Started readme journal for development
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3/6/2014
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Problem: Found out how the damping influences the calculated coefficients, logged results
into excel spreadsheet (sub:"Calc. Amp & Calc. damping"), need to separate variables

Solution: No true solution devised yet, for now I need to examine the mixture of the variables and
how to correct the errors introduced. For now I will treat the whole as a system of equations and try
to isolate each individual subsection.

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Observation: The calculated frequencies with the params (A=5, lam=-1, w=1, [delta_t=0.1, l : math.exp(lam1*x) * (A*math.sin(w1*x)) ])
are off by almost exactly -0.05 each time, investigate

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3/7/2014
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From logger pro it appears that Ac = (~0.5)(-lam)^(-1.406) + A, let's try A = Ac - 0.5(-lam)^(-1.406)
So: A = 2(norm)^(0.5) - 0.5(-lam)^(-1.406) ?

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While testing this a found large increases in lam for small lam. i.e. lam = 0.02 -> lamc = 0.07
This appears to be a constant increase of about 0.05 to lam, No idea why it's always between 0.0501 and 0.0507
So the best case seems to be (0.5*delta_t) + (ln(norm)/-delta_t)

I found that ∆A(7)/∆A(6) is indeed 7/6 therefore A is the coefficient in front of ∆A in [∆A as a function of lam]
So when I plot ∆A/A as a function of lam I find the line of best fit to be ~ 0.1892 * lam ^ -0.7903
Trying:
(Ac-A)/A = 0.2 lam^-0.8
Ac / (1 + 0.2 lam^-0.8) = A
failed

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3/8/2014
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So I asked Julian's help for finding a function which looks like the plot of ∆A/A in terms of lam, he proposed A/lam + B/e^lam,
this worked nearly perfectly, I then found that it was interpreting e as a variable, meaning the function is A/lam + BC^-lam,
or A/lam + B^(lam + d)
The values I find for A, B, and C were
A = 0.05751
B = 1.313
C = 127.0

or in A, B, d:
A = 0.05751
B = 127.0
d = 0.05615

While accurate, this was still not quite enough, I have adjusted it slightly to make:
A*lam^B + C^(d-lam) + D
this got a RMSE of 0.004214, much lower than the 0.02 of the previous function

This function appears to work very well for small values of lambda. It becomes less accurate when lam<-1.6ish

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Now I need to address the errors in the calculated damping with additional modes. It appears to be off by approximately 0.15
for the second mode for any value of lam2. Frequencies remain accurate. Ac appears accurate, though the Adjusted Ac (which relies
on lam2) is incorrect. Presumably due to this ∆=0.15

It appears that the change is dependent on the frequency, for example a frequency of 1.0 gives ∆=0.05... whereas a frequency of 2.0 gives ∆=0.15...
and freq 3.0 gives ∆=0.45...

So the function was more annoying than expected, I found A^(x^B + C) with:
A = 11060
B = 0.1719
C = -1.291
So.. I will implement this for now, but I need to figure out where these come from in the future
lc - l = A^(wc^B + C)
so l = wc - A^(wc^B + C)

I'll try A^(wc^B + C) + D to account for other nth order shit
A = 595.8
B = 0.2234
C = -1.389
D = -0.03057
this has RMSE 0.01001

This worked fairly well for ∆t = 0.1, however I have found that I get different values for the damping coefficients for different
∆t's. I need to look into this more tomorrow.

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3/9/2014
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