Lagrangian Interpolation
Docsity.com
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Lagrangian Interpolation Method for Finding Function Values and Derivatives, Slides of Mathematical Methods for Numerical Analysis and Optimization
The lagrangian interpolation method, which is used to find the value of a function at a given point based on its values at other points. Linear, quadratic, and cubic interpolation, and provides examples using a rocket velocity dataset. It also discusses the concept of interpolating polynomials and weighting functions.
Typology: Slides
2012/2013
1 / 18
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Lagrangian Interpolation Method for Finding Function Values and Derivatives and more Slides Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!
Lagrangian Interpolation
What is Interpolation?
Given (x
,y
), (x
,y
), …… (x
n
,y
n
), find the
value of ‘y’ at a value of ‘x’ that is not given.
Lagrangian Interpolation
Lagrangian interpolating polynomial is given by
∑
n
i
f n x Li x f xi
0
where ‘ n ’ in f n ( x ) stands for the n th order polynomial that approximates the function y = f ( x )
given at ( n + 1 ) data points as ( x (^) 0 , y 0 ) (, x 1 , y 1 ),......, ( xn − 1 , yn − 1 ) (, xn , yn ), and
∏ ≠
n
j i
j (^) i j
j i
x x
x x
L x
0
Li ( x ) is a weighting function that includes a product of ( n − 1 ) terms with terms of j = i
omitted.
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 0 10 227. 15 362. 20 517. 22.5 602. 30 901.
t v ( t )
Linear Interpolation (contd)
≠
= −
−
1
0
(^0 )
0 () j
j (^) j
j t t
t t L t 0 1
1 t t
t t
−
−
≠
= −
−
1
1
(^0 )
1 ( ) j
j (^) j
j t t
t t L t 1 0
0 t t
t t
−
−
( ) ( ) ( 1 ) 1 0
0 0 0 1
(^1) v t t t
t t v t t t
t t v t −
−
−
− = ( 517. 35 ) 20 15
15 ( 362. 78 ) 15 20
20 −
−
−
−
t t
( 517. 35 ) 20 15
16 15 ( 362. 78 ) 15 20
16 20 ( 16 ) −
−
−
− v =
= 0. 8 ( 362. 78 )+ 0. 2 ( 517. 35 )
= 393. 7 m/s.
Quadratic Interpolation
For the second order polynomial interpolation (also called quadratic interpolation), we
choose the velocity given by
=
=
2
0
( ) ( ) ( ) i
v t Li t vti
= L 0 (^) ( t ) v ( t 0 )+ L 1 ( t ) v ( t 1 )+ L 2 ( t ) v ( t 2 )
Quadratic Interpolation (contd)
This image cannot currently be displayed.
(^20010 12 14 16 18 )
250
300
350
400
450
500
517.35^550
y (^) s f range( ) f x( (^) desired)
10 x (^) s , range,x (^) desired 20
t 0 = 10 , v ( t 0 ) = 227. 04
t 1 = 15 , v ( t 1 )= 362. 78
t 2 = 20 , v ( t 2 )= 517. 35
≠
2
0 (^0 )
j j (^) j
j
t t
t t
L t
0 2
2 0 1
1
t t
t t
t t
t t
≠
2
1 (^0 )
j j (^) j
j
t t
t t
L t
1 2
2 1 0
0
t t
t t
t t
t t
≠
2
2 (^0 )
j j (^) j
j
t t
t t
L t
2 1
1 2 0
0
t t
t t
t t
t t
Quadratic Interpolation (contd)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )( ) ( )( ) ( )( )
19 m/s
08 227. 04 0. 96 362. 78 0. 12 527. 35
35 20 15
16 15
20 10
16 10
- 78 15 20
16 20
15 10
16 10
- 04 10 20
16 20
10 15
16 15 16
2 2 1
1
2 0
0 1 1 2
2
1 0
0 0 0 2
2
0 1
1
=
= − + +
−
−
−
− +
−
−
−
− +
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
v
v t t t
t t
t t
t t vt t t
t t
t t
t t vt t t
t t
t t
t t v t
The absolute relative approximate error obtained between the results from the first and second order polynomial is
∈ a
- 38410 %
100
19
19 393. 70
=
×
− ∈ a =
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 0 10 227. 15 362. 20 517. 22.5 602. 30 901.
t v ( t )
Cubic Interpolation (contd)
t (^) o = 10 , v ( t (^) o ) = 227. 04 t 1 = 15 , v ( t 1 ) = 362. 78
t 2 = 20 , v ( t 2 ) = 517. 35 t 3 = 22. 5 , v ( t 3 ) = 602. 97
≠
3
0 (^0 )
j j (^) j
j
t t
t t
L t
0 3
3 0 2
2 0 1
1
t t
t t
t t
t t
t t
t t
≠
3
1 (^0 )
j j (^) j
j
t t
t t
L t
1 3
3 1 2
2 1 0
0
t t
t t
t t
t t
t t
t t
≠
3
2 (^0 )
j j (^) j
j
t t
t t
L t
2 3
3 2 1
1 2 0
0
t t
t t
t t
t t
t t
t t
≠
3
3 (^0 )
j j (^) j
j
t t
t t
L t
3 2
2 3 1
1 3 0
0
t t
t t
t t
t t
t t
t t^20010 12 14 16 18 20 22
300
400
500
600
602.97^700
y (^) s f range( ) f x( (^) desired)
10 x (^) s , range,x (^) desired 22.
Comparison Table
Order of Polynomial 1 2 3
v(t=16) m/s 393.69 392.19 392.
Absolute Relative
Approximate Error
Distance from Velocity Profile
Find the distance covered by the rocket from t=11s to
t=16s?
( ) 4. 245 21. 265 0. 13195 0. 00544 , 2 3 v t = − + t + t + t 10 ≤ t ≤ 22. 5
16
11
s ( 16 ) s ( 11 ) v ( t ) dt
16
11
( 4. 245 21. 265 t 0. 13195 t^2 0. 00544 t^3 ) dt
16 11
2 3 4 ] 4
00544 3
13195 2
[ 4. 245 21. 265
t t t = − t + + +
= 1605 m
3 2 3 2
3 2 3 2