Handy Approximation Error Function
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Error Function Calculator
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Inverse Error Function
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Error Function Python
with us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and answer site for people studying math at any level http://wenku.baidu.com/view/d2a5e11ec5da50e2524d7ff1 and professionals in related fields. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Derivation of approximation of Error function up vote 1 down vote favorite 1 In Abramowitz and Stegun http://math.stackexchange.com/questions/1362037/derivation-of-approximation-of-error-function there are some formulas for approximation of Error funtion. I am intrested in the formulas $7.1.25$ to $7.1.28$, here is one of them ($7.1.26$). $$\operatorname{erf}(x)\approx 1 - e^{-x^2}\sum_{i=1}^5 a_i t^i +\epsilon(x),\quad 0\leq x<\infty,$$ where $t = 1/(1 + px)$. The coefficients $a_i$ and $p$ are some decimal numbers ($p = 0.3275911$, for example) and $|\epsilon(x)|\leq 1.5\cdot 10^{-7}$. The others are of similar form. How are these formulas derived? How to prove that the relative error $\epsilon(x)$ is really as small as they claim? What are the exact values of coefficients? In the bottom of the page with formulas it is said that Abramowitz and Stegun took these formulas from C. Hastings: Approximations for digital computers. Indeed, the formulas are present there (starting at page $167$), also, there are graphs of $\epsilon(x)$, but again, no derivation or exact values. Thank you. special-functions approximation error-function share|cite|improve this question edited Jul 15 '15 at 13:38 J. M. 53k5118254 asked Jul 15 '15 at 12:59 Antoine
error function and complimentary function used in communications are not exactly the same as the ones typically used in statistics. The relationship between the two is given at http://www.rfcafe.com/references/mathematical/erf-erfc.htm the bottom of the page. In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations. "In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations." error function - Wikipedia The Gaussian probability density function with mean = 0 and variance =1 is The error function erf(x) is defined as: Note that erf(0) = 0.5, and that erf(∞)=1. The complimentary error function erfc(x) is defined as: The following graph illustrates the region of the normal curve that is being integrated. For large values of x error function table (>3), the complimentary error function can be approximated by: The error in the approximation is about -2% for x=3, and -1% for x=4, and gets progressively better with larger values of x. Approximations RF Cafe visitor Ilya L. provided an approximation for the error function and complimentary error function that was published by Sergei Winitzki titled, "A handy approximation for the error function and its inverse." February 6, 2008. Here are the main results: Error function approximation: , where Complimentary error function: NOTE: I used to have an alternative approximation formula for the complimentary error function for large values of x, but decided to remove it since the source for it is not generally available to the public. It can be found as equation #13, on page 641, of IEEE Transactions on Communications volume COM-27, No. 3, dated March 1979. A subscription to the IEEE service is required to access the article. Try Using SEARCH to Find What You Need. >10,000 Pages Indexed on RF Cafe ! Copyright 1996 - 2016Webmaster: Kirt Blattenberger, BSEE - KB3UONFamily Websites: Airplanes and Rockets | Equine Kingdom All trademarks, co
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