HW02.pdf - Math 104A Homework#2 Instructor Lihui Chai...

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Math 104A Homework #2 * Instructor: Lihui Chai General Instructions: Please write your homework papers neatly. You need to turn in both full printouts of your codes and the appropriate runs you made. Write your own code, individually. Do not copy codes! 1. (a) Write the Lagrangian form of the interpolating polynomial P 2 ( x ) corresponding to the data in the table below: x j f ( x j ) 0 1 1 1 3 -5 (b) Use P 2 ( x ) you obtained in (a) to approximate f (2). 2. (optional) We proved in class that k f - P n k (1 + Λ n ) k f - P * n k , (1) where P n is the interpolating polynomial of f at the nodes x 0 , ..., x n , P * n is the best ap- proximation of f , in the maximum (infinity) norm, by a polynomial of degree at most n , and Λ n = n X j =0 l ( n ) j , (2) is the Lebesgue constant (here the l ( n ) j are the elementary Lagrange polynomials). (a) Write a computer code to evaluate the Lebesgue function L ( n ) ( x ) = n X j =0 l ( n ) j ( x ) , (3) associated to a given set of pairwise distinct nodes x 0 , ..., x n .
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