Error Function Erf Approximation
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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 excel error function erf − t 2 d t . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt
Erf Error Function Ti-89
{\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm
Q Function Erf
6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d
Gamma Function Approximation
t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1 (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ 6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}} 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 2 normal distribution erf (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\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 Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),\end − 0}} 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 8 (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
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 gaussian erf about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask Question mathematica erf _ Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join wiki erf 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 efficient and accurate approximation of error function https://en.wikipedia.org/wiki/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 Wiki suggests an approximation en.wikipedia.org/wiki/… –user17762 Jun 3 '11 at 2:37 possible duplicate of Definite integral of Normal Distribution –user17762 http://math.stackexchange.com/questions/42920/efficient-and-accurate-approximation-of-error-function 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 derivation of C99 erf...? –shaikh Jun 3 '11 at 2:45 @shaikh, oh, you have to read the code. See for instance the cephes library. –lhf Jun 3 '11 at 2:48 @shaikh: Or boost's implementation –ziyuang Jun 3 '11 at 2:51 ad
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