Hello All,

I used my latest best polynomial program to determine the best empirical-polynomial model to fit ln(x) for 1 <= x <= 10. The Excel VBA program performs power-value transformations on x and y (= ln(x)) before using these transformed values in polynomial fits. Thus, the program goes through a large number of transformed-polynomial models. The result is an 8th order polynomial:

ln(x) = a0 + a1*y + a2*y^2 + ... + a8*y^8

Where y = sqrt(x).

Here is the pseudo code with the values for the coefficients a0 through a8. The maximum absolute error for the fit I obtained is 1.54E-5. I took the values of x using the sequence 1, 1.01, 1.02, 1.03, ..., 9.99, 10.0.

function ln(x)

if x<=0 then raise error and exit

if x=1 then return 0

ln10 = 2.302585093

if x=10 then return ln10

n=0

if x>10

  do while x>10

    x=x/10

    n=n+1

  loop

elseif x<1

  do while x<1

    x=10*x

    n=n-1

  loop

end

y = sqrt(x)

z = -1.17472578289386e-03

z = z * y + 2.23235477256564e-02

z = z * y - 0.186434305125544

z = z * y + 0.899792871812638

z = z * y - 2.77773142836752

z = z * y + 5.74716024426924

z = z * y - 8.20298398274567

z = z * y + 8.74732523658923

z = z * y - 4.2482580027248

return z + n*ln10

end function

Enjoy!

Namir

Edited: 12 Dec 2012, 6:09 a.m.