Numerical Integration Of 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 complementary error function π ∫ 0 x e − t 2 d t . {\displaystyle {\begin −
Error Function Calculator
6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _
Error Function Table
9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) =
Inverse Error Function
2 π ∫ x ∞ e − t 2 d 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 error function matlab arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname 2 (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 r
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 error function excel more about Stack Overflow the company Business Learn more about hiring developers or error function python posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and answer error function properties 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 https://en.wikipedia.org/wiki/Error_function 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 11.7k52354 asked Jun 3 '11 http://math.stackexchange.com/questions/42920/efficient-and-accurate-approximation-of-error-function 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 106k5120271 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
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