HW03.pdf - Math 104A Homework#3 u2217 Instructor Ruimeng...

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Math 104A Homework #3 * Instructor: Ruimeng Hu General Instructions: Please write your homework papers neatly. You need to turn in both full printouts of your codes (a pdf file) and the appropriate runs you made (a jupyter notebook), on Gauchospace. Write your own code, individually. Do not copy codes! 1. (a) Equating the leading coefficient of in the Lagrange form of the interpolation polynomial P n ( x ) with that of the Newton’s form deduce that f [ x 0 , x 1 , ..., x n ] = n X j =0 f ( x j ) n Q k =0 k 6 = j ( x j - x k ) . (1) (b) Use (a) to conclude that divided differences are symmetric functions of their arguments, i.e. any permutation of x 0 , x 1 , ..., x n leaves the corresponding divided difference un- changed. 2. In Newton’s form of the interpolation polynomial we need to compute the coefficients, c 0 = f [ x 0 ], c 1 = f [ x 0 , x 1 ], ..., c n = f [ x 0 , x 1 , ..., x n ]. In the table of divided differences we proceed column by column and the needed coefficients are in the uppermost diagonal. A simple 1D array, c of size n + 1, can be used to store and compute these values. We just have to compute them from bottom to top to avoid losing values we have already computed. The following

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