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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?
There is an error term for the interpolating polynomial through n+1
points, namely, if p is the polynomial of degree n for which
p(x0)=f(x0),...,p(xn)=f(xn), then the difference between p(x) and f(x) is
f^(n+1)(z)/(n+1)! (x-x0)...(x-xn)
for some z between the min and max of x,x0,...,xn.
Using this, I think your formula will come out rather easily.
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