mws_gen_inp_ppt_lagrange(1)

# mws_gen_inp_ppt_lagrange(1) -...

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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 )...
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## 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.

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mws_gen_inp_ppt_lagrange(1) -...

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