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Solved: a. Show that H2n+1(x) is the unique polynomial of least degree agreeing with f

Chapter 3, Problem 11

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QUESTION:

a. Show that H2n+1(x) is the unique polynomial of least degree agreeing with f and f at x0, ... , xn. [Hint: Assume that P(x) is another such polynomial and consider D = H2n+1 P and D at x0, x1, ... , xn.] b. Derive the error term in Theorem 3.9. [Hint: Use the same method as in the Lagrange error derivation, Theorem 3.3, defining g(t) = f (t) H2n+1(t) (t x0)2 (t xn)2 (x x0)2 (x xn)2 [f (x) H2n+1(x)] and using the fact that g (t) has (2n + 2) distinct zeros in [a, b].]

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QUESTION:

a. Show that H2n+1(x) is the unique polynomial of least degree agreeing with f and f at x0, ... , xn. [Hint: Assume that P(x) is another such polynomial and consider D = H2n+1 P and D at x0, x1, ... , xn.] b. Derive the error term in Theorem 3.9. [Hint: Use the same method as in the Lagrange error derivation, Theorem 3.3, defining g(t) = f (t) H2n+1(t) (t x0)2 (t xn)2 (x x0)2 (x xn)2 [f (x) H2n+1(x)] and using the fact that g (t) has (2n + 2) distinct zeros in [a, b].]

ANSWER:

Step 1 of 3

(a)

Suppose  is another polynomial with  and  for  and the degree of  is at most .

 

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