Lecture 4 - Computer Science Department - University of Kentucky
Lecture 4 - Computer Science Department - University of Kentucky
Lecture 4 - Computer Science Department - University of Kentucky
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Error Analysis (1)<br />
If f”(x) exists and is continuous on [a,b], the error <strong>of</strong> the composite<br />
trapezoid rule T is<br />
for some ξ in (a,b)<br />
b<br />
b−a<br />
2<br />
2<br />
∫ f ( x)<br />
dx − T = − h f "( ξ ) = O(<br />
h )<br />
12<br />
a<br />
Pro<strong>of</strong>. We first prove the result for a=0, b= 1 and h=1. That is<br />
∫<br />
1<br />
0<br />
[ 1<br />
f<br />
(0) +<br />
f<br />
(1) ] = −<br />
f "(<br />
ξ<br />
)<br />
f ( x<br />
)<br />
dx −<br />
1<br />
2<br />
+<br />
12<br />
This simplified formula will be proved with the help <strong>of</strong> polynomial<br />
interpolation<br />
Define a polynomial <strong>of</strong> degree one that interpolates f at 0 and 1<br />
p ( x<br />
)<br />
=<br />
f<br />
(0)<br />
+<br />
[<br />
f<br />
(1) −<br />
f<br />
(0<br />
)]<br />
x<br />
14