User:Egm6341.s11.team2.franklin/hwk3

Problem 5: Evaluate the $$\displaystyle (n+1)^{th}$$ derivative of Lagrange Interpolation Error
''' This problem was solved without referring to S10 homework. '''

From the lecture slide Mtg 16-3

Problem Statement
Verify the following:


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E^{(n+1)}(x)=f^{(n+1)}(x)-0. $$     (5.1)
 * $$\displaystyle
 * $$\displaystyle
 * 
 * }

Solution
The Lagrange Interpolation error can be expressed as
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E(x)=f(x)-f_n(x) , $$     (5.2)
 * $$\displaystyle
 * $$\displaystyle
 * 
 * }

where


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f_n(x) \epsilon \mathcal{P}_n. $$     (5.3)
 * $$\displaystyle
 * $$\displaystyle
 * 
 * }

Because $$\displaystyle f_n(x) \epsilon \mathcal{P}_n $$, it is of the form


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f_n(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0. $$     (5.4)
 * $$\displaystyle
 * $$\displaystyle
 * 
 * }

Differentiating $$ \displaystyle f_n $$ n times gives


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f^{(n)}_n(x)=(n!)a_n , $$     (5.5)
 * $$\displaystyle
 * $$\displaystyle
 * 
 * }

which is a constant.

Differentiating the expression $$\displaystyle (n+1)$$ times gives


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E^{n+1}(x)=f^{n+1}(x)-f_n^{n+1}(x) , $$
 * $$\displaystyle
 * $$\displaystyle
 * (5.6)
 * }

where


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f_n^{n+1}(x) = 0. $$
 * $$\displaystyle
 * $$\displaystyle
 * (5.7)
 * }

Therefore,