A Simple Approximation Of The Error Function
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Normal Distribution
Text Content Citation Only Citation and Abstract Export Advanced search Close This document does not have an outline. JavaScript wolfram alpha is disabled on your browser. Please enable JavaScript to use all the features on this page. Computers & Chemical Engineering Volume 4, Issue 2, 1980, Pages 67-68 A simple approximation of the error function Author links open the overlay panel. Numbers correspond to the affiliation list which can be exposed by using the show more link. Opens overlay H.T. Karlsson ∗, Opens overlay I. Bjerle Division of Chemical Technology, Department of Chemical Engineering, Chemical Center, Lund Institute of Technology, P.O.B. 740, S-220 07 Lund 7, Sweden Received 20 February 1979, Available online 30 July 2001 Show more Choose an option to locate/access this article: Check if you have access through your login credentials or your institution. Check access Purchase Sign in using your ScienceDirect credentials Username: Password: Remember me Not Registered? Forgotten username or password? OpenAthens login Login via your institution Other institution login doi:10.1016/0098-1354(80)80015-9 Get rights and content AbstractA very simple approximation formula of the error function, with sufficient accuracy for engineering calculations, is proposed in this investigation. The presented form is compared with some of the less sophisticated approximations available in the literature. Aspects such as mnemonic form, computation time, accuracy and ease of inversion are considered. open in overlay Author to whom correspondence should be directed. Copyright © 1980 Published by Elsevier Ltd. ElsevierAbout ScienceDirectRemote accessShopping cartContact and supportTerms and conditionsPrivacy policyCookies are used by this site. For more information, visit the cookies page.Copyright © 2016 Elsevier B.V. or its licensors or contributors. ScienceDirect ® is a registered trademark of Elsevier B.V.RELX Group Close overlay Close Sign in using your ScienceDirect credentials Username: Password: Remember me Not Registered? Forgotten username or password? OpenAthens login L
Aims and Scope · Annual Issues · Article Processing Charges · Articles in Press · Author Guidelines · Bibliographic Information · Citations to this Journal · Contact Information · Editorial Board · Editorial Workflow · Free eTOC Alerts · Publication Ethics · Reviewers Acknowledgment · Submit a Manuscript · Subscription Information · Table of Contents Open Special Issues · Published Special Issues · Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Full-Text ePUB Full-Text XML Linked References Citations to this Article How to Cite this Article Mathematical Problems http://www.sciencedirect.com/science/article/pii/0098135480800159 in EngineeringVolume 2012 (2012), Article ID 124029, 22 pageshttp://dx.doi.org/10.1155/2012/124029Research ArticleHigh Accurate Simple Approximation of Normal Distribution IntegralHector Vazquez-Leal,1 Roberto Castaneda-Sheissa,1 Uriel Filobello-Nino,1 Arturo Sarmiento-Reyes,2 and Jesus Sanchez Orea11Electronic Instrumentation and Atmospheric Sciences School, University of Veracruz, Cto. Gonzalo Aguirre Beltrán S/N, Zona Universitaria Xalapa, 91000 Veracruz, VER, Mexico2Electronics Department, National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro No.1, 72840 Tonantzintla, PUE, MexicoReceived 8 September 2011; Revised https://www.hindawi.com/journals/mpe/2012/124029/ 15 October 2011; Accepted 18 October 2011Academic Editor: Ben T. Nohara Copyright © 2012 Hector Vazquez-Leal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.AbstractThe integral of the standard normal distribution function is an integral without solution and represents the probability that an aleatory variable normally distributed has values between zero and . The normal distribution integral is used in several areas of science. Thus, this work provides an approximate solution to the Gaussian distribution integral by using the homotopy perturbation method (HPM). After solving the Gaussian integral by HPM, the result served as base to solve other integrals like error function and the cumulative distribution function. The error function is compared against other reported approximations showing advantages like less relative error or less mathematical complexity. Besides, some integrals related to the normal (Gaussian) distribution integral were solved showing a relative error quite small. Also, the utility for the proposed approximations is verified applying them to a
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more http://math.stackexchange.com/questions/321569/approximating-the-error-function-erf-by-analytical-functions about hiring developers or posting ads 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 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 Approximating the error function erf error function 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 $|\Delta f|=0.0812$ and converge against each a simple approximation 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 of $10^{-3}$ usign e.g. $\tanh(k\cdot x)$ is an incredible
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