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Searching for an asymptotic to exp[0.5]
#31
(05/25/2014, 03:00 PM)sheldonison Wrote: ... Meanwhile, I have a new version I am experimenting with that will greatly reduce the discontinuity at the negative real axis, that may allow approximating both the Weiestrass zeros, and the error term for the convergence to the Kneser half iterate... but there will still be a discontinuity cut point at |L|; still working.

Well, version V works a lot lot better, but it I was hoping for something with more theoretical power. So there's this discontinuity for Kneser's half iterate at the negative real axis. Why not get rid of it, with some sort of mapping that still converges to the half iterate as real(z) increases? Well, I got rid of most of it; not all. Here exp^(0.5) refer's to Kneser's half iterate, with the real axis complex valued, following the cutpoints from above. Inside a circle of radius |L|, we just use the Kneser half iterate. Here is a log10/log10 chart of the relative error of v2,v3,v4,v5 vs the Kneser half iterate. At z=10^5, the error term is already -2.6E-27, where the old algorithm doesn't get that good until z=5.0E11.
   

Here is the algorithm for the version V half iterate approximation.


Now, half_v(-20) ~= -5.48222 - i3.778E-14, so the imaginary discontinuity is starting to get really insignificant. This is because at the negative real axis, we get

so we have cancelled most of the imaginary part, and are left with only the relatively small exp(half(z+3pi i)) term. The function is very nearly 2x the real part of the Kneser half iterate.

Now pretend this is an analytic function, and take the Taylor series of the function. To minimize errors, for a function with a very large magnitude range, use multiple Cauchy integrals as appropriate for different Taylor series terms. Also, it turns out the Taylor series function acts like a Laurent series, where the 1/x and 1/x^2 ... terms quickly decay to irrelevant as real(x) gets larger than 138, and then the oscillating takes over as the dominant error term, continually oscillating from slightly bigger than to slightly less than the Kneser half iterate. For the entire Taylor series, we drop the Laurent terms, and we have the Taylor series function below.
Code:
{halfv= 0.499100407682864282980350196595
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+x^279*  2.00800045291990379681136985289 E-3164
+x^280*  3.88852781952097451271692914060 E-3180
+x^281*  7.20045410377084607877619861129 E-3196
+x^282*  1.27509181724127031046616692488 E-3211
+x^283*  2.15964922735363615422176997757 E-3227
+x^284*  3.49895314766009754591249944461 E-3243
+x^285*  5.42323996238548370698411574285 E-3259
+x^286*  8.04262117412227823541530664764 E-3275
+x^287*  1.14131754981827406752997736453 E-3290
+x^288*  1.55001943773779939813788861284 E-3306
+x^289*  2.01484165993530008162353109238 E-3322
+x^290*  2.50708396515326564914382745069 E-3338
+x^291*  2.98656354216263237756653315717 E-3354
+x^292*  3.40643416590422644964249591711 E-3370
+x^293*  3.72052080393571704625140066739 E-3386
+x^294*  3.89164133408399578097076763129 E-3402
+x^295*  3.89885200481764336374706668619 E-3418
+x^296*  3.74166422420294635127661949707 E-3434
+x^297*  3.44005943228194981602191514912 E-3450
+x^298*  3.03032228244014506693574166427 E-3466
+x^299*  2.55788969580963050575575623475 E-3482
+x^300*  2.06915498791049377955618942408 E-3498
}

- Sheldon
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Messages In This Thread
RE: Searching for an asymptotic to exp[0.5] - by sheldonison - 05/29/2014, 11:09 PM

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