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ma174_hw6 - Assigned Monday Oct 31 2011 Due Date Friday Nov...

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Assigned: Monday Oct. 31, 2011 Due Date: Friday, Nov. 11, 2011 MATH 174/274: Homework VI “Polynomial Interpolation” Fall 2011 1. Consider the function f ( x ) = sin π 2 x + x 2 . (a) We would like to construct a polynomial P ( x ) that interpolates f ( x ) at the points x 0 = 0, x 1 = 2, and x 2 = 3. Compute all the necessary Lagrange polynomials and use these to construct the interpolating polynomial P ( x ). (b) Compute the error term R ( x ). (c) Interpolation: Compute the maximum bound on R ( x ) if we approximate f (1) by P (1). (d) Extrapolation: Compute the maximum bound on R ( x ) if we approximate f (4) by P (4). 2. SOURCE CODE: Let x 0 , x 1 , . . . , x n be distinct points with corresponding data values f 0 , f 1 , . . . , f n . Let P ( x ) be the unique polynomial of degree n which interpolates the data (i.e., P ( x i ) = f i for each i = 0 , . . . , n ). Write the following MATLAB routines: (a) F = DivDiff(x,f,n) - a function to compute the divided differences. (b) P = Horners(x,F,xbar) - a function which uses the divided differences to evaluate P ( x ) at the point xbar using Horner’s method. 1
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Assigned: Monday Oct. 31, 2011 Due Date: Friday, Nov. 11, 2011 3. Consider the function
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