EASY EXAMPLE.
Try to approximate
f(x)=x 2
with the synthesizer function
s(x)=a1+a2*x a1,a2
being free parameters, we try to optimize in the
interval x = 0..b with error integral method.
Assume b ( upper range of interval ) as a free parameter.
Aha p = 1,2 (a1,a2)
You can start at the beginning, but it is easier to use (1)=(2) equation
| b | b | ||
| ∫ | s ( x, ap )] * δs(ap)/δap dx= | ∫ | f(x)* δs(ap)/δap dx , p=0..n |
| a |
(1)
|
a |
(2)
|
we already have f(x) and s(x),
f(x)=x 2
s(x)=a1+a2*x
we have to compute ds(x) / da1 and ds(x) / da2 of s(x)=a1+a2*x :
ds(x) / da1 = 1 ( remember we are doing partial differentation
to a1 )
ds(x) / da2 = x ( same for a2, its easy isnt it :-)
Now the equations stepping through p :
| b | b | ||
| ∫ | (a1+a2*x)*1 dx= | ∫ | x 2*1 dx, p=1 |
| a | a |
| b | b | ||
| ∫ | (a1+a2*x)*x dx= | ∫ | x 2*x dx, p=2 |
| a | a |
This can be written shorter in an matrix form:
We dont choose a special value for b !
Instead solving and optimizing the integral for all values of b !!!
... leading to System:
Making some b,b**2 Division and using determinante
a1 = - b*b/6
a2 = b
So finally we have the SOLUTION:
s(x,b)=b*x - b*b/6 click
to get a graph of the solution !
Thats for every b a very good approximation for f(x) =x**2 !
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