# Pade approximation of gaussian distribution to given precision

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.

• 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. – Brian Borchers Jul 22 '16 at 0:13
• @BrianBorchers: edited the question with the reason why I need that. – Michael Jul 22 '16 at 0:30
• 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 $\int_{-\infty}^{c} e^{-x^2 }\mbox{erf}(x+a)dx$? – Brian Borchers Jul 22 '16 at 0:48
• 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. – Matt F. Jul 22 '16 at 4:23
• @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. – Michael Jul 22 '16 at 5:46