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msimon wrote:
> If a function f(x) in C^2(0,1) is approximated by a piecewise linear
> polynomial, how do I find a bound on the error?
>
> For a specific interval [x(j), x(j+1)], I know:
>
> p(x) = [f(x(j+1)) - f(x(j))] x / h_j + f(x(j))
>
> f(x) - p(x) = f(x) - [f(x(j+1)) - f(x(j))] x / h_j - f(x(j))
>
> where to go from here?
It is basically the Taylor Series with Remainder Term,
ie. a generalization of the Mean Value Theorem to
higher derivatives.
But before we go further, are we to assume a function
is continuous on the closed interval [0,1], with second
derivative continuous on the open interval (0,1)? The
notation you've used leaves me uncertain...
regards, chip
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