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Explicit Function Approximation
A closed form expression for F cannot be obtained for the Baker and Lonsdale equation (first equation in the Table). However, since the expression for F does not contain the parameter k, it can be numerically
"inverted." The numerical inversion was done with TableCurve 2D using the following procedure. First the expression for F on the left side of the equation was evaluated for 1000 equidistant F values from 0 to 100.
The columns containing the expression for F (which equals kt) and F were then reversed so that F is now the Y variable and kt is the X variable. All equations in TableCurve were then fit to this X,Y data set and the
equations ranked using the F statistic. The best fitting equation found (Fstat = 2.7x1014) was the following rational polynomial in fractional powers of x
where the coefficients are
a = 2.5788672e-6
b = -3.4434044
c = 244.94883
d = 3.9105658
e = -976.78997
f1= -1.50823
g = 1407.9333
h = 0.039306878
i = -862.63205
j = 0.0091845726
k1=187.88278
In this equation F is an explicit function of x (=kt) so it is used in the nonlinear curve fitter to estimate the parameter k. An analysis of the residuals found the
maximum absolute difference between the actual F and approximate F to be 0.0003 (at F= 99.9). The maximum absolute percentage difference was only 0.0% (at F = 0.1).
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