GAUSS ERROR INTEGRAL
Lets approximate the function f(x)=sin(x) with the function s(x)=C1*x + C3*x3
+ C5*x5
The calculation will not be easy any more. Do it with maple.
You think you already know a solution ? C1=1, C3= -1/3! and
C5=1/5!
Ok, but perhaps Gauss approximation is the better one ?
We ll compare it finally ...
We start at the same point as in the easy example:
| b | b | ||
| ∫ | s ( x, ap )] * δs(ap)/δap dx= | ∫ | f(x)* δs(ap)/δap dx , p=0..n |
| a |
(1)
|
a |
(2)
|
ds/dC0=x
ds/dC1=x3
ds/dC2=x5
So the system we have to solve for C1 C3 C5 is ...
| b | b | ||
| ∫ | (C5*x5+C3*x3+C1*x)*x dx = | ∫ | sin(x)*x dx , p=1 |
| a | a |
| b | b | ||
| ∫ | (C5*x5+C3*x3+C1*x)*x3 dx = | ∫ | sin(x)*x3 dx , p=2 |
| a | a |
| b | b | ||
| ∫ | (C5*x5+C3*x3+C1*x)*x5 dx = | ∫ | sin(x)*x5 dx , p=3 |
| a | a |
Its not easy to do and I have got to admit, solution
being a little bit long :-)
Thats because I solved it for all a and b
Choosing values for a and b (the optimizing interval) our function
gets much easier:
Example:
a=0 b=Pi
C1= 0.9878621255
C3= -0.1552714092
C5= 0.005643117749
Here the result
You see, computers are good for computing !
Setting a=0, we can plot a nice graphics. Showing the error (s(x,b)-sin(x))**2
We can even optimize up to b=4.5 without the error getting too big
So lets do that just for fun.
