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Q: FirstDegree Spline Accuracy Theorem  Cheney & Kincaid
Posted:
Dec 1, 2011 7:34 PM


Hi,
In Numerical Mathematics and Computing by Cheney & Kincaid they give the
FirstDegree Spline Accuracy Theorem and then say...
if f ' and f '' exist and are continuous, then more can be said, namely
 f(x)  p(x)  <= M1*h/2 (a <= x <= b)
 f(x)  p(x)  <= M2*h^2/8 (a <= x <= b)
I have not been able to derive these last two results. I have tried to use Taylors Theorem and the Mean Value Theorem, but I am not getting anywhere.
Any help much appreciated.
Here is the FirstDegree Spline Accuracy Theorem:
Let p be a firstdegree spline having knots a = x0 < x1 < ... < xn = b. If p interpolates a function f at these knots, then with h = max( x_i  x_(i1) ) we have
 f(x)  p(x)  <= w(f;h) (a <= x <= b)
w(f;h) is the modulus of continuity of f
Cheers, Brad



