Lagrangian Interpolation

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1 Lagrangian Inerpolaion Maor: All Engineering Maors Auhors: Auar Kaw, Jai Paul hp://numericalmehods.eng.usf.edu Transforming Numerical Mehods Educaion for STEM Undergraduaes hp://numericalmehods.eng.usf.edu

2 Lagrange Mehod of Inerpolaion hp://numericalmehods.eng.usf.edu

3 Wha is Inerpolaion? Gien x,y, x,y, x n,y n, find he alue of y a a alue of x ha is no gien. hp://numericalmehods.eng.usf.edu

4 Inerpolans Polynomials are he mos common choice of inerpolans because hey are easy o: Ealuae Differeniae, and Inegrae. 4 hp://numericalmehods.eng.usf.edu

5 Lagrangian Inerpolaion Lagrangian inerpolaing polynomial is gien by f n x n i L x f i x i where n in f n x sands for he gien a h n order polynomial ha approximaes he funcion y f x n daa poins as x y, x, y,..., x, y, x, y, n n n n, and L x i n x x x i i x L i x is a weighing funcion ha includes a produc of n erms wih erms of i omied. 5 hp://numericalmehods.eng.usf.edu

6 Example The upward elociy of a rocke is gien as a funcion of ime in Table. Find he elociy a 6 seconds using he Lagrangian mehod for linear inerpolaion. Table Velociy as a funcion of ime s m/s Figure. Velociy s. ime daa for he rocke example hp://numericalmehods.eng.usf.edu

7 Linear Inerpolaion L i i i L L y s f range f x desired , ν, ν x s x s, range, x desired x s 7 hp://numericalmehods.eng.usf.edu

8 hp://numericalmehods.eng.usf.edu 8 Linear Inerpolaion cond L L m/s.

9 hp://numericalmehods.eng.usf.edu 9 Quadraic Inerpolaion For he second order polynomial inerpolaion also called quadraic inerpolaion, we choos e he eloc iy gien by i i i L L L L

10 Example The upward elociy of a rocke is gien as a funcion of ime in Table. Find he elociy a 6 seconds using he Lagrangian mehod for quadraic inerpolaion. Table Velociy as a funcion of ime s m/s Figure. Velociy s. ime daa for he rocke example hp://numericalmehods.eng.usf.edu

11 Quadraic Inerpolaion cond, , 6. 78, L L L y s 57.5 f range f x desired x s, range, x desired hp://numericalmehods.eng.usf.edu

12 Quadraic Inerpolaion cond m/s The absolue relaie approximae error obained beween he resuls from he firs and second order polynomial is a a % hp://numericalmehods.eng.usf.edu

13 Cubic Inerpolaion For he hird order polynomial also called cubic inerpolaion, we choose he elociy gien by i L i i L L L L y s f range f x desired x s, range, x desired.5 hp://numericalmehods.eng.usf.edu

14 Example The upward elociy of a rocke is gien as a funcion of ime in Table. Find he elociy a 6 seconds using he Lagrangian mehod for cubic inerpolaion. Table Velociy as a funcion of ime s m/s Figure. Velociy s. ime daa for he rocke example hp://numericalmehods.eng.usf.edu

15 hp://numericalmehods.eng.usf.edu 5 Cubic Inerpolaion cond 4 7., o o 78 6., , , L ; L L ; L y s f range f x desired.5 x s range, x desired,

16 hp://numericalmehods.eng.usf.edu 6 Cubic Inerpolaion cond 9.6 m/s The absolue relaie approximae error obained beween he resuls from he firs and second order polynomial is a.69% a

17 Comparison Table Order of Polynomial 6 m/s Absolue Relaie Approximae Error %.69% 7 hp://numericalmehods.eng.usf.edu

18 Disance from Velociy Profile Find he disance coered by he rocke from s o 6s? ,. 5 6 s 6 s d d [ ] m hp://numericalmehods.eng.usf.edu

19 Acceleraion from Velociy Profile Find he acceleraion of he rocke a 6s gien ha ,. 5 d d, a d d a m / s 9 hp://numericalmehods.eng.usf.edu

20 Addiional Resources For all resources on his opic such as digial audioisual lecures, primers, exbook chapers, muliple-choice ess, workshees in MATLAB, MATHEMATICA, MahCad and MAPLE, blogs, relaed physical problems, please isi hp://numericalmehods.eng.usf.edu/opics/lagrange_ mehod.hml

21 THE END hp://numericalmehods.eng.usf.edu

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