mws_gen_inp_ppt_lagrange(1) -...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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.edu 3 What is Interpolation ? Given (x ,y ), (x 1 ,y 1 ), …… (x n ,y n ), find the value of ‘y’ at a value of ‘x’ that is not given. http://numericalmethods.eng.usf.edu 4 Interpolants Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate . http://numericalmethods.eng.usf.edu 5 Lagrangian Interpolation Lagrangian interpolating polynomial is given by ∑ = = n i i i n x f x L x f ) ( ) ( ) ( where ‘ n ’ in ) ( x f n stands for the th n order polynomial that approximates the function ) ( x f y = given at ) 1 ( + n data points as ( ) ( ) ( ) ( ) n n n n y x y x y x y x , , , ,......, , , , 1 1 1 1 − − , and ∏ ≠ = − − = n i j j j i j i x x x x x L ) ( ) ( x L i is a weighting function that includes a product of ) 1 ( − n terms with terms of i j = omitted. http://numericalmethods.eng.usf.edu 6 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) 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 t ) ( t v http://numericalmethods.eng.usf.edu 7 Linear Interpolation 10 12 14 16 18 20 22 24 350 400 450 500 550 517.35 362.78 y s f range ( ) f x desired ( ) x s 1 10 + x s 10 − x s range , x desired , ) ( ) ( ) ( 1 i i i t v t L t v = ∑ = ) ( ) ( ) ( ) ( 1 1 t v t L t v t L + = ( ) 78 . 362 , 15 = = t t ν ( ) 35 . 517 , 20 1 1 = = t t ν http://numericalmethods.eng.usf.edu 8 Linear Interpolation (contd) ∏ ≠ = − − = 1 ) ( j j j j t t t t t L 1 1 t t t t − − = ∏ ≠ = − − = 1 1 1 1 ) ( j j j j t t t t t L 1 t t t t − − = ) ( ) ( ) ( 1 1 1 1 t v t t t t t v t t t t t v − − + − − = ) 35 . 517 ( 15 20 15 ) 78 . 362 ( 20 15 20 − − + − − = t t ) 35 . 517 ( 15 20 15 16 )...
View Full Document

This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida - Tampa.

Page1 / 21

mws_gen_inp_ppt_lagrange(1) -...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online