A Handy Approximation For The Error Function And Its Inverse
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Inverse Error Function
百度文库 专业资料 上传文档 A handy approximation for the error function and eric5421上传于2010-11-12|暂无评价|0人阅读|0次下载|文档简介|举报文档 专业文档 分享至: × 阅读已结束,如果下载本文需要使用1下载券 下载 想免费下载本文? 立即加入VIP 免下载券下载文档 10万篇精选文档免费下 千本精品电子书免费看 定制HR最喜欢的简历 我要定制简历 下一篇 把文档贴到Blog、BBS或个人站等: 复制 预览 普通尺寸(450*500pix) 较大尺寸(630*500pix) 你可能喜欢 <%for(var i=0,len=data.length;i
that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e
Error Function Excel
− t 2 d t = 2 π ∫ 0 x e − inverse error function excel t 2 d t . {\displaystyle {\begin 鈭 2\operatorname 鈭 1 (x)&={\frac 鈭 0{\sqrt {\pi }}}\int _{-x}^ 鈦 error function python 9e^{-t^ 鈦 8}\,\mathrm 鈦 7 t\\&={\frac 鈦 6{\sqrt {\pi }}}\int _ 鈦 5^ 鈦 4e^{-t^ 鈦 3}\,\mathrm 鈦 2 t.\end 鈦 1}} The complementary error function, denoted erfc, is defined as http://wenku.baidu.com/view/d2a5e11ec5da50e2524d7ff1 erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 桅 8\operatorname 桅 7 (x)&=1-\operatorname 桅 6 (x)\\&={\frac 桅 5{\sqrt {\pi }}}\int _ 桅 4^{\infty }e^{-t^ 桅 3}\,\mathrm 桅 2 t\\&=e^{-x^ 桅 1}\operatorname 桅 0 https://en.wikipedia.org/wiki/Error_function (x),\end 鈦 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 桅 8 (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 桅 6 (x|x\geq 0)={\frac 桅 5{\pi }}\int _ 桅 4^{\pi /2}\exp \left(-{\frac 桅 3}{\sin ^ 桅 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 鈭 6\operatorname 鈭 5 (x)&=-i\operatorname 鈭 4 (ix)\\&={\frac 鈭 3{\sqrt {\pi }}}\int _ 鈭 2^ 鈭 1e^ 鈭 0}\,\mathrm 鈭 9 t\\&={\frac 鈭 8{\sqrt {\pi }}}e^ 鈭 7}D(x),\end 鈭 6}} where D(x) is the Dawson function (which can be used
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 http://math.stackexchange.com/questions/1362037/derivation-of-approximation-of-error-function more 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 Derivation of approximation of error function Error function up vote 1 down vote favorite 1 In Abramowitz and Stegun there are some formulas for approximation of Error funtion. I am intrested in the formulas $7.1.25$ to $7.1.28$, here is one of them ($7.1.26$). $$\operatorname{erf}(x)\approx 1 - e^{-x^2}\sum_{i=1}^5 a_i t^i +\epsilon(x),\quad 0\leq x<\infty,$$ where $t = 1/(1 + px)$. The coefficients $a_i$ and $p$ are some decimal numbers ($p = 0.3275911$, for example) and $|\epsilon(x)|\leq 1.5\cdot 10^{-7}$. The others are of similar form. inverse error function How are these formulas derived? How to prove that the relative error $\epsilon(x)$ is really as small as they claim? What are the exact values of coefficients? In the bottom of the page with formulas it is said that Abramowitz and Stegun took these formulas from C. Hastings: Approximations for digital computers. Indeed, the formulas are present there (starting at page $167$), also, there are graphs of $\epsilon(x)$, but again, no derivation or exact values. Thank you. special-functions approximation error-function share|cite|improve this question edited Jul 15 '15 at 13:38 J. M. 52.7k5118252 asked Jul 15 '15 at 12:59 Antoine 2,086622 These are, if memory serves, Chebyshev fits of a transformed version of the error function. There are nicer and more analytically tractable approximations now; you might want to search for the work of Serge Winitzki on this. –J. M. Jul 15 '15 at 13:40 I have seen Winitzki's work if you are referring to Uniform Approximations for Transcendental Functions, but I think that his $\epsilon$'s are bigger. Could you explain what is Chebyshev fit and what is a transformed version of the error function? –Antoine Jul 15 '15 at 13:47 That was one of his papers; there was another one that specially dealt with $\mathrm{erf}$. –J. M. Jul 15 '15 at 13:50 Let's call the polynom
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