mws_gen_inp_txt_lagrange - 05.05.1 Chapter 05.04 Lagrangian...

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Unformatted text preview: 05.05.1 Chapter 05.04 Lagrangian Interpolation After reading this chapter, you should be able to: 1. derive Lagrangian method of interpolation, 2. solve problems using Lagrangian method of interpolation, and 3. use Lagrangian interpolants to find derivatives and integrals of discrete functions. What is interpolation? Many times, data is given only at discrete points such as , , y x 1 1 , y x , ......, 1 1 , n n y x , n n y x , . So, how then does one find the value of y at any other value of x ? Well, a continuous function x f may be used to represent the 1 n data values with x f passing through the 1 n points (Figure 1). Then one can find the value of y at any other value of x . This is called interpolation . Of course, if x falls outside the range of x for which the data is given, it is no longer interpolation but instead is called extrapolation . So what kind of function x f should one choose? A polynomial is a common choice for an interpolating function because polynomials are easy to (A) evaluate, (B) differentiate, and (C) integrate, relative to other choices such as a trigonometric and exponential series. Polynomial interpolation involves finding a polynomial of order n that passes through the 1 n data points. One of the methods used to find this polynomial is called the Lagrangian method of interpolation. Other methods include Newtons divided difference polynomial method and the direct method. We discuss the Lagrangian method in this chapter. 05.05.2 Chapter 05.05 Figure 1 Interpolation of discrete data. The 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. The application of Lagrangian interpolation will be clarified using an example. Example 1 The upward velocity of a rocket is given as a function of time in Table 1. Table 1 Velocity as a function of time. t (s) ) ( t v (m/s) 0 0 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 , y x 1 1 , y x 2 2 , y x 3 3 , y x x f x y Lagrangian Interpolation 05.05.3 Determine the value of the velocity at 16 t seconds using a first order Lagrange polynomial. Solution For first order polynomial interpolation (also called linear interpolation), the velocity is given by 1 ) ( ) ( ) ( i i i t v t L t v ) ( ) ( ) ( ) ( 1 1 t v t L t v t L Figure 2 Graph of velocity vs. time data for the rocket example. 05.05.4 Chapter 05.05 Figure 3 Linear interpolation....
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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_txt_lagrange - 05.05.1 Chapter 05.04 Lagrangian...

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