Error Function Approximation
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Normal Distribution Approximation
Ask Question _ Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related gaussian approximation 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
Simple Approximation
erf by analytical functions up vote 12 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 error function values 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 of $10^{-3}$ usign e.g. $\tanh(k\cdot x)$ i
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Error Function Table
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Inverse Error Function Approximation
us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and answer site for people complementary error function approximation 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 http://math.stackexchange.com/questions/321569/approximating-the-error-function-erf-by-analytical-functions are voted up and rise to the top efficient and accurate approximation of error function up vote 2 down vote favorite I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$ reference-request special-functions approximation share|cite|improve this question edited Aug 27 '14 at 11:38 Jean-Claude Arbaut 11.4k52353 asked Jun 3 '11 at 2:32 shaikh 493619 http://math.stackexchange.com/questions/42920/efficient-and-accurate-approximation-of-error-function Wiki suggests an approximation en.wikipedia.org/wiki/… –user17762 Jun 3 '11 at 2:37 possible duplicate of Definite integral of Normal Distribution –user17762 Jun 3 '11 at 2:43 1 Related: stats.stackexchange.com/questions/7200/… –cardinal Jun 3 '11 at 8:30 possible duplicate of How to accurately calculate erf(x) with a computer? –J. M. Jul 23 '11 at 15:26 You will find implementations in most scientific libraries: cmlib, slatec, nswc, nag, imsl, harwell hsl... Also in gnu gsl, in R, probably octave and Scilab... You can also have a look at ACM TOMS Collected Algorithms. There are plenty of places to look for this. –Jean-Claude Arbaut Aug 27 '14 at 11:40 add a comment| 4 Answers 4 active oldest votes up vote 2 down vote accepted "Efficient and accurate" is probably contradictory... Have you tried the one listed in http://en.wikipedia.org/wiki/Error_function#Approximation_with_elementary_functions ? share|cite|improve this answer answered Jun 3 '11 at 2:39 lhf 105k5120270 yes, I have tried this. Its accuracy is up to 2 decimal places. Do we have more than this? –shaikh Jun 3 '11 at 2:40 @shaikh, C99 has an erf function, which should be quite accurate. –lhf Jun 3 '11 at 2:42 where can I find the de
institution loginHelpJournalsBooksRegisterJournalsBooksRegisterSign inHelpcloseSign in using your ScienceDirect credentialsUsernamePasswordRemember meForgotten username or password?Sign in via your institutionOpenAthens loginOther institution login Purchase Loading... Export http://www.sciencedirect.com/science/article/pii/0098135480800159 You have selected 1 citation for export. Help Direct export Save to Mendeley Save to RefWorks Export file Format RIS (for EndNote, ReferenceManager, ProCite) BibTeX Text Content Citation Only Citation and Abstract Export Advanced search Close This document does not have an outline. JavaScript is disabled on your browser. error function 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 error function approximation 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 c
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