peripatein
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Hi,
A quadratic piecewise interpolation is carried out for the function f(x)=cos(πx) for evenly distributed nodes in [0,1] (h=xi+1-xi, xi=ih, i=0,1,...,πh).
I am asked to bound the error.
I believe the error in this case is bounded thus:
|e(x)| ≤ [1/(n+1)!]max(f(n+1)(c))*max(∏[i=0,n](x-xi))
where c[itex]\in[/itex][0,1]
hence, yielding [1/3!]*π3*2h3/(3√3)
(1) First of all, is that correct?
(2) Next, in case this is correct, why is it bounded thus instead of as in the case for f(x)=e-x in [0,1] where, generally, it is bounded thus:
|e(x)| ≤ hn+1/(4*(n+1))
??
I'd sincerely appreciate some insight, please.
Homework Statement
A quadratic piecewise interpolation is carried out for the function f(x)=cos(πx) for evenly distributed nodes in [0,1] (h=xi+1-xi, xi=ih, i=0,1,...,πh).
I am asked to bound the error.
Homework Equations
The Attempt at a Solution
I believe the error in this case is bounded thus:
|e(x)| ≤ [1/(n+1)!]max(f(n+1)(c))*max(∏[i=0,n](x-xi))
where c[itex]\in[/itex][0,1]
hence, yielding [1/3!]*π3*2h3/(3√3)
(1) First of all, is that correct?
(2) Next, in case this is correct, why is it bounded thus instead of as in the case for f(x)=e-x in [0,1] where, generally, it is bounded thus:
|e(x)| ≤ hn+1/(4*(n+1))
??
I'd sincerely appreciate some insight, please.
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