Fall2011-HW9-soln - Homework 9 Solutions Math 371, Fall...

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Math 371, Fall 2011 1. (Interpolating Polynomials) The function f ( x ) = 1 / ( x + 1) is given at the four points x 0 = 1, x 1 = 2, x 2 = 3, x 3 = 4. (a) Write the interpolating polynomial in Lagrange form. ( x - 2)( x - 3)( x - 4) (1 - 2)(1 - 3)(1 - 4) 1 2 + ( x - 1)( x - 3)( x - 4) (2 - 1)(2 - 3)(2 - 4) 1 3 + ( x - 1)( x - 2)( x - 4) (3 - 1)(3 - 2)(3 - 4) 1 4 + ( x - 1)( x - 2)( x - 3) (4 - 1)(4 - 2)(4 - 3) 1 5 . (b) Write the interpolating polynomial in Newton form. 1 2 - 1 6 ( x - 1) + 1 24 ( x - 1)( x - 2) - 1 120 ( x - 1)( x - 2)( x - 3) (c) Evaluate f (1 . 5) and f (5) using the interpolating polynomial. Which approximate value is more accurate? f (1 . 5) 0 . 403125 , f (5) 0 . 133333 (d) Use the error formula to find an upper bound on the maximum error || f - p 3 || = max 1 x 4 | f ( x ) - p 3 ( x ) | . max 1 ξ 4 ± ± ± ± f (4) ( ξ ) 4! ± ± ± ± = max 1 ξ 4 1 ( ξ + 1) 5 = 1 32 . max 1 x 4 ( x - 1)( x - 2)( x - 3)( x - 4) = 1 . Therefore an upper bound on the error for 1 x 4 is 1 32 . 2. (Choice of Interpolation Points) This question investigates how different interpolation points af- fect the accuracy of the interpolation. Let f ( x ) = 1 20 x 2 +1 . The following two Matlab programs can be used to generate the plots of the interpolating polynomials. The program below computes the Lagrange interpolant.
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Fall2011-HW9-soln - Homework 9 Solutions Math 371, Fall...

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