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Apologies if the question is too elementary here.

For a certain computational application I need to approximate Gaussian distribution $e^{-x^2}$ with specific absolute precision (within $10^{-7}$ over $\mathbb{R}$), preferably with rational functions.

Alas, I'm not familiar with approximation theory. Google pointed me toward Pade approximation as the way to go. Alas, I still don't know how to derive Pade approximation for a given function, much less how to ensure the approximation would fit to the prescribed precision. Could you point me towards the relevant information?

EDIT:

Here is some background. For some engineering computations I need analytical expressions for certain integrals; in this particular case for $e^{-x^2} \operatorname{erf}(x+a)$, where $a$ is a parameter that changes throughout the computation. Mathematica was unable to provide an analytical expression of $\int e^{-x^2}\operatorname{erf}(x+a)\,dx$ and neither was I.

That caused me thinking how can I approximate well known functions such as $e^{-x^2}$ and $\operatorname{erf}(x+a)$ (and possibly a few others I may need) so that their products would be analytically integrable?

It's very well known that one can obtain an analytic expression for an integral of a rational function, and that products of rational functions are rational. Thus the idea: approximate the relevant functions with rational ones using Pade approximation and integrate the products of Pade approximations. That would basically replace the integral I want with an expression containing a bunch of rational functions and $\log$s and $\arctan$s.

I've got to tightly control the precision of the above approximations though to have the model perform as expected. Thus the question.

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  • $\begingroup$ Do you need to compute the Gaussian probability density or the comulative desnity (integral of the pdf from $-\infty$ to $x$) The former is trivial using an $\exp$ library function. The second is actually somewhat of a challenge. $\endgroup$
    – Brian Borchers
    Commented Jul 22, 2016 at 0:13
  • $\begingroup$ @BrianBorchers: edited the question with the reason why I need that. $\endgroup$
    – Michael
    Commented Jul 22, 2016 at 0:30
  • $\begingroup$ You'd do well to explain exactly what integral you actually want to evaluate. It seems unlikely that the strategy you're suggesting would be optimal. Is it just $\int_{-\infty}^{c} e^{-x^2 }\mbox{erf}(x+a)dx$? $\endgroup$
    – Brian Borchers
    Commented Jul 22, 2016 at 0:48
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    $\begingroup$ If all you need is the definite integral $\int_{-\infty}^{\infty} \exp(-x^2)\,\text{erf}(x+a)\,dx$, it has the closed form $\sqrt{\pi}\, \text{erf}(a/\sqrt{2})$. Mathematica does not know this, but it is 4.3.13 in Geller and Ng's paper at nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf. You can prove it by expanding erf as an integral of $y$, and then changing variables with a 45-degree rotation of the $xy$-plane. $\endgroup$
    – user44143
    Commented Jul 22, 2016 at 4:23
  • $\begingroup$ @BrianBorchers: Things like that, but possibly with different values of $c$ in the kernel of $e^{-x^c-y^c}$. The case $c=2$, a.k.a. Gaussian distribution, is the one that led to the question and is the most useful for this application. However, the case $c=1.5$, a.k.a. Holtsmark distribution, is also very useful in that context. $\endgroup$
    – Michael
    Commented Jul 22, 2016 at 5:46

2 Answers 2

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It would be worth looking at Schraudolph's sneaky trick if performance is your aim.

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In general, approximation with least maximal error is possible with the Remez Algorithm.

In your case I would suggest giving approximation with Chebyshev rational functions a try.

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