)
0 010 227.0415 362.7820 517.35
22.5 602.9730 901.67
t )(tv
http://numericalmethods.eng.usf.edu7
Linear Interpolation
10 12 14 16 18 20 22 24350
400
450
500
550517.35
362.78
y s
f range( )
f x desired
x s1
10x s0
10 x s range x desired
)()()(1
0ii
itvtLtv
)()()()( 1100 tvtLtvtL
78.362,15 00 tt
35.517,20 11 tt
http://numericalmethods.eng.usf.edu8
Linear Interpolation (contd)
1
00 0
0 )(
jj j
j
tt
tttL
10
1
tt
tt
1
10 1
1 )(
jj j
j
tt
tttL
01
0
tt
tt
)()()( 101
00
10
1 tvtt
tttv
tt
tttv
)35.517(1520
15)78.362(
2015
20
tt
)35.517(1520
1516)78.362(
2015
2016)16(
v
)35.517(2.0)78.362(8.0
7.393 m/s.
http://numericalmethods.eng.usf.edu9
Quadratic InterpolationFor the second order polynomial interpolation (also called quadratic interpolation), we
choose the velocity given by
2
0
)()()(i
ii tvtLtv
)()()()()()( 221100 tvtLtvtLtvtL
http://numericalmethods.eng.usf.edu10
Example The upward velocity of a rocket is given as a
function of time in Table 1. Find the velocity at t=16 seconds using the Lagrangian method for quadratic interpolation.
Table Velocity as a function of time
Figure. Velocity vs. time data for the rocket example
(s) (m/s)
0 010 227.0415 362.7820 517.35
22.5 602.9730 901.67
t )(tv
http://numericalmethods.eng.usf.edu11
Quadratic Interpolation (contd)
10 12 14 16 18 20200
250
300
350
400
450
500
550517.35
227.04
y s
f range( )
f x desired
2010 x s range x desired
,100 t 04.227)( 0 tv
,151 t 78.362)( 1 tv
,202 t 35.517)( 2 tv
2
00 0
0 )(
jj j
j
tt
tttL
20
2
10
1
tt
tt
tt
tt
2
10 1
1 )(
jj j
j
tt
tttL
21
2
01
0
tt
tt
tt
tt
2
20 2
2 )(
jj j
j
tt
tttL
12
1
02
0
tt
tt
tt
tt
http://numericalmethods.eng.usf.edu12
Quadratic Interpolation (contd)
m/s19.392
35.52712.078.36296.004.22708.0
35.5171520
1516
1020
101678.362
2015
2016
1015
101604.227
2010
2016
1510
151616
212
1
02
01
21
2
01
00
20
2
10
1
v
tvtt
tt
tt
tttv
tt
tt
tt
tttv
tt
tt
tt
tttv
The absolute relative approximate error obtained between the results from the first and second order polynomial is
a
%38410.0
10019.392
70.39319.392
a
http://numericalmethods.eng.usf.edu13
Cubic Interpolation
10 12 14 16 18 20 22 24200
300
400
500
600
700602.97
227.04
y s
f range( )
f x desired
22.510 x s range x desired
For the third order polynomial (also called cubic interpolation), we choose the velocity given by
3
0
)()()(i
ii tvtLtv
)()()()()()()()( 33221100 tvtLtvtLtvtLtvtL
http://numericalmethods.eng.usf.edu14
Example The upward velocity of a rocket is given as a
function of time in Table 1. Find the velocity at t=16 seconds using the Lagrangian method for cubic interpolation.
Table Velocity as a function of time
Figure. Velocity vs. time data for the rocket example
(s) (m/s)
0 010 227.0415 362.7820 517.35
22.5 602.9730 901.67
t )(tv
http://numericalmethods.eng.usf.edu15
Cubic Interpolation (contd) 04.227,10 oo tvt 78.362,15 11 tvt
35.517,20 22 tvt 97.602,5.22 33 tvt
3
00 0
0 )(
jj j
j
tt
tttL
30
3
20
2
10
1
tt
tt
tt
tt
tt
tt;
3
10 1
1 )(
jj j
j
tt
tttL
31
3
21
2
01
0
tt
tt
tt
tt
tt
tt
3
20 2
2 )(
jj j
j
tt
tttL
32
3
12
1
02
0
tt
tt
tt
tt
tt
tt;
3
30 3
3 )(
jj j
j
tt
tttL
23
2
13
1
03
0
tttt
tttt
tt
tt
10 12 14 16 18 20 22 24200
300
400
500
600
700602.97
227.04
y s
f range( )
f x desired
22.510 x s range x desired
http://numericalmethods.eng.usf.edu16
Cubic Interpolation (contd)
m/s06.392
97.6021024.035.517312.078.362832.004.2270416.0
97.602205.22
2016
155.22
1516
105.22
101635.517
5.2220
5.2216
1520
1516
1020
1016
78.3625.2215
5.2216
2015
2016
1015
101604.227
5.2210
5.2216
2010
2016
1510
151616
323
2
13
1
13
12
32
3
12
1
02
0
231
3
21
2
01
01
30
3
20
2
10
1
v
tvtt
tt
tt
tt
tt
tttv
tt
tt
tt
tt
tt
tt
tvtt
tt
tt
tt
tt
tttv
tt
tt
tt
tt
tt
tttv
The absolute relative approximate error obtained between the results from the first and second order polynomial is
a
%033269.0
10006.392
19.39206.392
a
http://numericalmethods.eng.usf.edu17
Comparison Table
Order of Polynomial
1 2 3
v(t=16) m/s 393.69 392.19 392.06
Absolute Relative
Approximate Error
--------0.38410
%0.033269
%
http://numericalmethods.eng.usf.edu18
Distance from Velocity Profile
Find the distance covered by the rocket from t=11s to
t=16s ?
,00544.013195.0265.21245.4)( 32 ttttv 5.2210 t
16
11
)()11()16( dttvss
16
11
32 )00544.013195.0265.21245.4( dtttt
1611
432
]4
00544.03
13195.02
265.21245.4[ttt
t
1605 m
)5727.2)(300065045()1388.4)(33755.7125.47(
)9348.1)(45008755.52()36326.0)(67505.10875.57()(2323
2323
tttttt
tttttttv
http://numericalmethods.eng.usf.edu19
Acceleration from Velocity Profile
,
Find the acceleration of the rocket at t=16s given that
32 00544.013195.0265.21245.4 tttdt
dtv
dt
dta
201632.026390.0265.21 tt
2)16(01632.0)16(26390.0265.21)16( a
665.29 2/ sm
,00544.013195.0265.21245.4)( 32 ttttv 5.2210 t
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
