Analytical Approximation For Error Function
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error function erf by analytical functions up vote 11 down vote favorite 2 The Error function $\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt$ shows up in many contexts, but can't be represented using elementary functions. I compared it with another function $f$ which also starts linearly, has $f(0)=0$ and converges against the constant value 1 fast, namely $\tanh{(x)} = \frac {e^x - e^{-x}} {e^x + e^{-x}}$. Astoningishly to me, I found that they never differ by more than $|\Delta gaussian approximation f|=0.0812$ and converge against each other exponentially fast! I consider $\tanh{(x)}$ to be the somewhat prettyier function, and so I wanted to find an approximation to $\text{erf}$ with "nice functions" by a short expression. I "naturally" tried $f(x)=A\cdot\tanh(k\cdot x^a-d)$ Changing $A=1$ or $d=0$ on it's own makes the approximation go bad and the exponent $a$ is a bit difficult to deal with. However, I found that for $k=\sqrt{\pi}\log{(2)}$ the situation gets "better". I obtained that $k$ value by the requirement that "norm" given by $\int_0^\infty\text{erf}(x)-f(x)dx,$ i.e. the difference of the functions areas, should valish. With this value, the maximal value difference even falls under $|\Delta f| = 0.03$. And however you choose the integration bounds for an interval, the area difference is no more than $0.017$. Numerically speaking and relative to a unit scale, the functions $\text{erf}$ and $\tanh{(\sqrt{\pi}\log{(2)}x)}$ are essentially the same. My question is if I can find, or if there are known, substitutions for this non-elementary function in terms of elementary ones. In the sense above, i.e. the approximation is compact/rememberable while the values are even better, from a numerical point of view. The purpose being for example, that if I see somewhere that for a computation I have to integrate erf, that I can think to myself "oh, yeah that's maybe complicated, but withing the bounds
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A Note on the Error Function Issue No. 04 - July/August (2010 vol. 12) ISSN: error function 1521-9615 pp: 84-88 DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/MCSE.2010.79 Mohankumar Nandagopal Soubhadra Sen Ajay Rawat ABSTRACT
A new exact representation of the error function of real arguments justifies an accurate analytical approximation for and simple analytical approximation.
INDEX TERMS analytics, error function CITATION Mohankumar Nandagopal, Soubhadra Sen, Ajay Rawat, "A Note on the Error Function", Computing in Science & Engineering, vol. 12, no. , pp. 84-88, July/August 2010, doi:10.1109/MCSE.2010.79 FULL ARTICLE PDF HTML BUY RSS Feed SUBSCRIBE CITATIONS ASCII BibTex Refworks Procite Refman EndNote SEARCH Articles by Mohankumar Nandagopal Articles by Soubhadra Sen Articles by Ajay Rawat SHARE Digg Facebook Google+ LinkedIn Reddit Tumblr Twitter Stumbleupon 85 ms (Ver )be down. Please try the request again. Your cache administrator is webmaster. Generated Fri, 30 Sep 2016 04:57:46 GMT by s_hv997 (squid/3.5.20)