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 = 2.7×1014) 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 0.03% (at F = 0.1).

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