3.2.interpolation lagrange

1
SUPPLEMENTARY LECTURE NOTES
Lagrange Interpolation
After reading this lecture, you should be able to:
1. derive Lagrangian method of interpolation,
2. solve problems using Lagrangian method of interpolation, and
3. use Lagrangian interpolants to find derivatives and integrals of discrete functions.
What is interpolation?
Many times, data is given only at discrete points such as  ,, 00 yx  11, yx , ......,  11,  nn yx ,
 nn yx , . So, how then does one find the value of y at any other value of x ? Well, a
continuous function  xf may be used to represent the 1n data values with  xf
passing through the 1n points (Figure 1). Then one can find the value of y at any other
value of x . This is called interpolation.
Of course, if x falls outside the range of x for which the data is given, it is no longer
interpolation but instead is called extrapolation.
So what kind of function  xf should one choose? A polynomial is a common
choice for an interpolating function because polynomials are easy to
(A) evaluate,
(B) differentiate, and
(C) integrate,
relative to other choices such as a trigonometric and exponential series.
Polynomial interpolation involves finding a polynomial of order n that passes
through the 1n data points. One of the methods used to find this polynomial is called the
Lagrangian method of interpolation. Other methods include Newton’s divided difference
polynomial method and the direct method. We discuss the Lagrangian method in this
lecture.

Numerical Analysis MATH351/352
2
Figure 1 Interpolation of discrete data.
The Lagrangian interpolating polynomial is given by


n
i
iin xfxLxf
0
)()()(
where n in )(xfn stands for the th
n order polynomial that approximates the function
)(xfy  given at 1n data points as        nnnn yxyxyxyx ,,,,......,,,, 111100  , and


 


n
ij
j ji
j
i
xx
xx
xL
0
)(
)(xLi is a weighting function that includes a product of 1n terms with terms of ij 
omitted. The application of Lagrangian interpolation will be clarified using an example.
Example 1
The upward velocity of a rocket is given as a function of time in Table 1.
Table 1 Velocity as a function of time.
t (s) )(tv (m/s)
0 0
10 227.04
15 362.78
20 517.35
22.5 602.97
30 901.67
 00, yx
 11, yx
 22, yx
 33, yx
 xf
x
y

Numerical Analysis Polynomial Interpolation - Lagrange
3
Determine the value of the velocity at 16t seconds using a first order Lagrange
polynomial.
Solution
For first order polynomial interpolation (also called linear interpolation), the velocity is
given by


1
0
)()()(
i
ii tvtLtv
)()()()( 1100 tvtLtvtL 
Figure 2 Graph of velocity vs. time data for the rocket example.

4
Figure 3 Linear interpolation.
Since we want to find the velocity at 16t , and we are using a first order polynomial, we
need to choose the two data points that are closest to 16t that also bracket 16t to
evaluate it. The two points are 150 t and 201 t .
Then
  78.362,15 00  tvt
  35.517,20 11  tvt
gives


 


1
0
0 0
0 )(
j
j j
j
tt
tt
tL
10
1
tt
tt





 


1
1
0 1
1 )(
j
j j
j
tt
tt
tL
01
0
tt
tt



Hence
)()()( 1
01
0
0
10
1
tv
tt
tt
tv
tt
tt
tv






2015),35.517(
1520
15
)78.362(
2015
20






 t
tt
)35.517(
1520
1516
)78.362(
2015
2016
)16(





v
)35.517(2.0)78.362(8.0 
 00, yx
 11, yx
 xf1
x
y

5
m/s69.393
You can see that 8.0)(0 tL and 2.0)(1 tL are like weightages given to the velocities at
15t and 20t to calculate the velocity at 16t .
Quadratic Interpolation
Figure 4 Quadratic interpolation.
Example 2
Table 2 Velocity as a function of time.
t (s) )(tv (m/s)
0 0
10 227.04
15 362.78
20 517.35
22.5 602.97
30 901.67
a) Determine the value of the velocity at 16t seconds with second order polynomial
interpolation using Lagrangian polynomial interpolation.
b) Find the absolute relative approximate error for the second order polynomial
approximation.
 00 , yx
 11, yx
 22, yx
 xf2
y
x

6
Solution
a) For second order polynomial interpolation (also called quadratic interpolation), the
velocity is given by


2
0
)()()(
i
ii tvtLtv
)()()()()()( 221100 tvtLtvtLtvtL 
Since we want to find the velocity at 16t , and we are using a second order polynomial,
we need to choose the three data points that are closest to 16t that also bracket 16t
to evaluate it. The three points are 20and,15,10 210  ttt .
Then
  04.227,10 00  tvt
  78.362,15 11  tvt
  35.517,20 22  tvt
gives


 


2
0
0 0
0 )(
j
j j
j
tt
tt
tL

















20
2
10
1
tt
tt
tt
tt


 


2
1
0 1
1 )(
j
j j
j
tt
tt
tL

















21
2
01
0
tt
tt
tt
tt


 


2
2
0 2
2 )(
j
j j
j
tt
tt
tL

















12
1
02
0
tt
tt
tt
tt
Hence
202
12
1
02
0
1
21
2
01
0
0
20
2
10
1
),()()()( ttttv
tt
tt
tt
tt
tv
tt
tt
tt
tt
tv
tt
tt
tt
tt
tv 
















































