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pset_V-06-2

# pset_V-06-2 - MMAE 350 D Rempfer Problem Set V Page 1 of 2...

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MMAE 350 D. Rempfer Problem Set V Page 1 of 2 Due Date: October 23, 2006 1. We want to calculate and compare two di ff erent interpolants of f ( x ) = 1 / (1 + 25 x 2 ) in the interval x [ - 1 , 1]. a. Write down the Lagrange interpolating polynomial you obtain for the points ( x i , f ( x i )), and plot it together with f ( x ), for x i ∈ {- 1 , - 0 . 5 , 0 , 0 . 5 , 1 } , i = 0 , 1 , 2 , . . . , 4. b. For the points x i ∈ {- 1 , - 0 . 75 , - 0 . 5 , - 0 . 25 , 0 , 0 . 25 , 0 . 5 , 0 . 75 , 1 } , i = 0 , 1 , 2 , . . . , 8, derive a linear system of equations describing the coe ffi cients a i of the interpolating polynomial p ( x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + . . . . Pick the highest-order polynomial possible. Solve the system of equations, and plot the resulting interpolating polynomial together with the functions from a . Comment on your findings. c. Calculate a cubic-spline interpolant using the same set of points as in a , and then plot the result together with f , following these steps: * The second derivatives of the cubic spline s ( x ) at the nodes are given by the equation ( x i - x i - 1 ) s ( x i - 1 ) + 2( x i + 1 - x i - 1 ) s ( x i ) + ( x i + 1 - x i ) s ( x i + 1 ) = 6 f ( x i + 1 ) - f ( x i ) x i + 1 - x i + 6 f ( x i - 1 ) - f ( x i ) x i - x i - 1 .

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pset_V-06-2 - MMAE 350 D Rempfer Problem Set V Page 1 of 2...

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