hw5 - Homework 5 Math 104A Fall 2010 Due on Tuesday...

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Homework 5 – Math 104A, Fall 2010 Due on Tuesday, November 9th, 2010 1. Given x i , i = 0 , 1 , . . . , n , consider the Lagrange polynomials L n,j for j = 0 , 1 , . . . , n . Prove that n j =0 L n,j ( x ) = 1 for all x R . 2. The following data is taken from a polynomial. What is its degree? 3. Consider the function e x on [0 , b ] and its approximation by an interpo- lating polynomial. For n 1, let h = b/n , x j = jh , j = 0 , 1 , . . . , n ; and let p n ( x ) be the n th degree polynomial interpolating e x on the nodes x 0 , . . . , x n . Prove that max 0 x b | e x - p n ( x ) | → 0 as n → ∞ . 4. Let x 0 , x 1 , . . . , x n be distinct real points, and consider the following interpolation problem. Choose a function P n ( x ) = n j =0 c j e jx such that P n ( x i ) = y i , i = 0 , 1 , . . . , n, with the { y j } given data. Show there is a unique choice of c 0 , c 1 , . . . , c n . 5. Show there is a unique cubic polynomial p ( x ) for which p ( x 0 ) = f ( x 0 ); p ( x 2 ) = f ( x 2 ); p ( x 1 ) = f ( x 1 ); p ( x 1 ) = f ( x 1 ) , where f ( x ) is a given function and x 0 = x 2 . Derive a formula for p ( x ), analogous to the Lagrange formula for ordinary interpolation.
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