Re: Low accuracy approximation to the ellipsoid surface area, using only additions and multiplications

On Aug 2, 11:17 am, Rene Grothmann <mga...@xxxxxx> wrote:
Very nice and interesting. I tested your older and better
approximation (using the p-norm) too, and it works with relative
errors less than 1.07%.

Then I tried to approximate a on a grid of 5000 data using any
quadratic function generated by 1,x,y,xy,x^2,y^2 (approximating
EllipseSurface(1,y,x)), and could not get better than you did with
only 1,x,y,xy. The maximal relative error on this grid was 3.3%. The
best coefficients were

 6.931488697637e-005
      6.065501329042
      3.438170041422
    -0.8104612329251
    0.05209633277943
       3.93853279598

for the above basis functions.

These computations were carried out with Euler Math Toolbox (with
elliptic functions added). The minimization was done with Nelder-Mead.

Rene Grothmann

Thank you very much, Rene, for your response.

Maybe I should share my practical experiences with optimizing the
above approximation:

I also used Nelder-Mead for optimization, and I found that the values
of the coefficients and the max. |rel. error| converged very slowly
with increasing resolution of the discrete semiaxis search space.

However, I also found extrapolation to infinite search space to be
feasible:

Nelder-Mead optimization was repeated for various resolutions of the
discrete semiaxis space used for sampling the max. |rel.error|.

For each independent optimization, the error search involved
combinations of integer value semiaxes, (B=1..A, C=0..B), where the
major semiaxis A was set to a fixed value which effectively
determined the resolution of that particular search space.

Consequently, both the values of the coefficients and the max. |rel.
error| effectively could be regarded as functions of search space
resolution A.

Then, fortunately, it turned out that any of these functions f, to a
high accuracy, obeys the relation:

f(a) = f(infinity) + C/a

C being a constant.

Best regards,
Knud
.

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