Information Sampling
The use of an exponential guiding function is detailed in Expanding the
integrand.doc#InfoSampling. Sometimes the integral of interest is
  t  t0  2 
I   exp   
f t dt
  w    




(1)
Let
t  t0
dt '  dt / w
w
t  t0  t ' w  t (t ', w, t0 )
t'
(2)

I  w  exp  t '2  f  t  t ', w, t0   dt '
(3)

y
t'
1
(4)
exp  t '2  dt
(5)
2

1
dy 
 exp   x  dx  A  t '

Igau

So that
1
I   w f  t  t '( y ), w, t0   dy 
0
yi 
w
N
N
 f t t '( y ), w, t  
i 1
i
0
(6)1
1
 i  1/ 2 
N
The value of t’(yi) needs to be found.
1
I have assumed that all derivatives with respect to y are zero at both ends of the integration regin. When this is
not true the sum can be treated according to the methpds given in ..\..\integration\Midpoint Traprule.doc.
Figure 1 AiGau(t)
The method used to produce t(y) in the example in ..\..\WaveFunction\Resonances\TWave1.zip is to
find AiGau for 1000 evenly spaced values of x. The values on the vertical axis are y, those on the
horizontal axis are t. The value of t(yi) is found by using locate. For [ ..\..\interpolation\Locate.doc] to
find values
y j  yi  y j 1
(7)
Then Uelag.for Is used to form Lagrange polynomials in y that allow interpolation in t.
[..\..\interpolation\Lagrange.doc]
Figure 2 Error estimate in Aigau(x)
Figure 3 t'(y)
The use of AiGau to produce a free energy wave packet is described along with links to the code in
..\..\WaveFunction\Resonances\TravellingWaves.doc -- code is in
..\..\WaveFunction\Resonances\TWave1.zip – read the note.