Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.
|
|
Math Forum
»
Discussions
»
sci.math.*
»
sci.math
Notice: We are no longer accepting new posts, but the forums will continue to be readable.
Topic:
Q: First-Degree Spline Accuracy Theorem - Cheney & Kincaid
Replies:
0
|
 |
|
|
Q: First-Degree Spline Accuracy Theorem - Cheney & Kincaid
Posted:
Dec 1, 2011 7:34 PM
|
|
Hi,
In Numerical Mathematics and Computing by Cheney & Kincaid they give the
First-Degree 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 First-Degree Spline Accuracy Theorem:
Let p be a first-degree spline having knots a = x0 < x1 < ... < xn = b. If p interpolates a function f at these knots, then with h = max( x_i - x_(i-1) ) we have
| f(x) - p(x) | <= w(f;h) (a <= x <= b)
w(f;h) is the modulus of continuity of f
Cheers, Brad
|
|
|
|