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
Lagrange Mehod of Inerpolaion hp://numericalmehods.eng.usf.edu
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
Inerpolans Polynomials are he mos common choice of inerpolans because hey are easy o: Ealuae Differeniae, and Inegrae. 4 hp://numericalmehods.eng.usf.edu
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
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 7.4 5 6.78 57.5.5 6.97 9.67 6 Figure. Velociy s. ime daa for he rocke example hp://numericalmehods.eng.usf.edu
Linear Inerpolaion 57.5 55 L i i i L L y s f range f x desired 5 45 5, ν, ν 6. 78 57. 5 6.78 4 5 4 6 8 4 x s x s, range, x desired x s 7 hp://numericalmehods.eng.usf.edu
hp://numericalmehods.eng.usf.edu 8 Linear Inerpolaion cond L L 57.5 5 5 6.78 5 57.5 5 5 6 6.78 5 6 6.57.5.86.78 9.7 m/s.
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
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 7.4 5 6.78 57.5.5 6.97 9.67 Figure. Velociy s. ime daa for he rocke example hp://numericalmehods.eng.usf.edu
Quadraic Inerpolaion cond, 7. 4 5, 6. 78, 57. 5 L L L y s 57.5 f range f x desired 7.4 55 5 45 4 5 5 4 6 8 x s, range, x desired hp://numericalmehods.eng.usf.edu
Quadraic Inerpolaion cond 6 5 6 5 6 7.4 6.78 57.5.8 7.4.96 6.78. 57.5 9.9 m/s 6 6 5 5 6 6 5 5 The absolue relaie approximae error obained beween he resuls from he firs and second order polynomial is a a 9.9 9.7 9.9.84% hp://numericalmehods.eng.usf.edu
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 6.97 7 6 y s f range f x desired 5 4 7.4 4 6 8 4 x s, range, x desired.5 hp://numericalmehods.eng.usf.edu
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 7.4 5 6.78 57.5.5 6.97 9.67 4 Figure. Velociy s. ime daa for he rocke example hp://numericalmehods.eng.usf.edu
hp://numericalmehods.eng.usf.edu 5 Cubic Inerpolaion cond 4 7., o o 78 6., 5 5 57., 97 6..5, L ; L L ; L 4 6 8 4 4 5 6 7 6.97 7.4 y s f range f x desired.5 x s range, x desired,
hp://numericalmehods.eng.usf.edu 6 Cubic Inerpolaion cond 9.6 m/s 6.97.4 57.5. 6.78.8 7.4.46 6.97.5 6 5.5 5 6.5 6 57.5.5.5 6 5 5 6 6 6.78.5 5.5 6 5 6 5 6 7.4.5.5 6 6 5 5 6 6 The absolue relaie approximae error obained beween he resuls from he firs and second order polynomial is a.69% 9.6 9.9 9.6 a
Comparison Table Order of Polynomial 6 m/s 9.69 9.9 9.6 Absolue Relaie Approximae Error --------.84%.69% 7 hp://numericalmehods.eng.usf.edu
Disance from Velociy Profile Find he disance coered by he rocke from s o 6s? 57.5 47.5 4.45.65.95.544,. 5 6 s 6 s d 6 4.45.65.95.544 d 87.5 7.5 675.66 75 4.88 5.5 45 4 [ 4.45.65.95.544 ] 4 875 65 6 45.948.577 8 65 m hp://numericalmehods.eng.usf.edu
Acceleraion from Velociy Profile Find he acceleraion of he rocke a 6s gien ha 4.45.65.95.544,. 5 d d, 4.45.65.95.544 a d d.65.69.6 a 6.65.696.66 9.665 m / s 9 hp://numericalmehods.eng.usf.edu
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
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