Difference between revisions of "AuZex Approximation"
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| + | #redirect [[AuZex Approximation]] |
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| − | [[File:AuZexLamPlotT.jpg|440px|thumb|Fig.1. Plot of [[AuZex]] (thick curve) in comparison to [[LambertW]] (thin curve) ]] |
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| − | [[File:AuZexMapT.jpg|420px|thumb|Fig.2. [[Complex map]], $u\!+\!\mathrm i = \mathrm {AuZex}(x\!+\!\mathrm i y)$]] |
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| − | This article describes the approximations of [[AuZex]], which is [[inverse function]] of [[SuZex]] and [[Abel function]] of [[zex]]. The explicit plot of function [[AuZex]] is shown in figure 1. Its complex map is shown in figure 2. This map should be compared to the similar maps for the approximations below. |
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| − | ==Background== |
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| − | [[AuZex]] is [[Inverse function]] of [[SuZex]], so, in wide ranges of values of $z$, the relations |
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| − | (1) $ ~ ~ ~ \mathrm{SuZex}\Big( \mathrm{AuZex}(z) \Big) = z $ |
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| − | and |
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| − | (2) $ ~ ~ ~ \mathrm{AuZex}\Big( \mathrm{SuZex}(z) \Big) = z $ |
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| − | should hold. In particular, $~\mathrm{AuZex}(1)=0$. |
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| − | Also, [[AuZex]] satisfies the [[Abel equation]] |
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| − | (3) $ ~ ~ ~ \mathrm{AuZex}\Big( \mathrm{zex}(z) \Big) = \mathrm{AuZex}(z) +1$ |
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| − | in this case, [[zex]] appears as [[transfer function]], and [[AuZex]] is its Abel function. |
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| − | Iterations of equation (3) gives the relations |
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| − | (4) $ ~ ~ ~ \mathrm{AuZex}\Big( \mathrm{zex}^n(z) \Big) = \mathrm{AuZex}(z) +n$ |
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| − | This can be re-writtern also as |
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| − | (5) $ ~ ~ ~ \mathrm{AuZex}\Big( \mathrm{LambertW}^n(z) \Big) = \mathrm{AuZex}(z) - n$ |
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| − | as [[LambertW]] is [[inverse function]] of [[zex]]. |
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| − | The relations above indicate ways to construct the efficient approximation of [[AuZex]], covering the whole complex plane, and make the efficient (fast and precise) implementation. |
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| − | ==Taylor expansion at unity== |
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| − | <div class="thumb tright"><div style="width:15em;"> |
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| − | Coefficients in expansion (3) |
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| − | : $\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\! \displaystyle |
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| − | \begin{array}{r|r} |
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| − | n & c_n~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\ \hline |
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| − | 0 & 0.0000000000000000 \\ |
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| − | 1 & 1.4011764331478447\\ |
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| − | 2 & -1.2313176379841106\\ |
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| − | 3 & 1.1612567820116564\\ |
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| − | 4 & -1.1231269305776580\\ |
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| − | 5 & 1.0992876544297898\\ |
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| − | 6 & -1.0830479804216504\\ |
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| − | 7 & 1.0713113178859344\\ |
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| − | 8 & -1.0624516150969114 |
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| − | \end{array} |
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| − | $ |
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| − | </div></div> |
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| − | Some tens of coefficients of the Taylor expansion of function [[AuZex]] can be evaluated just inverting the Taylor expansion of $\mathrm{SuZex}(z)$ at $z\!=\!0$; that leads to the approximation |
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| − | (3) $ ~ ~ ~ \displaystyle \mathrm{AuZex}(1+t)\approx\mathrm{AuZt}_N(1\!+\!t)=\sum_{n=1}^N c_n t^n$ |
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| − | Approximatoins for the first eight coefficients $c$ are shown in the table at right. |
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| − | The [[Complex map]] of this approximation AuZt with $N\!=\!32$ is shown in Figure 3. |
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| − | The agreement |
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| − | (4) $ ~ ~ ~ \displaystyle |
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| − | A(z)=-\lg\left( |
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| − | \frac |
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| − | {\left|\mathrm{SuZex}\Big(\mathrm{AuZt}_{32}(z)\Big)- z\right|} |
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| − | {\left|\mathrm{SuZex}\Big(\mathrm{AuZt}_{32}(z)\Big)\right|+|z|} |
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| − | \right) |
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| − | $ |
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| − | is shown in figure 4. |
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| − | (sorry, the figures are not yet loaded) |
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| − | Iterations of function [[LambertW]] applied to the argument of function AuZt give the approximation |
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| − | (5) $ ~ ~ ~ \displaystyle |
