mws_gen_inp_ppt_spline(1)

mws_gen_inp_ppt_spline(1) -...

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Unformatted text preview: http://numericalmethods.eng.usf.edu 1 Spline Interpolation Method Major: All Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates Spline 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 Why Splines ? 2 25 1 1 ) ( x x f + = Table : Six equidistantly spaced points in [-1, 1] Figure : 5 th order polynomial vs. exact function x 2 25 1 1 x y + =-1.0 0.038461 -0.6 0.1 -0.2 0.5 0.2 0.5 0.6 0.1 1.0 0.038461 http://numericalmethods.eng.usf.edu 6 Why Splines ? Figure : Higher order polynomial interpolation is a bad idea-0.8-0.4 0.4 0.8 1.2-1-0.5 0.5 1 x y 19th Order Polynomial f (x) 5th Order Polynomial http://numericalmethods.eng.usf.edu 7 Linear Interpolation Given ( ) ( ) ( )( ) n n n n y x y x y x y x , , ,......, , , , 1 1 1 1 , fit linear splines to the data. This simply involves forming the consecutive data through straight lines. So if the above data is given in an ascending order, the linear splines are given by ( ) ) ( i i x f y = Figure : Linear splines http://numericalmethods.eng.usf.edu 8 Linear Interpolation (contd) ), ( ) ( ) ( ) ( ) ( 1 1 x x x x x f x f x f x f + = 1 x x x ), ( ) ( ) ( ) ( 1 1 2 1 2 1 x x x x x f x f x f + = 2 1 x x x . . . ), ( ) ( ) ( ) ( 1 1 1 1 + = n n n n n n x x x x x f x f x f n n x x x 1 Note the terms of 1 1 ) ( ) ( i i i i x x x f x f in the above function are simply slopes between 1 i x and i x . http://numericalmethods.eng.usf.edu 9 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 linear splines....
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