)35.517(
)1520)(1020(
)1516)(1016(
)78.362(
)2015)(1015(
)2016)(1016(
)04.227(
)2010)(1510(
)2016)(1516(
)16(








v
)35.517)(12.0()78.362)(96.0()04.227)(08.0( 

7
m/s19.392
b) The absolute relative approximate error a for the second order polynomial is
calculated by considering the result of the first order polynomial (Example 1) as the
previous approximation.
100
19.392
69.39319.392


a
%38410.0
Example 3
Table 3 Velocity as a function of time
t (s) )(tv (m/s)
0 0
10 227.04
15 362.78
20 517.35
22.5 602.97
30 901.67
a) Determine the value of the velocity at 16t seconds using third order Lagrangian
polynomial interpolation.
b) Find the absolute relative approximate error for the third order polynomial
approximation.
c) Using the third order polynomial interpolant for velocity, find the distance covered by the
rocket from s11t to s16t .
d) Using the third order polynomial interpolant for velocity, find the acceleration of the
rocket at s16t .
Solution
a) For third order polynomial interpolation (also called cubic interpolation), the velocity is
given by


3
0
)()()(
i
ii tvtLtv
)()()()()()()()( 33221100 tvtLtvtLtvtLtvtL 

8
Figure 5 Cubic interpolation.
Since we want to find the velocity at 16t , and we are using a third order polynomial, we
need to choose the four data points closest to 16t that also bracket 16t to evaluate it.
The four points are 20,15,10 210  ttt and 5.223 t .
Then
  04.227,10 00  tvt
  78.362,15 11  tvt
  35.517,20 22  tvt
  97.602,5.22 33  tvt
gives


 


3
0
0 0
0 )(
j
j j
j
tt
tt
tL

























30
3
20
2
10
1
tt
tt
tt
tt
tt
tt


 


3
1
0 1
1 )(
j
j j
j
tt
tt
tL

























31
3
21
2
01
0
tt
tt
tt
tt
tt
tt


 


3
2
0 2
2 )(
j
j j
j
tt
tt
tL
 00, yx
 11, yx
 22, yx
 33, yx
 xf3
x
y

9

























32
3
12
1
02
0
tt
tt
tt
tt
tt
tt


 


3
3
0 3
3 )(
j
j j
j
tt
tt
tL

























23
2
13
1
03
0
tt
tt
tt
tt
tt
tt
Hence
303
23
2
13
1
03
0
2
32
3
12
1
02
0
1
31
3
21
2
01
0
0
30
3
20
2
10
1
),()(
)()()(
ttttv
tt
tt
tt
tt
tt
tt
tv
tt
tt
tt
tt
tt
tt
tv
tt
tt
tt
tt
tt
tt
tv
tt
tt
tt
tt
tt
tt
tv


































































































)97.602(
)205.22)(155.22)(105.22(
)2016)(1516)(1016(
)35.517(
)5.2220)(1520)(1020(
)5.2216)(1516)(1016(
)78.362(
)5.2215)(2015)(1015(
)5.2216)(2016)(1016(
)04.227(
)5.2210)(2010)(1510(
)5.2216)(2016)(1516(
)16(











v
)97.602)(1024.0()35.517)(312.0()78.362)(832.0()04.227)(0416.0( 
m/s06.392
b) The absolute percentage relative approximate error, a for the value obtained for
)16(v can be obtained by comparing the result with that obtained using the second order
polynomial (Example 2)
100
06.392
19.39206.392


a
%033269.0
c) The distance covered by the rocket between s11t to s16t can be calculated from
the interpolating polynomial as
5.2210),97.602(
)205.22)(155.22)(105.22(
)20)(15)(10(
)35.517(
)5.2220)(1520)(1020(
)5.22)(15)(10(
)78.362(
)5.2215)(2015)(1015(
)5.22)(20)(10(
)04.227(
)5.2210)(2010)(1510(
)5.22)(20)(15(
)(













t
ttt
ttt
tttttt
tv
)97.602(
)5.2)(5.7)(5.12(
)20)(15025(
)35.517(
)5.2)(5)(10(
)5.22)(15025(
)78.362(
)5.7)(5)(5(
)5.22)(20030(
)04.227(
)5.12)(10)(5(
)5.22)(30035(
22
22











tttttt
tttttt

10
)5727.2)(300065045()1388.4)(33755.7125.47(
)9348.1)(45008755.52()36326.0)(67505.10875.57(
2323
2323


tttttt
tttttt
,00544.013195.0265.21245.4 32
ttt  5.2210  t
Note that the polynomial is valid between 10t and 5.22t and hence includes the limits
of 11t and 16t .
So

16
11
)()11()16( dttvss
 
16
11
32
)00544.013195.0265.21245.4( dtttt
16
11
432
4
00544.0
3
13195.0
2
265.21245.4 






ttt
t
m1605
d) The acceleration at 16t is given by
    16
16 
 t
tv
dt
d
a
Given that
32
00544.013195.0265.21245.4)( ttttv  , 5.2210  t
   tv
dt
d
ta 
 32
00544.013195.0265.21245.4 ttt
dt
d

2
01632.026390.0265.21 tt  , 5.2210  t
2
)16(01632.0)16(26390.0265.21)16( a
2
m/s665.29
Note: There is no need to get the simplified third order polynomial expression to conduct
the differentiation. An expression of the form

























30
3
20
2
10
1
0 )(
tt
tt
tt
tt
tt
tt
tL
gives the derivative without expansion as
 )(0 tL
dt
d

















































10
1
30
3
30
3
20
2
20
2
10
1
tt
tt
tt
tt
tt
tt
tt
tt
tt
tt
tt
tt

3.2.interpolation lagrange

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3.2.interpolation lagrange