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| − | \mathrm{AuZex}(z) \approx \mathrm{AuZt}_n( \mathrm{LambertW}^n(z))+n$ |
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| − | The complex maps of these approximations are shown in figure 5. While the efficient implementation for function [[LambertW]] is available, the iterations for integer $n$ in (5) cause no problems. |
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| − | ==Asymptotic expansion== |
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| − | The approximations by (5) cover the most of the complex plane, except the region in vicinity of the origin of coordinates. |
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| − | For this region, the asymptotic expansion below van be used: |
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| − | (6) $ ~ ~ ~ \displaystyle |
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| − | \mathrm{AuZex}(z) \approx \mathrm{AsZa}_N\Big(\mathrm{LambertW}^m(z)\Big)+x_1+m$ |
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| − | where |
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| − | (7) $ ~ ~ ~ \displaystyle \mathrm{AsZa}_N(t)=\frac{-1}{t} + \frac{1}{2} \ln(t) + \sum_{n=1}^N ~ b_n\, t^n$ |
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| − | The coefficients $b$ of this expansion can be generated by [[Mathematica]] with code |
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| − | So1[z_, a_] := Extract[Extract[Solve[z, a], 1], 1] |
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| − | zex[z_] = z Exp[z] |
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| − | Clear[b]; |
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| − | g[n_,z_] = -1/z + Log[z]/2 + Sum[b[m] z^m, {m, 1, n}] |
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| − | For[k=1,k<64, |
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| − | b[k]=ReplaceAll[b[k],So1[Coefficient[Series[g[k,zex[z]]-g[k,z]-1,{z,0,k+1}],z^(k+1)]==0, b[k]]]; |
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| − | Print[b[k]]; k++] |
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| − | Coefficients $b_n$ for $n\!=\!1..9$ are shown in the table |
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| − | $\begin{array}{c|ccl} |
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| − | n & b_n & &{\rm ~approximation ~ of~ ~} b_n\\ \hline \\ |
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| − | 1 & -1/6 &\approx&-0.1666666666666666667\\ |
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| − | 2 & 1/16 &\approx& ~ ~ ~ 0.0625\\ |
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| − | 3 & -19/540&\approx& -0.0351851851851851852 \\ |
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| − | 4 & 1/48&\approx& ~ ~ ~ 0.0208333333333333333 \\ |
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| − | 5 & -41/4200&\approx& -0.0097619047619047619\\ |
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| − | 6 & 37/103680&\approx& ~ ~ ~ 0.00035686728395061728\\ |
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| − | 7 & 18349/3175200&\approx&~ ~ ~ 0.005778848576467624 \\ |
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| − | 8 & -443/80640 &\approx& -0.005493551587301587\\ |
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| − | 9 & 55721/21555072&\approx& -0.002585052835824441 |
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| − | \end{array} |
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| − | $ |
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| − | <!-- |
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| − | Complex map of function AsZa is shown in figure. |
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| − | The agreement |
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| − | (8) |
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| − | is shown in figure |
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| − | !--> |
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| − | ==Implementation of AuZex== |
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| − | The approximations above cover the whole complex plane. On the base of these approximations, the [[complex double]] function AuZex is implemented. The implementation is available as [[AsZex.cin]]. |
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| − | This implementation provides of order of 15 correct decimal digits, and the errors are comparable to the rounding errors at the [[Complex double]] arithmetics. |
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| − | Similar approach is used to implement two [[Abel function]]s of the exponential to to base $b=\exp^2(-1)=\exp(1/\mathrm e)\approx 1.444667861$ |
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| − | <ref> |
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| − | http://tori.ils.uec.ac.jp/PAPERS/2012e1eMcom2590.pdf |
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| − | H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of Computation, 2012 February 8. ISSN 1088-6842(e) ISSN 0025-5718(p) |
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| − | <!-- http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf Journal version (the registration may be required) !--> |
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| − | </ref>. In that case, the [[transfer function]] $\exp_b$ also has derivative unity at its fixed point $L=\mathrm e\approx 2.718$ |
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| − | ==References== |
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| − | <references/> |
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| − | [[Category:AuZex]] |
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| − | [[Category:Abel function]] |
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| − | [[Category:Zex]] |
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| − | [[Category:LambertW]] |
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| − | [[Category:Articles in English]] |
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Latest revision as of 06:58, 1 December 2018
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