Approximation Error Function
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that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ 0 x e − t 2 d t . {\displaystyle {\begin −
Approximation Q Function
2\operatorname − 1 (x)&={\frac − 0{\sqrt {\pi }}}\int _{-x}^ 9e^{-t^ 8}\,\mathrm 7 t\\&={\frac approximation gamma function 6{\sqrt {\pi }}}\int _ 5^ 4e^{-t^ 3}\,\mathrm 2 t.\end 1}} The complementary error function, denoted erfc, is defined as
Approximation Normal Distribution
erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin Φ 8\operatorname Φ 7 error function table (x)&=1-\operatorname Φ 6 (x)\\&={\frac Φ 5{\sqrt {\pi }}}\int _ Φ 4^{\infty }e^{-t^ Φ 3}\,\mathrm Φ 2 t\\&=e^{-x^ Φ 1}\operatorname Φ 0 (x),\end 9}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname Φ 8 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / inverse error function approximation 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname Φ 6 (x|x\geq 0)={\frac Φ 5{\pi }}\int _ Φ 4^{\pi /2}\exp \left(-{\frac Φ 3}{\sin ^ Φ 2\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin − 6\operatorname − 5 (x)&=-i\operatorname − 4 (ix)\\&={\frac − 3{\sqrt {\pi }}}\int _ − 2^ − 1e^ − 0}\,\mathrm − 9 t\\&={\frac − 8{\sqrt {\pi }}}e^ − 7}D(x),\end − 6}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 4 (x)} is real when x is real. When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: w ( z ) = e − z 2 erfc ( − i z ) = erfcx ( − i z ) . {\displaystyle w(z)=e^{-z^ 2}\operatorname 1 (-iz)=\operatorname 0 (-iz).} Contents 1 The name "error function" 2 Properties 2.1 Taylor series 2.2 Derivative and integral 2.3 Bürmann series 2.4 Inverse functions 2.5 Asymptotic expansion 2.6 Continued fraction expansion 2.7 Integral of error function with
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Complementary Error Function Approximation
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Quadratic Approximation Error Bound
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 https://en.wikipedia.org/wiki/Error_function can answer The best answers 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 11.3k52253 asked Jun 3 http://math.stackexchange.com/questions/42920/efficient-and-accurate-approximation-of-error-function '11 at 2:32 shaikh 493619 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 105k5118267 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 quit
institution loginHelpJournalsBooksRegisterJournalsBooksRegisterSign inHelpcloseSign in using your ScienceDirect credentialsUsernamePasswordRemember meForgotten username or password?Sign in via your institutionOpenAthens loginOther institution login Purchase Loading... Export You have selected 1 citation for export. Help Direct export Save to Mendeley Save to RefWorks http://www.sciencedirect.com/science/article/pii/0098135480800159 Export file Format RIS (for EndNote, ReferenceManager, ProCite) BibTeX Text Content Citation Only http://www.ams.org/mcom/1966-20-096/S0025-5718-1966-0203907-1/S0025-5718-1966-0203907-1.pdf Citation and Abstract Export Advanced search Close This document does not have an outline. JavaScript 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 error function 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 error function approximation 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 Login via your institution Other institution login Other users also viewed these articles Do not show again