Unformatted text preview: Homework 9 due Thursday December 9 before 13:30 in corresponding group folder on door of oﬃce CM110 1. Find the Hermite interpolating cubic for 1 / (1 + x 2 ), based on the values of this function and its derivative at − 1 and 1. ——————————————————————————————— We use the Hermite interpolation formula for f ( x ) = (1+ x 2 )1 at the nodes x = − 1 and x 1 = 1. So f ( x ) = − 2 x (1 + x 2 )2 and f ( ± 1) = 1 / 2, f ( ± 1) = ∓ 1 / 2. Taking x = − 1 and x 1 = 1, calculations reveal that h ( x ) = (1 − x 2 )(1 − x ) 4 , h 1 ( x ) = ( x 2 − 1)(1 + x ) 4 , h ( x ) = ( x + 2)(1 − x ) 2 4 , h 1 ( x ) = (2 − x )(1 + x ) 2 4 . So using the Hermite interpolation polynomial p 3 ( x ) = 1 2 (1 − x 2 )(1 − x ) 4 − 1 2 ( x 2 − 1)(1 + x ) 4 + 1 2 ( x + 2)(1 − x ) 2 4 + 1 2 (2 − x )(1 + x ) 2 4 = 1 4 ( 3 − x 2 ) ....
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This note was uploaded on 02/09/2011 for the course MATH 2051 taught by Professor Dr.a.wacher during the Fall '11 term at Durham.
 Fall '11
 Dr.A.Wacher
 Numerical Analysis, Derivative

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