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Cauchy integral also for b< e^(1/e)? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Cauchy integral also for b< e^(1/e)? (/showthread.php?tid=248) Pages:
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Cauchy integral also for b< e^(1/e)? - bo198214 - 03/03/2009 Hi Dmitrii, why not apply a similar Cauchy-integral technique for bases Instead of letting the parameter In the version But similarly we can compute the values on the horizontal line So summarized we had the recursion formula: Though the question is whether this is faster than the direct limit formula. RE: Cauchy integral also for b< e^(1/e)? - Kouznetsov - 03/22/2009 bo198214 Wrote:why not apply a similar Cauchy-integral technique for basesBecause, in the case If you mention, even for b=e, in the final implementation, I use the truncated series instead of the numerical approximation of the integrals. It is at least 2 orders of magnitude faster. There is difference, to get the figure within few seconds or tomorrow, ah? Now I feel that I can evaluate any tetration with 14 digits, and I work making the evaluation fast; and the algorithm - simple and portable (the same pieces of code run with C++, Mathematica and Maple). If you implement the integral formulas, we may compare the results with my expansion. I expect, of order of 14 digits will coincide. In the recursion formula, I would specify the base, to avoid confusions: If you want your implementation to converge, in the last integral, Be careful with the order of updates of the estimates of the values of the function: the speed of convergence (if at all) may depend on this order. RE: Cauchy integral also for b< e^(1/e)? - Kouznetsov - 03/24/2009 bo198214 Wrote:why not apply a similar Cauchy-integral technique for bases Yes. Please, apply it, reproduce the fugure http://en.citizendium.org/wiki/Image%3AHolomorphicTetrationBaseSqrt2v01.jpg and compare the values to those by the algorithm (updated version) posted at http://en.citizendium.org/wiki/Tetration/Code/HolomorphicBaseSqrt2v01 I expect we can get the agreement with 14 decimal digits.. Who wants to write a paper about it? RE: Cauchy integral also for b< e^(1/e)? - Kouznetsov - 03/27/2009 Ansus Wrote:How did you evaluate Taylor series coefficients with Cauchy integral?I have algorithm I express I differentiate this expression with respect to Usually, I use the circular contour of integration, centered at the point of expansion, with radius slightly smaller than distance to the closest singularity. This allows to work with good precision, close to the limit due to the rounding errors of the complex double variables. RE: Cauchy integral also for b< e^(1/e)? - bo198214 - 03/27/2009 See also Wikipedia about Cauchy's formula. There is the explicit formula for the RE: Cauchy integral also for b< e^(1/e)? - bo198214 - 03/27/2009 Ansus Wrote:What about this equation? Looks damn good. Simpler than Dmitrii's default formula. RE: Cauchy integral also for b< e^(1/e)? - bo198214 - 03/27/2009 Ansus Wrote:Yes, but that formula is recursive. But that does not matter as long we have once computed f (on the imaginary axis). RE: Cauchy integral also for b< e^(1/e)? - Kouznetsov - 03/28/2009 Ansus Wrote:But how to compute it on the imaginary axis?for b< e^(1/e), the algorithm of the evaluation on the imaginary axis through the integral Cauchi equation is not yet ready. While, you may recover the function from the asymptotic behavior. I believe, it is even better than Cauchi. For b=sqrt(2), the expansion is F(z)=2- u - v_2 u^2 -v_3 u^3... where u=u(z)=exp((z-t)Q) Substitute the expansion into equation F(z+1)=exp_b(F(z)). Calculate the Taylor expansion of F(z+1)-exp_b(F(z)) with respect to u, Taking into account that u(z+1)=Qu(z). Set to zero the coefficients in this expansion. These equations determine Q, v_2, v_3, ... Recover t from the condition F(0)=1. Then, using F(z-1)=log_b(F(z)) , you can evaluate F everywhere, not only along the imaginary axis. You may use the representation through the asymptotic for evaluation of few first derivatives. If you want a preise evaluation of many coefficients of the Taylor expansion, better, use the Cauchi; you may evaluate a hundred coefficients within a second. RE: Cauchy integral also for b< e^(1/e)? - Kouznetsov - 03/29/2009 Ansus Wrote:Because they did not allow to plot the map of tetration.Quote:r b< e^(1/e), the algorithm of the evaluation on the imaginary axis through the integral Cauchi equation is not yet ready.There is a plenty of formulas for tetration on this forum. Why not to use them? Because they do not give 14 decimal digits. Because they are slow in the evaluation. Ansus Wrote:Use formulas for the derivatives through the Cauchi integrals from Wiki.Quote:better, use the Cauchi; you may evaluate a hundred coefficients within a second.How? Chose the circular contour of integration. Distribute some thoudand points uniformly at this contour. Spend halfsecond of CPU time to evaluate function in these points. Spend another halfsecond to approximate the Cauchi integrals with the finite sums and get the coefficients in the Taylor expansion. Compare the truncated Taylor expansion to the values by some of algorithms you have mentioned. RE: Cauchy integral also for b< e^(1/e)? - andydude - 03/29/2009 Ansus Wrote:Which Wiki? I think he means Citizendium. |